the cohomology of sl(3 z - purdue universityhopf.math.purdue.edu/henn/sl3.pdf · 2005-01-03 · z=2...

The cohomology of SL(3,Z[1/2])

Hans-Werner Henn

Abstract

We compute the cohomology of SL(3,Z[1/2]) with coefficients inthe prime fields and in the integers. On the way we obtain the co-homology of certain mod - 2 congruence subgroups of SL(3,Z) withcoefficients in Fp for p > 2. Finally we compute the cohomology ofGL(3,Z[1/2]).

Contents

1 Introduction 2

2 Contractible spaces with actions of SL(3,Z) and SL(3,Z[1/2]) 82.1 The symmetric space and the Bruhat-Tits-building . . . . . . 82.2 Well-rounded lattices and the deformation retractions . . . . . 102.3 The space W0/SO(3) . . . . . . . . . . . . . . . . . . . . . . . 112.4 Γi - equivariant cell structures on Z . . . . . . . . . . . . . . . 17

2.4.1 The case i = 0 . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 The cases i = 1 and i = 2 . . . . . . . . . . . . . . . . 19

2.5 Symmetries of well-rounded quadratic forms . . . . . . . . . . 202.6 The equivalence relations ∼i on the spaces Γi ×Di . . . . . . 23

2.6.1 3 - cells . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6.2 2 - cells . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6.3 1 - cells . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.4 0 - cells . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 The homology of the quotient spaces 343.1 Quotients of (X∞,X∞,s(i)) by Γi . . . . . . . . . . . . . . . . . 353.2 Quotients of X∞,s(i) by Γi . . . . . . . . . . . . . . . . . . . . 413.3 Quotients of X∞ by Γi . . . . . . . . . . . . . . . . . . . . . . 413.4 Quotients by SL(3,Z[1/2]) . . . . . . . . . . . . . . . . . . . . 45

1

2 Hans-Werner Henn

4 The cohomology of SL(3,Z[1/2]) 494.1 Mod - 2 cohomology . . . . . . . . . . . . . . . . . . . . . . . 494.2 Mod - 3 cohomology . . . . . . . . . . . . . . . . . . . . . . . 534.3 Higher torsion in the integral cohomology . . . . . . . . . . . . 65

5 The cohomology of GL(3,Z[1/2]) 67

1 Introduction

So far there exist only very few complete computations of integral or mod- p cohomology rings of arithmetic or more generally S - arithmetic groups.Among the known results we mention the calculations for SL(2,Z) (whichis straightforward from the well-known amalgamated product decomposi-tion SL(2,Z) ∼= Z/6 ∗Z/2 Z/4), of SL(2,Z[1/2]) [Mi] and that of SL(3,Z)[So]. Soule’s computation is already fairly involved; e.g. he obtains thatthe integral cohomology ring of SL(3,Z), after localization at the prime 2, isgenerated by 7 elements which are subject to 22 relations. His result suggeststhat the answer for SL(n,Z) would not be easily digestable (one should addthat it also seems to be completely out of reach at this point).

From a conceptual point of view the complexity of the answer in Soule’scalculation can also be explained by Quillen’s work [Q] which says amongother things that the minimal prime ideals in the mod - p cohomology ringH∗(Γ;Fp) of an S - arithmetic group Γ are in one to one correspondence withthe conjugacy classes of maximal elementary abelian p - subgroups of Γ. (Werecall that an elementary abelian p - group is a group isomorphic to (Z/p)k forsome natural number k.) From this point of view those cases in which thereexists a unique conjugacy class of maximal elementary abelian p - subgroupslook more favourable than others. In the case of SL(n,Z) or GL(n,Z) it isvery difficult to determine the precise number of conjugacy classes of maximalelementary abelian p - subgroups (this is essentially a problem of the integralrepresentation theory of elementary abelian p - groups) and thus the mod -p cohomology of these groups must be complicated. The situation improvesif one inverts p and adjoins p - th roots of unity. In particular in the caseof SL(n,Z[1/2]) and GL(n,Z[1/2]) every elementary abelian 2 - subgroup isdiagonalizable and there is a unique maximal one up to conjugacy.

This observation was presumably the basis of Quillen’s conjecture (p. 591of [Q]), which in the case of H∗(GL(n,Z[1/2]);F2) claims that the inclu-sion of rings Z[1/2] ⊂ R (and identifying H∗(GL(n,Z[1/2]);F2) as usualwith the mod 2 - cohomology of the classifying space BGL(n,Z[1/2])) makesH∗(GL(n,Z[1/2]);F2) into a free, in particular a torsion free module over

The cohomology of SL(3,Z[1/2]) 3

the polynomial ring F2[w1, ..., wn] ∼= H∗(BGL(n,R);F2) with wi denoting asusual the i - th universal Stiefel - Whitney class. In [HLS] it was shown thattorsion-freeness implies that the restriction map ρn : H∗(GL(n,Z[1/2]);F2)−→ H∗(Dn;F2) (with Dn

∼=∏n

i=1(Z[1/2])× ∼=∏n

i=1(Z × Z/2) denoting thesubgroup of diagonal matrices of GL(n,Z[1/2])) is injective. Quillen also re-marked that with his conjecture a calculation of H∗(GL(n,Z[1/2]);F2) shouldbe within reach. In fact, the image Im ρn has been computed by Mitchell.In order to state his result we identify the classes wi with their image underrestriction in H∗(Dn;F2) ∼= F[x1, ..., xn] ⊗ E(a1, ..., an) (with E as usual de-noting an exterior algebra, and with all generators of dimension 1), namelywith the i - th elementary symmetric polynomial in the variables xi. Wealso need classes ei ∈ H2i−1(Dn;F2): they are the symmetrizations of theelements x2

1...x2i−1ai with respect to the canonical action of the symmetric

group Sn on n letters.

Now Mitchell’s result reads as follows.

Theorem 1.1 [Mi] Im ρn ∼= F2[w1, ..., wn]⊗ E(e1, ..., e2n−1) . 2

Note that with this result Quillen’s conjecture would imply an isomorphismH∗(GL(∞,Z[1/2]);F2) ∼= F2[w1, w2, ...] ⊗ E(e1, e3, ...) and hence the Dwyer- Friedlander version [DF] of the Lichtenbaum - Quillen conjecture at p = 2.Unfortunately Quillen’s conjecture was too optimistic. Dwyer has recentlyshown.

Theorem 1.2 [D] The restriction map ρn is not injective for all n. 2

The only previous complete computation of H∗(GL(n,Z[1/2]);F2) was thatof [Mi] for n = 2, and in this case ρn turned out to be injective. Somequalitative information on the size of the kernel of ρn as n grows is providedin [H2]. Dwyer shows, in fact, that ρ32 is not injective, so that the case n = 3becomes an interesting test case in which one also has a nice candidate,namely Im ρ3, for the answer.

In fact, one of the main results of this paper shows that this candidate iscorrect.

Theorem 1.3 The restriction homomorphism maps H∗(GL(3,Z[1/2]);F2)isomorphically onto the subalgebra F2[w1, w2, w3]⊗E(e1, e3, e5) of H∗(D3;F2).

This result is really an easy consequence of the following companion resultfor SL(3,Z[1/2]). We denote its subgroup of diagonal matrices by SD3. Notethat the restriction map from H∗(D3;F2) to H∗(SD3;F2) kills the elementsw1 and e1. Let vi be the image of wi and d2i−1 the image of e2i−1, i = 2, 3.

4 Hans-Werner Henn

Theorem 1.4 The restriction homomorphism maps H∗(SL(3,Z[1/2]);F2)isomorphically onto the subalgebra F2[v2, v3]⊗E(d3, d5) of H∗(SD3;F2).

We remark that the corresponding result does not hold in the same way forn = 2, i.e. the restriction map is not an isomorphism in this case, althoughthere is an abstract isomorphism H∗(SL(2,Z[1/2]);F2) ∼= F2[v2]⊗E(d3) [Mi].

How can Theorem 1.4 be proved? The standard approach would be to take asuitable finite dimensional contractible space X on which Γ := SL(3,Z[1/2])acts properly and with finite isotropy groups (there is a canonical such can-didate, namely the product of the symmetric space SL(3,R)/SO(3) and theBruhat-Tits-building for SL(3,Q2), see Section 2.1 below). Then one wouldtake the Borel construction EΓ×Γ X as model for the classifying space BΓand study its mod - 2 cohomology H∗Γ(X;F2) via the cohomology spectralsequence of the map EΓ ×Γ X −→ Γ\X. If X has the structure of a Γ- CW - complex then the E1 - term of this spectral sequence is given asEs,t

1 =⊕

σHt(Γσ;F2) where σ runs through a set of representatives of the Γ

- orbits of s - dimensional cells of X and Γσ denotes the isotropy group ofσ. This is how Soule studied the cohomology of SL(3,Z) [So]. However, inour case the space X looks too complicated to make this spectral sequencemanageable: in Section 2.6 we actually analyze the canonical X above andwe essentially produce a Γ - equivariant deformation retract with finitelymany Γ - orbits of cells; however, finite means 474 (!) orbits (see the table atthe beginning of Section 3) and so this standard approach looks unfeasible.Instead we use a more manageable “centralizer spectral sequence”

Es,t2∼= lims

E∈A∗(Γ) Ht(CΓ(E);F2) =⇒ Hs+t

Γ (Xs;F2)

converging to the mod - 2 cohomology of the Borel - construction of the 2- singular locus Xs, i.e. the subspace of X consisting of all points whoseisotropy group contains an element of order 2. Here A∗(Γ) is the category ofelementary abelian 2 - subgroups of Γ, lims is the s - th derived functor ofthe inverse limit functor and CΓ(E) is the centralizer in Γ of the elementaryabelian 2 - subgroup E ⊂ Γ. This spectral sequence is based on a homotopycolimit decomposition of EΓ×ΓXs and was introduced in [H1]. In this paperwe also evaluated this spectral sequence and obtained the following resultin which Σ denotes as usual the suspension functor, e.g. Σ4Fp denotes thegraded Fp - vectorspace which is trivial in all dimensions except in dimension4 where it is Fp.

Theorem 1.5 [H1] Let Γ = SL(3,Z[1/2]) and let X be any mod - 2 acyclicfinite dimensional Γ - CW - complex for which the stabilizers of all cells are


finite. Then there is a short exact sequence

0 −→ Σ4F2 −→ H∗Γ(Xs;F2)ρ−→ F2[v2, v3]⊗ E(d3, d5) −→ 0

in which ρ is an algebra homomorphism. Furthermore, if π denotes the pro-jection map from EΓ×ΓXs to the classifying space BΓ then the composition

H∗(Γ;F2)π∗−→ H∗Γ(Xs;F2)

ρ−→ F2[v2, v3]⊗ E(d3, d5) ⊂ H∗(SD3;F2)

agrees with the restriction homomorphism of 1.4. 2

Now for ∗ exceeding dimX, the dimension of X, we have isomorphismsH∗Γ(Xs;F2) ∼= H∗Γ(X;F2) ∼= H∗(Γ;F2) and hence Theorem 1.5 is also a com-putation of H∗(Γ;F2) in large dimensions. In fact, X can be chosen to be ofdimension 5 (see [BS] or Section 2 below) and Theorem 1.5 gives encouragingevidence for Theorem 1.4.

In this paper we complete the proof of Theorem 1.4 by computing forthe canonical space X mentioned above, the relative groups H∗Γ(X,Xs;F2)and the boundary homomorphism of the appropriate long exact cohomologysequence. Note that, because the isotropy groups outside of Xs are finite oforder prime to 2, we have the following isomorphisms for the relative groups:H∗Γ(X,Xs;F2) ∼= H∗(Γ\(X,Xs);F2).

As a byproduct of our investigations we obtain the following results whichare of independent interest. In these results we abbreviate SL(3,Z[1/2]) byΓ, SL(3,Z) by Γ0, and we denote the subgroup of SL(3,Z) consisting ofall matrices whose first column agrees with the first standard basis vectormodulo 2 by Γ1, and the subgroup of all matrices which are upper triangularmodulo 2 by Γ2.

Theorem 1.6 Let X∞ denote the symmetric space SL(3,R)/SO(3), X2 theBruhat-Tits-building of SL(3,Q2), X = X∞ × X2 and let p be any prime.Then the reduced cohomology of the quotient spaces by the obvious action ofthe respective groups is given as follows:

a) H∗(Γ0\X∞;Fp) = 0

b) H∗(Γ1\X∞;Fp) = 0

c) H∗(Γ2\X∞;Fp) = Σ3Fpd) H∗(Γ\X ;Fp) = Σ5Fp .

For p > 3 there are no elements of order p in these groups (because thereare obviously no elements of order p in SL(3,Q)) and hence we obtain thefollowing Corollary. For SL(3,Z) this was already known by [So] and forSL(3,Z)[1/2] by [Mo]. The results for Γ1 and Γ2 are compatible with theEuler chararacteristic computations in [Mo].

6 Hans-Werner Henn

Corollary 1.7 Assume p > 3. Then

a) H∗(Γ0;Fp) = 0

b) H∗(Γ1;Fp) = 0

c) H∗(Γ2;Fp) = Σ3Fpd) H∗(Γ;Fp) = Σ5Fp . 2

Theorem 1.8 Let X∞, X2, X and p be as in the previous theorem. Then weget the following relative cohomology groups (where (X∞,s(i) denotes the 2 -singular locus of X∞ with respect to the action of Γi, and Xs the 2 - singularlocus of X with respect to the action of Γ):

a) H∗(Γ0\(X∞,X∞,s(0));Fp) = 0

b) H∗(Γ1\(X∞,X∞,s(1));Fp) = Σ2(Fp)2

c) H∗(Γ2\(X∞,X∞,s(2));Fp) = Σ3Fp ⊕ Σ2(Fp)6

d) H∗(Γ\(X ,Xs);F2) = Σ5F2 .

(Observe that we restrict to the case p = 2 for the last part of the Theorem.)

The next result together with Theorem 1.5 and the last part of Theorem1.8 finishes the proof of Theorem 1.4.

Proposition 1.9 The boundary homomorphism

H4Γ(Xs;F2) −→ H5

Γ(X ,Xs;F2)

is an epimorphism.

With the help of Theorem 1.6 we are also able to compute the mod - 3cohomology. Again this was known for SL(3,Z) by [So].

Theorem 1.10 There are isomorphisms of F3 - algebras (without unit) whichin the case of a), b) and d) are induced by restrictions to appropriate sub-groups:

a) H∗(Γ0;F3) ∼=∏2

i=1 H∗(S3;F3)

b) H∗(Γ1;F3) ∼=∏2

i=1 H∗(S3;F3)

c) H∗(Γ2;F3) ∼= Σ3F3

d) H∗(Γ;F3) is isomorphic to the subalgebra of∏2

i=1 H∗(S3×Z;F3) which

can be characterized as follows: it is all of∏2

i=1 H∗(S3 × Z;F3) except in

degrees 1 and 4; in degree 1 it is trivial, and in degree 4 it is of dimension 3and is generated by the image of the Bockstein of H3 and one further elementwhich restricts non-trivially to both factors.


The paper is organized as follows: In Section 2 we recall the symmetricspace X∞ and the Bruhat - Tits building X2. We discuss the Soule - Lannesmethod of replacing the symmetric space by a smaller space Z for which thequotients by Γ0 = SL(3,Z) and the congruence subgroups Γ1 and Γ2 arecompact. The bulk of this long section is then devoted to patiently workingout an explicit cell structure of the quotients Γi\Z, i = 0, 1, 2, in fact even a Γi- equivariant cell structure on Z. This is straightforward but it is crucial forthe remainder of the paper; for i = 0 it is a variation of Soule’s investigations[So]. In Section 3 we use these cell structures to prove Theorem 1.6 andTheorem 1.8 as well as the corresponding results for the cohomology of thequotients of the singular locus X∞,s(i) resp. Xs. This is quite an elaboratecalculation but apart from the last part of Theorem 1.8 it is straightforwardgiven the results in Section 2. The last part of Theorem 1.8 is more tricky andto settle it we use low dimensional information on H∗Γ(Xs;F2) as provided byTheorem 1.5. In Section 4 we apply the results of Section 2 and Section 3 andderive the remaining results listed in this introduction. We also determinethe height of torsion in H∗(SL(3,Z[1/2]);Z) (Proposition 4.15). In Section5 we compute H∗(GL(3,Z[1/2]);Fp) for primes p > 2 (Proposition 5.1, 5.2)and for p = 2, i.e. we derive Theorem 1.3.

Acknowledgements: During the research presented in this paper the authorwas supported by a Heisenberg fellowship of the DFG. The author is happyto acknowledge numerous discussions with Jean Lannes which stimulated hisinterest in the cohomology of SL(3,Z[1/2]). He also thanks Bob Oliver forhelpful discussions in connection with the proof of Proposition 1.9.

8 Hans-Werner Henn

2 Contractible spaces with actions of SL(3,Z)

and SL(3,Z[1/2])

2.1 The symmetric space and the Bruhat-Tits-building

We start by recalling the contractible spaces on which our groups act withfinite stabilizer groups.

The symmetric space. The space Q(n) of positive definite quadraticforms on Rn is equipped with an action of the multiplicative group R+ ofpositive real numbers, given by (rq)(x) = rq(x) for r ∈ R+, q ∈ Q(n) andx ∈ Rn. The quotient will be denoted by X∞(n), or simply by X∞ if n isclear from the context. The space X∞(n) is contractible because Q(n) is aconvex open cone in Rn2

. Furthermore, X∞(n) can be identified with thesymmetric space of SL(n,R), i.e. the space of left cosets SL(n,R)/SO(n),via the map which sends a matrix A to the equivalence class of the positivedefinite quadratic form q, given by q(x) = ||A−1x|| where || || denotes theeuclidean norm in Rn. The group SL(n,Z) acts on this coset space from theleft, and this action is proper, i.e. if C ⊂ X∞(n) is compact then there areonly finitely many g ∈ SL(n,Z) for which gC ∩ C 6= ∅; in particular theisotropy groups of the action are all finite.

The Bruhat-Tits-building. The group SL(n,Z[1/2]) acts on the cosetspace X∞(n) as well. However, in this case the action is not proper. In orderto get a contractible space with proper action, the space X∞(n) has to beenlarged by the appropriate Bruhat-Tits-building X2(n) (or simply X2 if nis clear from the context) for the group SL(n,Q2). As reference for more onthis bulding we recommend [B2]. We recall here only some basic properties.

The space X2(n) is an (n−1)-dimensional simplicial complex which can bedescribed as follows: an n - dimensional 2 - adic lattice L is a Z2 - submoduleof Qn2 which is free of rank n. The group Q×2 of units in Q2 acts on the set ofall such lattices via scalar multiplication, and the set of equivalence classes isthe set of vertices in X2(n). A finite subset {l0, l1, ..., ln} of vertices spans an n- dimensional simplex in X2(n) if and only if there are representative latticesLi in the class of li for i = 0, ..., n such that L0 ( L1 ( ... ( Ln ( 1

2L0.

The space X2(n) is contractible (see Section V.8 and Theorem VI.3 in [B2]).Furthermore the set of all 2 - adic lattices can be identified with the setof left cosets GL(n,Q2)/GL(n,Z2) via the map which sends a matrix A tothe lattice A(Zn2 ). The natural left action of SL(n,Q2) on this coset spaceinduces a simplicial left action of SL(n,Q2) on X2(n) and the quotient ofX2(n) by the action of SL(n,Q2) is an (n− 1) - dimensional simplex ∆n−1.


Furthermore SL(n,Z[1/2]) is dense in SL(n,Q2) and therefore the quotientof X2(n) by the action of SL(n,Z[1/2]) agrees with the quotient by the groupSL(n,Q2).

The group SL(n,Z[1/2]) embedds diagonally as a discrete subgroup intoSL(3,R)×SL(n,Q2) and acts properly on the contractible space X := X∞×X2.

The projection maps and congruence subgroups. From now on weconcentrate on the case n = 3. We will be interested in the SL(3,Z[1/2]) -equivariant projection map p : X −→ X2.

With respect to the action ofGL(3,Z[1/2]) all vertices in X2 fall into a singleorbit and hence their isotropy groups (in SL(3,Z[1/2])) are conjugate inthe larger group GL(3,Z[1/2]), in particular they are abstractly isomorphic;similarly with simplices of dimension one. For the vertex l0 correspondingto the standard lattice L0 (which is spanned over Z3

2 by the standard basisvectors e1,e2 and e3, i.e. L0 = 〈e1, e2, e3〉), the isotropy group is SL(3,Z2) ∩SL(3,Z[1/2]) = SL(3,Z) =: Γ0. For the edge consisting of the set {l0, l1}with l0 the class of L0 and l1 the class of the lattice L1 = 〈1

2e1, e2, e3〉, the

isotropy group is the subgroup Γ1 of Γ0 consisting of matrices whose firstcolumn is equal to e1 modulo 2; for the two-dimensional simplex spannedby the set {l0, l1, l2} with l2 the class of the lattice L2 = 〈1

2e1,

12e2, e3〉, the

isotropy group is the subgroup Γ2 of Γ0 consisting of all matrices which areupper triangular modulo 2. For simplicity of notation we will write Γ insteadof SL(3,Z[1/2]).

The “fibres” of the map (which is induced by p)

p : EΓ×Γ X −→ Γ\X2∼= ∆2

over the 0 -, 1 - resp. 2 - dimensional simplices respectively are homotopy -equivalent to the classifying spaces BΓ0, BΓ1 and BΓ2 respectively. We willhave to study the mod - 2 (co)homology spectral sequence of p as well asthat of the map (which is also induced by p)

p : Γ\X −→ Γ\X2∼= ∆2 .

In particular we need to understand the “fibres” of p, i.e. the quotientsΓi\X∞, i = 0, 1, 2. These quotients are not compact and in the next sectionwe recall the Soule-Lannes method of finding a deformation retract of Γi\X∞which is compact, even a finite 3 - dimensional complex (see [A]).

10 Hans-Werner Henn

2.2 Well-rounded lattices and the deformation retrac-tions

Well-rounded lattices. We note that Γi\X∞ ∼= Γi\(SL(3,R)/SO(3)) mayalso be obtained as quotient of Γi\SL(3,R) by the right action of SO(3).Now the right SO(3) - space GL(3,Z)\GL(3,R) can be identified with thespace of all integral lattices in R3 (via the correspondence which sends amatrix g to the lattice g−1(Zn)), and the space Γ0\SL(3,R) can be identifiedwith the space of equivalence classes (with respect to scalar multiplication)of integral lattices L in R3, or equivalently with the space of integral latticeswhose minimal vectors are of length 1, i.e. for which m(L) := min{‖x‖|x ∈L−{0}} = 1. We will denote this latter space by L0. Note that, in terms oflattices, the right action of SO(3) on Γ0\SL(3,R) is given by L · g := g−1Lfor L ∈ L0 and g ∈ SO(3).

Similarly the space Γ1\SL(3,R) can be identified with the space L1 of pairs(L0, L1) of lattices such that m(L0) = 1 and L0 ( L1 ( 1

2L0, and the space

Γ2\SL(3,R) can be identified with the space L2 of triples (L0, L1, L2) oflattices such that m(L0) = 1 and L0 ( L1 ( L2 ( 1

2L0.

We recall that a lattice L in R3 is called well-rounded if its set of minimalvectors, i.e. {x ∈ L − {0}|‖x‖ = m(L)} spans R3. For i = 0, 1, 2 let Wi

denote the subspace of Li consisting of all tuples (L0, ..., Li) for which L0 iswell-rounded.

The deformation retractions. There is a beautiful geometric argumentwhich shows that Wi is an SO(3) - equivariant deformation retract of Li,hence Wi/SO(3) is a deformation retract of Li/SO(3) ∼= Γi\X∞. We recallthe construction ([A]).

For i = 0, 1, 2 and 1 ≤ p ≤ 3 let Wpi be the set of tuples (L0, ..., Li) of

lattices such that the dimension of the subspace of R3 spanned by the set ofminimal vectors in L0 is at least p. Then W1

i = Li, W3i =Wi and therefore

it suffices to show that Wp+1i is an SO(3) - equivariant deformation retract

of Wpi for p = 1, 2. So assume that the set of minimal vectors in L0 spans a

subspace U of dimension q ≥ p. If q > p then nothing happens to our tuplein the next step of the deformation. Otherwise, consider a radial contractinghomotopy in the subspace U⊥ of R3 perpendicular to U , and extend linearlyto a deformation of R3 by leaving U fixed. This defines a deformation Lj(t),0 < t ≤ 1 of lattices (for 0 ≤ j ≤ i) with Lj(1) = Lj and there will be amaximal t0 with 0 < t0 < 1 for which L0(t0) has a new vector of minimallength 1. The corresponding tuple (L0(t0), ..., Li(t0)) of lattices lies in Wp+1

i

and is the image under the next step in the deformation. It is easy to seethat these constructions describe continuous SO(3) - equivariant maps which


combine to give an SO(3) - equivariant deformation retraction from Li toWi

and induce a deformation retraction from Γi\X∞ ∼= Li/SO(3) to Wi/SO(3).We will see in the next section that the spaces Wi/SO(3) are compact andof dimension 3.

We can do even a bit better: the SO(3) - equivariant deformation re-traction of L0 can be lifted to give a left SL(3,Z) - equivariant and rightSO(3) - equivariant deformation retraction of SL(3,R) onto the subspaceY := {g ∈ SL(3,R)|g−1(Zn) is a wellrounded lattice}. Dividing out by theSO(3) - action gives a left SL(3,Z) - space Z and an SL(3,Z) - equivariantdeformation retraction from X∞ to Z. The space Z will also be called thespace of (equivalence classes of) well - rounded quadratic forms.

The remainder of Section 2 is devoted to a detailed analysis of the spacesΓi\Z ∼=Wi/SO(3), in particular we will exhibit explicit finite cell structureson them.

2.3 The space W0/SO(3)

Our first task is to understand the space Γ0\Z ∼= W0/SO(3). This spaceagrees with Soule’s deformation retract of the space Γ0\SL(3,R)/SO(3) [So];however, our point of view is a bit different in so far as we emphasize latticesrather than quadratic forms, i.e. we prefer to think in terms of W0/SO(3),the space of wellrounded 3 - dimensional lattices L with m(L) = 1, modulothe action of SO(3).

We will see in a moment that in dimension 3 (unlike in higher dimensions)the sublattice spanned by any set of 3 linearly independent vectors of minimallength in a well-rounded lattice L is all of L, and therefore L is (up to theaction of SO(3)) determined by m(L) and the 3 scalar products betweenthese vectors. We will analyze which of these 3 - tuples occur in this wayand which tuples give the same lattice, up to the action of SO(3). Thisanalysis will lead to an explicit description of the spaces Wi/SO(3). In thissection we will first concentrate on the case i = 0. Our first step is given bythe following Lemma.

Lemma 2.1 Suppose L ⊂ R3 is a well-rounded lattice and let v1, v2 andv3 be linearly independent vectors of minimal length m(L) in L. Then thesublattice L′ spanned by these vectors is all of L.

Proof. By scaling and rotating L we may assume that m(L) = 1, v1 =(1, 0, 0) and v2, v3 have the form: v2 = (a, x, 0) and v3 = (b, y, z). Assumethere exists w = (w1, w2, w3) ∈ L− L′. By adding a suitable vector in L′ we

12 Hans-Werner Henn

may assume that |w3| ≤ 12|z| ≤ 1

2, |w2| ≤ 1

2|x| ≤ 1

2and |w1| ≤ 1

2. But then

||w|| < 1 and we obtain a contradiction to the assumption that m(L) = 1. 2

The next two results will enable us to give an explicit description of thespace W0/SO(3). They will be proved together.

Proposition 2.2 Suppose v1, v2 and v3 are linearly independent vectors oflength 1 in R3 with scalar products a = 〈v1, v2〉, b = 〈v1, v3〉 and c = 〈v2, v3〉.Assume that a ≥ 0 and b ≥ 0. Then the lattice L spanned by v1, v2 and v3 iswell-rounded with m(L) = 1 if and only if

1. c ≥ 0 and a, b, c ≤ 12, or

2. c ≤ 0, a, b, |c| ≤ 12

and a+ b− c ≤ 1.

Clearly, the assumption on a and b can be assured by replacing, if necessary,one of the vectors vi by its negative.

Proposition 2.3 Suppose v1, v2 and v3, a, b, c and L are as in Proposition2.2. Furthermore assume a ≥ b ≥ |c|. Then the set of minimal vectors in Lcontains ±v1,±v2,±v3 and in addition only the following vectors:

1. ±(v1 − v2) if a = 12, b 6= 1

2and a+ b− c 6= 1.

2. ±(v1 − v2 − v3) if a 6= 12, b 6= 1

2and a+ b− c = 1.

3. ±(v1 − v2) and ±(v1 − v3) if a = b = 12

and c 6= 0, 12.

4. ±(v1 − v2) and ±(v1 − v2 − v3) if a = 12, b 6= 1

2and a+ b− c = 1.

5. ±(v1 − v2), ±(v1 − v3) and ±(v2 − v3) if a = b = c = 12.

6. ±(v1 − v2), ±(v1 − v3) and ±(v1 − v2 − v3) if a = b = 12

and c = 0.

Again the assumption on a, b and c can always be assured by permutingthe vectors vi and passing to negatives if necessary.

Proof. 1. Let us first consider the case c ≥ 0.

Consider a vector w in L and write

w = n1v1 + n2v2 + n3v3, ni ∈ Z, i = 1, 2, 3 .

Then

||w||2 = n21 + n2

2 + n23 + 2an1n2 + 2bn1n3 + 2cn2n3 , (2.1)

or equivalently

||w||2 = a(n1 + n2)2 + b(n1 + n3)2 + c(n2 + n3)2

+(1− a− b)n21 + (1− a− c)n2

2 + (1− b− c)n23. (2.2)

If a > 12

then n1 = −n2 = 1, n3 = 0 gives a vector w with ||w|| = 2− 2a < 1.The same argument for b and c shows that, if m(L) = 1, then b, c ≤ 1

2. Now

assume that a, b, c ≤ 12. We distinguish different cases.


1.1. At least one ni = 0, w.l.o.g. n3 = 0. Then we obtain

||w||2 = n21 + n2

2 + 2an1n2 = a(n1 + n2)2 + (1− a)n21 + (1− a)n2

2. (2.3)

Because 1− a ≥ 12

and 1− b ≥ 12

it is clear from (2.3) that ||w||2 ≥ 1 unlessw = 0.

We also observe that the only vectors of length 1 in L with n3 = 0 arethe vectors ±v1, ±v2, and if a = 1

2, the vector ±(v1 − v2). Similarly, the

only vectors with n2 = 0 are the vectors ±v1, ±v3, and if b = 12, the vector

±(v1 − v3). The only vectors with n1 = 0 are the vectors ±v2, ±v3, and ifc = 1

2, the vector ±(v2 − v3).

1.2. We may now assume that all ni 6= 0. Then at least one of the sumsn1 +n2, n1 +n3, n2 +n3 must be different from 0. If precisely one of the sumsis non-zero, say n2 + n3, then n1 = −n3, n1 = −n2 and |n2 + n3| ≥ 2 and(2.2) yields ||w||2 ≥ 3− 2a− 2b+ 2c ≥ 1; equality holds iff c = 0, a = b = 1

2,

n1 = −n2 = −n3 = ±1, i.e. w = ±(v1 − v2 − v3). If at least two of the sumsare non-zero, say n1 + n3 and n2 + n3, then |n1 + n3| ≥ 2 and |n2 + n3| ≥ 2and (2.2) yields ||w||2 ≥ 3− 2a+ 2b+ 2c > 1, in particular there are no suchvectors of length 1.

2. Now consider the case c ≤ 0. Then we write

||w||2 = a(n1 + n2)2 + b(n1 + n3)2 − c(n2 − n3)2

+(1− a− b)n21 + (1− a+ c)n2

2 + (1− b+ c)n23. (2.4)

As before we see that a, b, |c| ≤ 12

is necessary for L to satisfy m(L) = 1.Now assume these inequalities hold. Again we distinguish different cases.

2.1. If at least one ni = 0 and w 6= 0, then we see as above that ||w||2 ≥ 1and we only obtain additional vectors of length 1 iff a = 1

2resp. b = 1

2resp.

c = −12, namely the vectors ±(v1 − v2) resp. ±(v1 − v3) resp. ±(v2 + v3).

2.2. We may now assume that all ni 6= 0. Consider the sums n1+n2, n1+n3,n2 − n3. We subdivide into further cases. In case all sums are zero we haven1 = −n2 = −n3 and from (2.4) we obtain again ||w||2 ≥ 3 − 2a − 2b + 2c.By taking n1 = −n2 = −n3 = ±1 we see that the condition a+ b− c ≤ 1 isnecessary for L to satisfy m(L) = 1, and there are further vectors of length1 iff a+ b− c = 1, namely the vectors ±(v1 − v2 − v3).

If two sums are zero, then the third one is as well, hence we may next assumethat at most one sum is zero, hence at least two of the terms |n1 + n2| and|n1 + n3|, |n2 − n3| are ≥ 2. In case |n1 + n2| and |n1 + n3| are ≥ 2, (2.4)yields ‖w‖2 ≥ 3 + 2a+ 2b+ 2c > 1, in particular there are no such vectors oflength 1. The other two cases are analogous. 2

14 Hans-Werner Henn

After these preparations we can now describe the space W0/SO(3). Con-sider the following subspace D0 of R3 (see figure 1):

D0 := {(a, b, c) ∈ R3| |c| ≤ b ≤ a ≤ 1

2, and a+ b− c ≤ 1 if c ≤ 0} .

We define a map

γ : D0 −→ Y = {g ∈ SL(3,R)|g−1(Zn) is a wellrounded lattice}

by sending the triple (a, b, c) to the unique matrix γ(a, b, c) with the follow-ing properties: γ(a, b, c) is (up to a scalar multiple guaranteeing γ(a, b, c) ∈SL(3,R)) the inverse of the matrix whose i - th column is the basis vec-tor vi, where v1 = (1, 0, 0), v2 = (a, x, 0), v3 = (b, y, z) and x, y and z areuniquely determined by the requirements x ≥ 0, ab + xy = c, z ≥ 0 and||vi|| = 1 for i = 1, 2, 3. By construction and Proposition 2.2 the latticeγ(a, b, c)−1(Zn) is well - rounded, hence γ(a, b, c) ∈ Y . Let λ : D0 −→ W0

denote the composition of γ with the canonical projection Y −→ W0; thenλ(a, b, c) is the well - rounded lattice spanned by the vectors v1, v2 and v3.Note that by construction a = 〈v1, v2〉, b = 〈v1, v3〉 and c = 〈v2, v3〉. Fi-nally let ϕ : D0 −→ W0/SO(3) be the composition of λ with the canonicalprojection W0 −→W0/SO(3). Clearly all these maps are continuous.

Finally we define an equivalence relation ∼ on D0 by declaring the points(1

2, b, c) with 0 ≤ c ≤ 1

2b equivalent to (1

2, b, b − c) and equivalent to (1

2, b −

c,−c) (cf. figure 1).

Theorem 2.4 The map ϕ : D0 −→W0/SO(3) is onto and induces a home-omorphism ϕ : D0/∼ −→W0/SO(3).

In the proof we will make repeated use of the following elementary fact.

Lemma 2.5 Assume v1, v2, v3 and v1′, v2

′, v3′ are two sets of linearly

independent vectors of length 1 in R3 such that 〈vi, vj〉 = 〈vi′, vj ′〉 for all1 ≤ i < j ≤ 3. Then there exist unique rotations R, S ∈ O(3) such thatRvi = vi

′ and Svi = −vi′ for i = 1, 2, 3, and either R or S is in SO(3). 2

Proof of Theorem. That ϕ is onto can be seen as follows. Assume we aregiven a well-rounded lattice L with minimal vectors of length 1. By Lemma2.1 we can find spanning vectors w1, w2 and w3 in L of length 1, and after asuitable permutation (and passing to additive inverses, if necessary) we mayassume that the scalar products a, b and c are as in 2.2 and 2.3. Then Lemma2.5 implies ϕ(a, b, c) = [L] where [L] denotes the image of L in W0/SO(3).


a

b

c

C = (12, 1

2, 1

2)

O = (0, 0, 0) D = (12, 1

2, 1

4)

B = (12, 1

2, 0)A = (1

2, 0, 0)

E = (12, 1

4,−1

4) F = (1

3, 1

3,−1

3)

-

6

��

��

��

��

��

��

��

��

@@@@@@@@@@�

��

��

AAAAAAAAAAAAAAAAAAA

Figure 1: The space D0 and the equivalence relation ∼ . The equivalencerelation is given by identifying the triangle ∆ABD with the triangles ∆ACDand ∆ABE via reflections at the edges AD and AB.

16 Hans-Werner Henn

Next we show that equivalent triples have the same image under ϕ so thatϕ induces a continuous map ϕ. So assume 0 ≤ c ≤ 1

2b and consider the

lattice λ(12, b, c). This has (at least) 4 pairs of minimal vectors, namely

±v1 = ±(1, 0, 0), ±v2 = ±(a, x, 0), ±v3 = ±(b, y, z) and ±(v1 − v2). (Herex, y and z are as before.) Then it is straightforward to check that thescalar products between the vectors v′1 := v1, v′2 := v1 − v2 and v′3 := v3

are (12, b, b − c) and those between the vectors v′′1 := v2 − v1, v′′2 := v2 and

v′′3 := −v3 are (12, b − c,−c) and Lemma 2.5 implies again that the image

under ϕ of these triples agree.

Now we turn to injectivity of ϕ. As D0/ ∼ is compact and W0/SO(3) isHausdorff, this will show that ϕ is a homeomorphism and finish the proof.So assume ϕ(a, b, c) = ϕ(a′, b′, c′). By assumption the corresponding latticesL := λ(a, b, c) and L′ := λ(a′, b′, c′) agree up to a rotation R ∈ SO(3), i.e.L = RL′. In particular, L and L′ have the same number of minimal vectors,the vectors Rv′1, Rv′2, Rv′3 form a set of linearly independent vectors of length1 in L and the triple (a′, b′, c′) occurs as a triple of scalar products betweenthree linearly independent vectors of length 1 of L. We have to show thatthis happens only if (a, b, c) and (a′, b′, c′) are equivalent under the relation∼.

In the “generic” case, i.e. if a 6= 12

and a+b−c 6= 1, L has only the minimalvectors ±v1, ±v2 and ±v3 (cf. Proposition 2.3), and in this case it is obviousthat the triple of scalar products is uniquely determined by L and by thecondition a ≥ b ≥ |c|.

Now assume (a, b, c) 6= (a′, b′, c′) and we have 6 pairs of minimal vectors.By Proposition 2.3 this can only happen if w.l.o.g. (a, b, c) = (1

2, 1

2, 1

2) and

(a′, b′, c′) = (12, 1

2, 0). However, these points are clearly equivalent under ∼.

Next assume we have precisely 5 pairs of minimal vectors in L. By Proposi-tion 2.3 and because ϕ is constant on ∼ - equivalence classes we may assumethat our triples are of the form (1

2, 1

2, c) and (1

2, 1

2, c′) with 0 < c, c′ ≤ 1

4and

we have to show c = c′. The lattice L = λ(12, 1

2, c) has the following pairs

of minimal vectors of length 1: ±v1, ±v2, ±v3, ±(v1 − v2) and ±(v1 − v3),and the triple (1

2, 1

2, c′) must occur as a triple of scalar products of 3 linearly

independent vectors taken from those 5 pairs. It is now straightforward tocheck that this can happen only if c = c′.

Finally assume we have exactly 4 pairs of linearly independent vectors oflength 1 in L = λ(a, b, c). Again by Proposition 2.3 and because ϕ is constanton ∼ - equivalence classes we may assume that the triple (a, b, c) satisfieseither a = 1

2and 0 ≤ c ≤ 1

2b < 1

4, or a 6= 1

2, c ≤ 0 and a + b − c = 1. We

have to show that L determines uniquely a triple of this form. In the firstcase the set of minimal vectors consists of ±v1, ±v2, ±v3, ±(v1 − v2), in the


second case of ±v1, ±v2, ±v3, ±(v1 − v2 − v3). Again it is straightforwardto see that all triples of linearly independent vectors taken from those setswhich lead to scalar products of the required form, lead indeed to the samescalar products, and thus the proof is complete. 2

2.4 Γi - equivariant cell structures on Z2.4.1 The case i = 0

We recall the map γ : D0 −→ Y which sends d = (a, b, c) ∈ D0, up to ascalar multiple, to the inverse of the matrix whose i - th colum is the vectorvi specified in the last section (cf. the discussion before Theorem 2.4). Thecomposition of the map γ : D0 −→ Y with the quotient map πZ : Y −→Y/SO(3) ∼= Z will be denoted by ψ0. Note that ψ0(d) is the equivalenceclass of the positive definite quadratic form for which the scalar productsbetween the standard basis vectors e1, e2, e3 are given by 〈ei, ej〉 = 〈vi, vj〉for i ≤ i, j ≤ 3, i.e. 〈ei, ei〉 = 1 for i = 1, 2, 3, 〈e1, e2〉 = a, 〈e1, e3〉 = band 〈e2, e3〉 = c. In particular this map is injective and a homeomorphismfrom D0 to ψ0(D0). The Γ0 - equivariant extension Γ0 × D0 −→ Z whichsends (g, d) to gψ0(d) will still be denoted by ψ0. Let ∼0 be the equivalencerelation on Γ0 ×D0 induced by the map ψ0, i.e. defined by (g, d) ∼0 (g′, d′)iff ψ0(g, d) = ψ0(g′, d′). Then ∼0 is Γ0 - equivariant, i.e. if (g, d) ∼0 (g′, d′)then (hg, d) ∼0 (hg′, d′) for every h ∈ Γ0.

Let gAD ∈ Γ0 be given by gAD(e1) = −e1, gAD(e2) = −e1+e2 and gAD(e3) =−e3, and let gAB ∈ Γ0 be given by gAB(e1) = e2 − e1, gAB(e2) = e2 andgAB(e3) = −e3. Then we have the following result which is a refinement ofTheorem 2.4.

Theorem 2.6 The equivalence relation ∼0 on Γ0 ×D0 induced by the mapψ0 : Γ0 ×D0 −→ Z is the smallest Γ0 - equivariant equivalence relation gen-erated by the following elementary relations: (g, d) and (g′, d′) are elementaryequivalent if either

1. g′ = 1, d = d′ and g belongs to the isotropy group Hd ⊂ Γ0 of the (classof the) quadratic form ψ0(d).

2. g′ = 1, d = (12, b, c), 0 ≤ c ≤ 1

2b, and either

d′ = (1

2, b, b− c), g = gAD , or

d′ = (1

2, b− c,−c), g = gAB .

18 Hans-Werner Henn

Furthermore the induced map Ψ0 : Γ0×D0/ ∼0−→ Z is a homeomorphismof Γ0 - spaces.

Proof. First we observe that points of Γ0 ×D0 which are elementary equiv-alent are mapped to the same point in Z under ψ0. This is trivial for thefirst elementary relation. For the second one it follows because by defini-tion of gAD and Lemma 2.5 we have gADγ(1

2, b, c) ∈ γ(1

2, b, b − c)SO(3),

i.e. gADψ0(12, b, c) = ψ0(1

2, b, b − c). Similarly, gAB is defined such that

gABγ(12, b, c) ∈ γ(1

2, b − c,−c)SO(3), i.e. gABψ0(1

2, b, c) = ψ0(1

2, b − c,−c).

It follows that ψ0 induces a map Ψ0 as claimed.

Furthermore ψ0 induces (on passing to the quotients with respect to theactions of Γ0) the surjection ϕ of Theorem 2.4. In particular, if follows thatψ0 and hence Ψ0 is surjective. Next assume that ψ0(g, d) = ψ0(g′, d′). ThenTheorem 2.4 shows that d ∼ d′ and by definition of ∼0 we may thereforeassume that d = d′. But then we clearly have g−1g′ ∈ Hd and by Γ0 -equivariance of ∼0 we see that (g, d) ∼0 (g′, d′) and injectivity of Ψ0 follows.

Finally it is easy to see that the map Ψ0 is an open map and hence a home-omorphism (e.g. by using that the actions of Γ0 on Γ0 ×D0/ ∼0 and Z areproper, and that the induced map on the quotient spaces is a homeomorphismby Theorem 2.4). 2

Cell structures on D0, D0/ ∼ and a Γ0 - equivariant cell structureon Z. Theorem 2.6 allows us to establish a Γ0 - equivariant cell structureon Z in terms of a cell structure on D0 resp. on D0/ ∼. We start with cellstructures on D0 and D0/ ∼ (see figure 1).

0. The 0 - dimensional cells of D0 are the vertices O = (0, 0, 0), A = (12, 0, 0),

B = (12, 1

2, 0), C = (1

2, 1

2, 1

2), D = (1

2, 1

2, 1

4), E = (1

2, 1

4,−1

4) and F = (1

3, 1

3, 1

3).

On D0/ ∼ this gives 5 cells which will still be labelled O, A, B ∼ C, D ∼ Eand F .

1. The 1 - dimensional cells of D0 are the edges OC, OF , OA, EF , BF ,AB, AC, AD, AE, BD, CD and BE. On D0/ ∼ this gives 8 cells labelledOC, OF , OA, EF , BF , AB ∼ AC, AD ∼ AE and BD ∼ CD ∼ BE.

2. The 2 - dimensional cells ofD0 are the quadrangles OAEF (characterizedby b = −c) and OCBF (a = b), and the triangles OAC (b = c), BEF(a + b − c = 1) and ABD, ACD and ABE. On D0/ ∼ this gives 5 cellslabelled OAEF , OCBF , OAC, BEF and ABD ∼ ACD ∼ ABE.

3. D0 has one cell of dimension 3, namely the interior of D0, and this givesalso one cell for D0/ ∼.

It follows easily from Proposition 2.3 (see also Section 2.5 below) that theisotropy groups Hd of the action of Γ0 on Z at ψ0(d) are constant within the


interior of each cell e of D0 and this is the reason for the choice of our cellstructure on D0. If we denote this isotropy group by He then Theorem 2.6shows that Z has an equivariant cell structure with one orbit (Γ0/He)× e ofcells for each equivalence class of cells in D0/ ∼. The attaching maps can beread off from figure 1.

2.4.2 The cases i = 1 and i = 2

We consider the right Γ0 - spaces Γi\Γ0. In case i = 1 this coset spacecan be identified with the set S1 of non-zero vectors in (F3

2 − {0}) and incase i = 2 with the set S2 of pairs consisting of a line in F3

2 and a planein F3

2 containing the line, i.e. with the set of complete flags in F32. In

fact, there is a canonical left action of Γ0 on the sets Si, and if we con-vert this into a right action in the usual way via s · g := g−1s, then themap Γ0 −→ S1, g 7→ g−1(e1) mod 2 induces an isomorphism of right Γ0 -spaces Γ1\Γ0 −→ S1; similarly the map Γ0 −→ S2 which sends g to the flag(〈g−1(100)〉 ⊂ 〈g−1(100), g−1(010)〉 mod 2) induces an isomorphism of rightΓ0 - spaces Γ2\Γ0 −→ S2. (Here 〈〉 denotes the subgroup generated by theelements within the brackets and 100, 010 are standard basis vectors in F3

2.)The sets D0 × Si will be denoted by Di.

Now we choose representatives for the right cosets of Γi in Γ0. Such achoice of a representative gs for each s ∈ Si gives an explicit Γi - equivarianthomeomorphism

Γi ×D0 × Si −→ Γ0 ×D0, (g, d, s) 7→ (ggs, d) .

In order to obtain a Γi - equivariant cell structure on Z we will carry overthe Γ0 - equivariant equivalence relation ∼0 on Γ0×D0 to a Γi - equivariantequivalence relation ∼i on Γi×Di. For this we note that the isotropy groupsHd act from the right on the coset spaces Si. Likewise the matrices gAB andgAD act from the right on Si. The following result is now a straightforwardconsequence of Theorem 2.6.

Theorem 2.7 The equivalence relation ∼i on Γi ×Di induced by the map

ψi : Γi ×Di −→ Z, (g, d, s) 7→ ggsψ0(d)

is the smallest Γi - equivariant equivalence relation generated by the follow-ing elementary relations: (g, d, s) and (g′, d′, s′) are elementary equivalent ifeither

1. g′ = 1, d = d′, there exists an element h ∈ Hd with s = s′h (in particulars and s′ belong to the same Hd - orbit with respect to the right action of Hd

on the set Si) and g is determined by ggs = gs′h.

20 Hans-Werner Henn

2. g′ = 1, d = (12, b, c), 0 ≤ c ≤ 1

2b and either

d′ = (1

2, b, b− c), s = s′gAD, ggs = gs′g

AD , or

d′ = (1

2, b− c,−c), s = s′gAB, ggs = gs′g

AB .

Furthermore the induced map Ψi : Γi×Di/ ∼i−→ Z is a homeomorphism ofΓi - spaces, the Γi - equivariant equivalence relation ∼i induces an equivalencerelation (denoted by ∼(i)) on the quotient Di of Γi×Di such that the induced

map Ψ : Di/ ∼(i)−→ Γi\Z is a homeomorphism. 2

Γi - equivariant cell structures on Z and cell structures on Γi\Z.Theorem 2.7 yields Γi - equivariant cell strucures on the space Z (and thenordinary cell structures on the quotients Γi\Z). The indexing set for the Γi- orbits of cells on Z (resp. the cells on the quotients Γi\Z) are equivalenceclasses of pairs (e, s) with e a cell in D0 and s ∈ Si, with the equivalencerelation generated by the following elementary relations: (e, s) ∼i (e′, s′) iffeither

1. e = e′ and s and s′ are in the same He - orbit, or

2a. e is ABD or a face of it, e′ is ACD or the corresponding face of it ands = s′gAD, or

2b. e is ABD or a face of it, e′ is ABE or the corresponding face of it ands = s′gAB.

The Γi - orbits of cells of Z are then of the form Γi/H(e,s) × (e, s) where(e, s) runs through a set of representatives of equivalence classes of such pairsand the isotropy group H(e,s) of the cell (e, s) is given by Γi ∩ gsHegs

−1, i.eagrees up to conjugation by gs with gs

−1Γigs∩He which is the isotropy groupof s with respect to the right action of He on Si. The attaching maps canagain be read off from figure 1.

In the next two sections we will make this concrete, i.e. we will describe inexplicit form the isotropy groups He, their actions on the sets Si, and alsothe effect of the action of gAB and gAD on Si.

2.5 Symmetries of well-rounded quadratic forms

In order to make the equivariant cell structure of the spaces Γi\Z concretewe need to determine the isotropy groups H(a,b,c) of the action of Γ0 on Z atψ0(a, b, c). Of course, H(a,b,c) preserves the length of vectors and the scalar


products between them (both taken, of course, with respect to a representa-tive quadratic form of (the equivalence class of quadratic forms) ψ0(a, b, c)),and hence H(a,b,c) acts on the set of minimal vectors in the standard lattice.These sets have been determined in Theorem 2.4 (we just have to replacethe letter v by e everywhere). The standard basis vectors are always mini-mal vectors and so H(a,b,c) is determined by this action. It is clear that thegroups H(a,b,c) are constant in the interior of each cell of D0 and this givesthe justification for the choice of our cell structure on D0.

The case of the 3 - dimensional cell is particularly simple. If (a, b, c) is in itsinterior then we have only the 3 standard basis vectors and their negativesin the set of minimal vectors and it is easy to check that H(a,b,c) = {1}.

Tables 1, 2 and 3 below give the isotropy groups on the open cells of D0 ofdimension 2, 1 and 0. In fact it will be enough for us to take one cell fromeach ∼ - equivalence class of cells. The first column lists the name of the cell,the second one the set of minimal vectors on the standard lattice with respectto (a representative quadratic form of) ψ0(a, b, c) if (a, b, c) is an interior pointof the appropriate cell and the third column gives the isotropy group. Thelast column describes the action of the isotropy group on the tuple (e1, e2, e3)of minimal vectors explicitly; the 3 - tuples in this column are the images ofthe tuple (e1, e2, e3) under the action of appropriate generators. The proofsare straightforward and are left to the reader.

We use the following notation in these tables: for the symmetric group onn - letters we write Sn, o denotes the wreath product construction, and thedihedral group with n elements is denoted by Dn.

Table 1: Symmetries on the 2-dimensional cells

Cell Minimal vectors Isotropy Generators

ABD±e1,±e2,±e3,

trivial±(e1 − e2)

OAC ±e1,±e2,±e3 Z/2 (−e2,−e1,−e3)OAEF ±e1,±e2,±e3 Z/2 (e2, e1,−e3)OCBF ±e1,±e2,±e3 Z/2 (−e1,−e3,−e2)

BEF±e1,±e2,±e3 Z/2× Z/2 (−e2,−e1, e1 − e2 − e3)

±(e1 − e2 − e3) (−e3, e1 − e2 − e3,−e1)

22 Hans-Werner Henn


Cell Minimal vectors Isotropy Generators

OC ±e1,±e2,±e3 S3(−e1,−e3,−e2)

(−e2,−e1,−e3)

OF ±e1,±e2,±e3 S3(−e1,−e3,−e2)

(e2, e1,−e3)

OA ±e1,±e2,±e3 Z/2× Z/2 (−e2,−e1,−e3)

(e2, e1,−e3)

AB±e1,±e2,±e3, Z/2 (e2 − e1, e2,−e3)±(e1 − e2)

AD±e1,±e2,±e3, Z/2 (−e1, e2 − e1,−e3)±(e1 − e2)

BD±e1,±e2,±e3, Z/2× Z/2 (−e1,−e3,−e2)

±(e1 − e2),±(e1 − e3) (−e1, e3 − e1, e2 − e1)

BF±e1,±e2,±e3,

D8BEF symmetries,

±(e1 − e2 − e3) (−e1,−e3,−e2)

EF±e1,±e2,±e3

D8BEF symmetries,

±(e1 − e2 − e3) (e2, e1,−e3)


Cell Minimal vectors Isotropy Description

O ±e1,±e2,±e3 S4∼= (Z/2)2 oS3 symmetry of a cube;

index 2 in (Z/2)3 oS3

permuting the set of pairs{±e1}, {±e2}, {±e3}

C ±e1,±e2,±e3 S4∼= (Z/2)2 oS3 Z/2× Z/2 generated by:

±(e1 − e2) (e2 − e3, e1 − e3,−e3)±(e1 − e3) (e3 − e2,−e2, e1 − e2);±(e2 − e3) S3 symmetry as on OC

F ±e1,±e2,±e3 S4∼= (Z/2)2 oS3 Z/2× Z/2 action as on BEF;

±(e1 − e2 − e3) S3 symmetry as on OFA ±e1,±e2,±e3 D12 {±e3} is invariant.

±(e1 − e2) Standard action on the regularplanar hexagon formed by±e1,±e2,±(e1 − e2)

D ±e1,±e2,±e3 D8 Z/2× Z/2 action as on BD;±(e1 − e2) additional generator:±(e1 − e3) (−e1, e2 − e1,−e3)


2.6 The equivalence relations ∼i on the spaces Γi ×DiBy Theorem 2.7 the equivalence relations ∼i, as well as Γi - equivariant cellstructures on Z and ordinary cell structures on Γi\Z, are determined by theright actions of the isotropy groups Hd ⊂ Γ0, d ∈ D, on the sets Si togetherwith the right action of gAB and gAD on these sets. Clearly, the associatedleft action of Hd on the sets Si has identical orbits and isotropy groups as theright action; in the case of gAB and gAD the left and right actions are evenidentical because both elements agree with their own inverses; we prefer towork with the left actions.

As remarked above the isotropy groups and hence their actions are constanton the cells of D and we consider cell by cell separately. In our analysis theelements in S2 will be labelled by pairs consisting of a non-zero vector in F3

2

and a plane in F32 containing this vector, e.g. 010x = z, 011x = 0, . . . . The

plane with equation x + y + z = 0 will be abbreviated by Σ = 0. We willalso abbreviate (ABD, 100y = z) by ABD100y = z and so on. The proofsin this section are all straightforward and are left to the reader.

2.6.1 3 - cells

By Theorem 2.7 and by Section 2.5 there are no identifications in the interiorof the 3 - cells. As there is only one 3 - cell in D0, the 3 - cells in Γ1\Z willbe labelled just by the non-zero vectors F3

2. So there are seven 3 - cells inΓ1\Z, labelled:

100, 010, 001, 110, 101, 011, 111 .

Similarly there are 21 cells of dimension 3 in Γ1\Z which will be labelled:

100y = 0, 100z = 0, 100y = z010x = 0, 010z = 0, 010x = z,001x = 0, 001y = 0, 001x = y,

110z = 0, 110x = y, 110Σ = 0,101y = 0, 101x = z, 101Σ = 0,011x = 0, 011y = z, 011Σ = 0,111x = y, 111x = z, 111y = z.

The isotropy group of the 3 - cell is trivial, so there is nothing else to do inthis case.

2.6.2 2 - cells

1. ABD By Section 2.5 and Theorem 2.7 all the relations involving these 2- cells are of type 2, i.e. the following cells become equivalent.

24 Hans-Werner Henn

ABDs∼iACDgADs∼iABEgABs . (2.5)

Clearly the isotropy groups are trivial for all these cells.

We will now make the maps gAD and gAB explicit. We recall that bydefinition these matrices induce the linear maps on F3

2 given by

gAD(100) = 100, gAD(010) = 110, gAD(001) = 001 ,

gAB(100) = 110, gAB(010) = 010, gAB(001) = 001 .

Explicit knowledge of the action of these maps on the sets Si will be usedrepeatedly later on and therefore these maps are explicitly described in tables4 and 5 below.

Table 4: Action of gAD and gAB on S1

s 100 010 001 110 101 011 111

gADs 100 110 001 010 101 111 011

gABs 110 010 001 100 111 011 101

Table 5: Action of gAD and gAB on S2

s 100y = 0 100z = 0 100y = z 010x = 0 010z = 0 010x = z

gADs 100y = 0 100z = 0 100y = z 110x = y 110z = 0 110Σ = 0

gABs 110x = y 110z = 0 110Σ = 0 010x = 0 010z = 0 010x = z

s 001x = 0 001y = 0 001x = y 110z = 0 110x = y 110Σ = 0

gADs 001x = y 001y = 0 001x = 0 010z = 0 010x = 0 010x = z

gABs 001x = 0 001x = y 001y = 0 100z = 0 100y = 0 100y = z

s 101y = 0 101x = z 101Σ = 0 011x = 0 011y = z 011Σ = 0

gADs 101y = 0 101Σ = 0 101x = z 111x = y 111y = z 111x = z

gABs 111x = y 111x = z 111y = z 011x = 0 011Σ = 0 011y = z

s 111x = y 111x = z 111y = z

gADs 011x = 0 011Σ = 0 011y = z

gABs 101y = 0 101x = z 101Σ = 0


2. OAC In the interior of these cells all relations are of type 1. In otherwords we have to determine the action of the group HOAC

∼= Z/2 on thevector space F3

2. By table 1 the action of the non-trivial element h ∈ HOAC

on F32 is given by

h100 = 010, h010 = 100, h001 = 001 .

Hence we get the following orbits and isotropy groups for the action on S1:

Isotropy groups {1} {1} Z/2 Z/2 Z/2Orbits 100, 010 101, 011 001 110 111

For the action on S2 we obtain:

Isotropy groups {1} {1} {1} {1} {1}Orbits 100y = 0 100z = 0 100y = z 101y = 0 101x = z

010x = 0 010z = 0 010x = z 011x = 0 011y = z

Isotropy groups {1} {1} {1} Z/2 Z/2Orbits 101Σ = 0 001x = 0 111x = z 001x = y 111x = y

011Σ = 0 001y = 0 111y = z

Isotropy groups Z/2 Z/2 Z/2Orbits 110x = y 110z = 0 110Σ = 0

3. OAEF Again all relations are of type 1. Furthermore the action ofHOAEF on F3

2 is the same as in the case of OAC. Therefore we obtain thesame list of orbits and isotropy groups.

4. OCBF Once again all relations are of type 1. By table 1 the action ofthe non-trivial element h ∈ HOCBF on F3

2 is given by

h100 = 100, h010 = 001, h001 = 010

and we get the following orbits and isotropy groups for the action on S1 resp.S2:

Isotropy groups {1} {1} Z/2 Z/2 Z/2Orbits 010, 001 110, 101 100 011 111

26 Hans-Werner Henn

Isotropy groups {1} {1} {1} {1} {1}Orbits 010x = 0 010z = 0 010x = z 110z = 0 110x = y

001x = 0 001y = 0 001x = y 101y = 0 101x = z

Isotropy groups {1} {1} {1} Z/2 Z/2Orbits 110Σ = 0 100y = 0 111x = y 100y = z 111y = z

101Σ = 0 100z = 0 111x = z

Isotropy groups Z/2 Z/2 Z/2Orbits 011x = 0 011y = z 011Σ = 0

5. BEF All relations are of type 1. By table 1 the action of two generatorsh1 and h2 of HBEF

∼= Z/2× Z/2 on F32 is given by

h1100 = 010, h1010 = 100, h1001 = 111 ,

h2100 = 001, h2010 = 111, h2001 = 100 .

Hence we get the following orbits and (types of) isotropy groups for the actionon S1 resp. S2:

Isotropy groups {1} Z/2 Z/2 Z/2Orbits 100, 111, 010, 001 110 101 011

Isotropy groups {1} {1} {1} Z/2 Z/2Orbits 100y = 0 100z = 0 100y = z 110x = y 101x = z

111x = z 111x = y 111y = z 110z = 0 101y = 0010x = z 010z = 0 010x = 0001y = 0 001x = y 001x = 0

Isotropy groups Z/2 Z/2× Z/2 Z/2× Z/2 Z/2× Z/2Orbits 011y = z 110Σ = 0 101Σ = 0 011Σ = 0

011x = 0

2.6.3 1 - cells

1. OC There are only relations of type 1. By table 1 the action of twogenerators h1 and h2 of HOC

∼= S3 on F32 is given by

h1100 = 100, h1010 = 001, h1001 = 010,

h2100 = 010, h2010 = 100, h2001 = 001,


and we get the following orbits and isotropy groups for the actions on S1

resp. S2:

Isotropy groups Z/2 Z/2 S3

Orbits 100, 010, 001 110, 101, 011 111

Isotropy groups {1} Z/2 Z/2Orbits 100y = 0, 100z = 0 100y = z 110x = y

010x = 0, 010z = 0 010x = z 101x = z

001x = 0, 001y = 0 001x = y 011y = z

Isotropy groups Z/2 Z/2 Z/2Orbits 110z = 0 110Σ = 0 111x = y

101y = 0 101Σ = 0 111x = z

011x = 0 011Σ = 0 111y = z

2. OF Again there are only relations of type 1, and furthermore the actionof HOF on F3

2 is the same as in the case of OC. Therefore we obtain the samelist of orbits and isotropy groups.

3. OA There are only relations of type 1. By table 1 the action of twogenerators h1 and h2 of HOA

∼= Z/2× Z/2 on F32 is given by

h1100 = 100, h1010 = 010, h1001 = 001 ,

h2100 = 010, h2010 = 100, h2001 = 001 .

Hence we get the same orbits as in the case of the 2 - cells OAC resp. OAEF .However, the isotropy groups are now larger: the trivial ones get replaced byZ/2 generated by h1h2, the Z/2 gets replaced by Z/2× Z/2.

4. AB and AC There are relations of both types. Those of type 2 leadto ABs ∼i ACgADs and are described by tables 4 and 5. As far as relationsof type 1 are concerned we can concentrate on the edge AB. By table 2 theaction of the non-trivial element h ∈ HAB on F3

2 is here given by

h100 = 110, h010 = 010, h001 = 001 .

Hence we get the following orbits and isotropy groups for the action on S1

resp. S2 (the orbits for AB and those for AC in the same column correspondto each other via gAD; the same conventions will hold in later tables of thissection):

Isotropy groups {1} {1} Z/2 Z/2 Z/2Orbits for AB 100, 110 101, 111 010 001 011

Orbits for AC 100, 010 101, 011 110 001 111

28 Hans-Werner Henn

Isotropy groups {1} {1} {1} {1} {1}Orbits for AB 100y = 0 100z = 0 100y = z 101y = 0 101Σ = 0

110x = y 110z = 0 110Σ = 0 111x = y 111y = z

Orbits for AC 100y = 0 100z = 0 100y = z 101y = 0 101x = z

010x = 0 010z = 0 010x = z 011x = 0 011y = z

Isotropy groups {1} {1} {1} Z/2 Z/2Orbits for AB 101x = z 001x = y 011Σ = 0 010x = 0 010z = 0

111x = z 001y = 0 011y = z

Orbits for AC 101Σ = 0 001x = 0 111x = z 110x = y 110z = 0011Σ = 0 001y = 0 111y = z

Isotropy groups Z/2 Z/2 Z/2Orbits for AB 010x = z 001x = 0 011x = 0Orbits for AC 110Σ = 0 001x = y 111x = y

5. AD and AE There are again relations of both types. Those of type2 lead to ADs ∼i AEgABs and are described by tables 4 and 5. As far asrelations of type 1 are concerned we can concentrate on the edge AD. Bytable 2 the action of the non-trivial element h ∈ HAD on F3

2 is here given by

h100 = 100, h010 = 110, h001 = 001 .


resp. S2:

Isotropy groups {1} {1} Z/2 Z/2 Z/2Orbits for AD 110, 010 111, 011 001 100 101

Orbits for AE 100, 010 101, 011 001 110 111

Isotropy groups {1} {1} {1} {1} {1}Orbits for AD 110x = y 110z = 0 110Σ = 0 111x = y 111x = z

010x = 0 010z = 0 010x = z 011x = 0 011Σ = 0

Orbits for AE 100y = 0 100z = 0 100y = z 101y = 0 101x = z

010x = 0 010z = 0 010x = z 011x = 0 011y = z

Isotropy groups {1} {1} {1} Z/2 Z/2Orbits for AD 111y = z 001x = 0 101x = z 100y = 0 100z = 0

011y = z 001x = y 101Σ = 0

Orbits for AE 101Σ = 0 001x = 0 111x = z 110x = y 110z = 0011Σ = 0 001y = 0 111y = z

Isotropy groups Z/2 Z/2 Z/2Orbits for AD 100y = z 001y = 0 101y = 0Orbits for AE 110Σ = 0 001x = y 111x = y


6. BD, CD and BE There are again relations of both types. Those oftype 2 lead to BDs ∼i BEgABs resp. BDs ∼i CDgADs and are described bytables 4 and 5. As far as relations of type 1 are concerned we can concentrateon the edge BD. By table 2 the action of two generators h1 and h2 ofHBD

∼= Z/2× Z/2 on F32 is here given by

h1100 = 100, h1010 = 001, h1001 = 010 ,

h2100 = 100, h2010 = 101, h2001 = 110 .


resp. S2:

Isotropy groups {1} Z/2× Z/2 Z/2× Z/2 Z/2× Z/2Orbits for BD 010, 001 100 011 111

101, 110

Orbits for CD 110, 001 100 111 011101, 010

Orbits for BE 010, 001 110 011 101111, 100

Isotropy groups {1} {1} {1} Z/2 Z/2Orbits for BD 010x = 0 010z = 0 010x = z 100y = 0 011x = 0

001x = 0 001y = 0 001x = y 100z = 0 011Σ = 0101Σ = 0 101y = 0 101x = z

110Σ = 0 110z = 0 110x = y

Orbits for CD 110x = y 110z = 0 110Σ = 0 100y = 0 111x = y

001x = y 001y = 0 001x = 0 100z = 0 111x = z

101x = z 101y = 0 101Σ = 0010x = z 010z = 0 010x = 0

Orbits for BE 010x = 0 010z = 0 010x = z 110x = y 011x = 0001x = 0 001x = y 001y = 0 110z = 0 011y = z

111y = z 111x = y 111x = z

100y = z 100z = 0 100y = 0

Isotropy groups Z/2 Z/2× Z/2 Z/2× Z/2 Z/2× Z/2Orbits for BD 111x = y 100y = z 011y = z 111y = z

111x = z

Orbits for CD 011x = 0 100y = z 111y = z 011y = z

011Σ = 0

Orbits for BE 101y = 0 110Σ = 0 011Σ = 0 101Σ = 0101x = z

30 Hans-Werner Henn

7. BF There are only relations of type 1. By table 1 and table 2 we knowthe action of three generating involutions of HBF

∼= D8 on F32

h1100 = 010, h1010 = 100, h1001 = 111 ,

h2100 = 001, h2010 = 111, h2001 = 100 ,

h3100 = 100, h2010 = 001, h2001 = 010 .


resp S2:Isotropy groups Z/2 Z/2× Z/2 D8

Orbits 100, 111, 010, 001 110, 101 011

Isotropy groups {1} Z/2 Z/2Orbits 100y = 0, 111x = z 100y = z 110x = y

010x = z, 001y = 0 111y = z 110z = 0100z = 0, 111x = y 010x = 0 101x = z

001x = y, 010z = 0 001x = 0 101y = 0

Isotropy groups Z/2× Z/2 Z/2× Z/2 D8

Orbits 110Σ = 0 011y = z 011Σ = 0101Σ = 0 011x = 0

8. EF Again there are only relations of type 1 and by table 1 and table 2we know the action of three generating involutions of HEF

∼= D8 on F32

h1100 = 010, h1010 = 100, h1001 = 111 ,

h2100 = 001, h2010 = 111, h2001 = 100 ,

h3100 = 010, h3010 = 100, h3001 = 001 .


resp. S2:

Isotropy groups Z/2 Z/2× Z/2 D8

Orbits 100, 111, 010, 001 101, 011 110

Isotropy groups {1} Z/2 Z/2Orbits 100y = 0, 111x = z 100z = 0 101x = z

010x = z, 001y = 0 111x = y 101y = 0010x = 0, 111y = z 010z = 0 011y = z

100y = z, 001x = 0 001x = y 011x = 0


Orbits 101Σ = 0 110x = y 110Σ = 0011Σ = 0 110z = 0


2.6.4 0 - cells

1. O There are only relations of type 1. By table 3 the action of HO∼= S4

factors through an action of S3, and this action agrees with that in thecase of the edge OC. Therefore we get the same orbits; the isotropy groups“grow” by Z/2×Z/2, more precisely the trivial isotropy group gets replacedby Z/2× Z/2, Z/2 gets replaced by D8 and S3 by S4.

2. B and C There are relations of both types. Those of type 2 lead toBs ∼i CgADs and are described by table 4 and 5. As far as relations oftype 1 are concerned we can concentrate on C. By table 3 the action ofHC∼= S4

∼= Z/2×Z/2 oS3 is described as follows: for the generators h1 andh2 of Z/2× Z/2 we have

h1100 = 011, h1010 = 101, h1001 = 001 ,

h2100 = 011, h2010 = 010, h2001 = 110 .

The action of S3 is again as in the case of the edge OC. Hence we get thefollowing orbits and isotropy groups for the action on S1 resp. S2:

Isotropy groups Z/2× Z/2 S4

Orbits for C 100, 010, 001, 110, 101, 011 111

Orbits for B 100, 110, 001, 010, 101, 111 011


Orbits for C 100y = 0, 100z = 0 100y = z 111x = y

010x = 0, 010z = 0 010x = z 111x = z

001x = 0, 001y = 0 001x = y 111y = z

110z = 0, 110Σ = 0 110x = y

101y = 0, 101Σ = 0 101x = z

011x = 0, 011Σ = 0 011y = z

Orbits for B 100y = 0, 100z = 0 100y = z 011x = 0010x = z, 010z = 0 010x = 0 011y = z

001y = 0, 001x = y 001x = 0 011Σ = 0110z = 0, 110x = y 110Σ = 0101y = 0, 101x = z 101Σ = 0111x = y, 111x = z 111y = z

32 Hans-Werner Henn

3. F Here there are only relations of type 1. By table 3 the action ofHF∼= S4

∼= Z/2×Z/2 oS3 is described as follows: for the generators h1 andh2 of Z/2× Z/2 we have

h1100 = 010, h1010 = 100, h1001 = 111 ,

h2100 = 001, h2010 = 111, h2001 = 100 .

The action of S3 is again as in the case of the edge OC resp. OF . Hence weget the following orbits and isotropy groups for the action on S1 resp. S2:

Isotropy groups S3 D8

Orbits 100, 010, 001, 111 110, 101, 011


Orbits 100y = 0, 010x = 0 110x = y 110Σ = 0100z = 0, 010z = 0 110z = 0 101Σ = 0100y = z, 010x = z 101x = z 011Σ = 0001x = 0, 111x = y 101y = 0001y = 0, 111x = z 011x = 0001x = y, 111y = z 011y = z

4. A Again there are only relations of type 1. By table 3 the actionof HA

∼= D12 factors through an action of S3 and permutes the elements100, 010 and 110 while 001 is fixed under the action. Therefore we get thefollowing orbits and isotropy groups for the action on S1 resp. S2:


Orbits 100, 110, 010 101, 111, 011 001

Isotropy groups Z/2 Z/2× Z/2 Z/2× Z/2Orbits 101x = z, 101Σ = 0 100y = z 100y = 0

011y = z, 011Σ = 0 110Σ = 0 110x = y

111x = z, 111y = z 010x = z 010x = 0

Isotropy groups Z/2× Z/2 Z/2× Z/2 Z/2× Z/2Orbits 100z = 0 101y = 0 001x = 0

110z = 0 011x = 0 001x = y

010z = 0 111x = y 001y = 0


5. D and E There are relations of both types. Those of type 2 lead toDs ∼i EgABs and are described by table 4 and 5. As far as relations of type1 are concerned we can concentrate on D. By table 3 we know the action ofthree generating involutions of HD

∼= D8

h1100 = 100, h1010 = 001, h1001 = 010 ,

h2100 = 100, h2010 = 101, h2001 = 110 ,

h3100 = 100, h3010 = 110, h3001 = 001 .


resp. S2:


Orbits for D 110, 101, 010, 001 111, 011 100

Orbits for E 100, 111, 010, 001 101, 011 110

Isotropy groups {1} Z/2 Z/2Orbits for D 010x = 0, 010x = z 010z = 0 011x = 0

001x = 0, 001x = y 001y = 0 011Σ = 0101x = z, 101Σ = 0 110z = 0 111x = y

110x = y, 110Σ = 0 101y = 0 111x = z

Orbits for E 100y = 0, 100y = z 100z = 0 101y = 0010x = 0, 010x = z 010z = 0 101x = z

001x = 0, 001y = 0 001x = y 011x = 0111x = z, 111y = z 111x = y 011y = z


Orbits for D 100y = 0 011y = z 100y = z

100z = 0 111y = z

Orbits for E 110z = 0 101Σ = 0 110Σ = 0110x = y 011Σ = 0

34 Hans-Werner Henn

3 The homology of the quotient spaces

Let p be any prime. In this section we will compute the mod - p cohomologyresp. homology of the quotients of X∞, X∞,s(i) and the pair (X∞,X∞,s(i)) bythe groups Γi, and also the cohomology of the quotients of X , Xs and (X ,Xs)by Γ := SL(3,Z[1/2]); in particular we prove Theorem 1.6, Corollary 1.7 andTheorem 1.8.

In Sections 2.4 and 2.6 we described cell structures on the spaces Γi\Z 'Γi\X∞. Let Zs(i) be the 2 - singular locus of Z with respect to the actionof Γi, i = 0, 1, 2, so that Γi\Zs(i) ' Γi\X∞,s(i). We will use the resultsof Section 2 to give the boundary homomorphisms of the chain complexesC∗(Γi\(Z,Zs(i))) and C∗(Γi\Zs(i)) (with integral coefficients) in an explicitform. Then we compute the homology groups of interest from these com-plexes. As these complexes are quite big our computations will be simplifiedby Euler characteristic considerations. We summarize the discussion of Sec-tion 2 relevant for the Euler characteristic χ in the following table.

0-cells 1-cells 2-cells 3-cells number of all cells χ

Γ0\Z 5 8 5 1 19 1

Γ0\Zs(0) 5 8 4 0 17 1

Γ0\(Z,Zs(0)) 0 0 1 1 2 0

Γ1\Z 13 31 26 7 77 1

Γ1\Zs(1) 13 26 12 0 51 −1

Γ1\(Z,Zs(1)) 0 5 14 7 26 2

Γ2\Z 24 72 69 21 186 0

Γ2\Zs(2) 23 49 21 0 93 −5

Γ2\(Z,Zs(2)) 1 23 48 21 93 5

In order to determine the incidence matrices, i.e. the boundary homo-morphisms in the relevant cellular chain complexes, we will have to chooseorientations for our cells. We will choose the orientation of the edges andtriangles in D0 in accordance with the ordering of the vertices in their namesso that for example [ACD] = −[ADC] and for the boundary of [ABD] weobtain [AB] + [BD] − [AD]. (Here [ACD], [ABD] etc. denote the basiselements in the chain complex given by the cells ACD,ABD etc.; similarnotation will be used below.) The 2 - dimensional cell OAEF is orientedsuch that its boundary is [OA] + [AE] + [EF ]− [OF ]; likewise with OCBF .The 3 - dimensional cell in D can then be oriented such that its boundary isgiven by [OAEF ]− [ABD]− [ADC]− [AEB]− [BEF ]− [OCBF ]− [OAC].

Then we get an orientation of the cells (e, s) in D0 × Si (by choosing the


orientation of e) and finally we get induced orientations for the cells in Γi\Z.For example in C∗(Γ0\Z) we obtain [ABD] = [ACD] = −[ADC].

3.1 Quotients of (X∞,X∞,s(i)) by Γi

We will compute the homology of the homotopy-equivalent quotients of thepairs (Z,Zs(i)) by Γi.

1. Γ0 : There is only one 2 - and one 3 - dimensional cell in Γ0\(Z,Zs(0)) andit is clear that the boundary map ∂3 : C3 −→ C2 in the cellular chain complexC∗(Γ0\(Z,Zs(0)) is an isomorphism. This implies part a) of Theorem 1.8.

2. Γ1 : Using the description of Γ1\Z that we gave in Section 2.4 and 2.6 itis straightforward to check that the boundary maps ∂2 and ∂3 in the cellularcomplex C∗(Γ1\(Z,Zs(1)) are given by the matrices in tables 6 and 7 below.In these matrices the columns and rows are labelled by cells in Γ1\(Z,Zs(1)),i.e. by equivalence classes of “nonsingular” cells in D1 (cf. Section 2.4.2),and we have chosen representatives from equivalence classes where necessary.Furthermore all zero entries in these matrices have been omitted.

One sees at once that ∂3 has trivial kernel, i.e. H3(Γ1\(Z,Zs(1));Fp) = 0and that ∂2 is onto, i.e. H1(Γ1\(Z,Zs(1));Fp) = 0. Then the Euler charac-teristic argument implies that H2(Γ1\(Z,Zs(1));Fp) ∼= (Fp)2 and we obtainpart b) of Theorem 1.8. For later use we specify two 2 - dimensional cycleswhich form a basis of H2. We can take the cycles

[ABD100]− [OAC100] and [ABD011] + [OAEF101] . (3.1)

3. Γ2 : Now consider the complex C∗(Γ2\(Z,Zs(2))). First of all it is clearthat ∂1 : C1 −→ C0 is onto and hence we obtain H0(Γ2\(Z,Zs(2));Fp) = 0.Furthermore, using our description in Section 2.6 again, it is straightforwardto check that ∂2 and ∂3 are given by the matrices in tables 8 - 11 below.

These matrices show that the kernel of ∂3 is of dimension 1 and is generatedby the cycle:

[100y = 0]−[100z = 0]−[010x = 0]+[010z = 0]+[001x = 0]−[001y = 0] .(3.2)

In particular we get H3(Γ2\(Z,Zs(2));Fp) ∼= Fp. (Here [100y = 0] etc.denote the 3 - dimensional cells in Γ2\Z corresponding to the elements100y = 0 etc. in S2.) Furthermore, the image of ∂2 is of dimension 22,i.e. H1(Γ2\(Z,Zs(2));Fp) = 0, and then the Euler characteristic argumentimplies H3(Γ2\(Z,Zs(2));Fp) ∼= (Fp)6 and hence part c) of Theorem 1.8.

Again for later use we specify six 2 - dimensional cycles whose homologyclasses form a basis of H2. We can take the cycles

c1 : = [OAC100y = z]− [ABD100y = z] (3.3)

36 Hans-Werner Henn

c2 : = [OAC100z = 0] + [ABD100y = 0]− [ABD100z = 0]−−[OAC100y = 0] (3.4)

c3 : = −[ABD001y = 0] + [BEF100z = 0] + [OAC001x = 0] +

+[ABD100y = 0]− [OAC100y = 0] + [OCBF111x = y] (3.5)

c4 : = [ABD011x = 0] + [OAEF101y = 0] (3.6)

c5 : = [ABD011y = z]− [OAC111x = z] + [OAEF101Σ = 0] (3.7)

c6 : = [ABD011Σ = 0]− [OAC111x = z] + [OAEF101x = z] . (3.8)

Table 6: The boundary homomorphism C2 → C1 for Γ1\(Z,Zs(1))

ABD BEF OAC OAEF OCBF

100 010 001 110 101 011 111 100 100 101 100 101 010 110

BD 010 1 1 1 1 1

AB 100 1 1 1

101 1 1 1

AD 110 −1 −1 1

111 −1 −1 1

Table 7: The boundary homomorphism C3 → C2 for Γ1\(Z,Zs(1))

100 010 001 110 101 011 111

ABD 100 1

010 1

001 1

110 1 1 −1

101 1

011 1

111 1 1 −1

BEF 100 −1 −1 −1 −1

OAC 100 −1 −1

101 −1 −1

OAEF 100 1 1

101 1 1

OCBF 010 −1 −1

110 −1 −1

The

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37

Table 8: The boundary homomorphism C2 → C1 for Γ2\(Z,Zs(2)); Part I

ABD BEF

100 010 001 110 101 011 111 100

y = 0 z = 0 y = z x = 0 z = 0 x = z x = 0 y = 0 x = y z = 0 x = y Σ = 0 y = 0 x = z Σ = 0 x = 0 y = z Σ = 0 x = y x = z y = z y = 0 z = 0 y = z

BD 010 x = 0 1 1 1 1 1

010 z = 0 1 1 1 1 1

010 x = z 1 1 1 1 1

AB 001 y = 0 1 1

011 y = z 1 1

100 y = 0 1 1

100 z = 0 1 1

100 y = z 1 1

101 y = 0 1 1

101 Σ = 0 1 1

101 x = z 1 1

AD 001 x = 0 −1 −1

101 x = z −1 −1

110 x = y −1 −1

110 z = 0 −1 −1

110 Σ = 0 −1 −1

111 x = y −1 −1

111 x = z −1 −1

111 y = z −1 −1

OC 100 y = 0

OF 100 y = 0

EF 100 y = 0 1 1

BF 100 y = 0 −1 −1

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Table 9: The boundary homomorphism C2 → C1 for Γ2\(Z,Zs(2)); Part II

OAC OAEF OCBF

100 101 001 111 100 101 001 111 010 110 100 111

y = 0 z = 0 y = z y = 0 x = z Σ = 0 x = 0 x = z y = 0 z = 0 y = z y = 0 x = z Σ = 0 x = 0 x = z x = 0 z = 0 x = z z = 0 x = y Σ = 0 y = 0 x = y

BD 010 x = 0 −1 1 1 −1

010 z = 0

010 x = z 1 −1 −1 1

AB 001 y = 0 1

011 y = z 1

100 y = 0 1

100 z = 0 1

100 y = z 1

101 y = 0 1

101 Σ = 0 1

101 x = z 1

AD 001 x = 0 1

101 x = z 1

110 x = y 1

110 z = 0 1

110 Σ = 0 1

111 x = y 1

111 x = z 1

111 y = z 1

OC 100 y = 0 −1 −1 −1 1 1 1

OF 100 y = 0 −1 −1 −1 −1 −1 −1

EF 100 y = 0 1 1 1 1

BF 100 y = 0 1 1 1 1

The

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[1/2])

39

Table 10: The boundary homomorphism C3 → C2 for Γ2\(Z,Zs(2)); Part I

100 010 001 110 101 011 111

y = 0 z = 0 y = z x = 0 z = 0 x = z x = 0 y = 0 x = y z = 0 x = y Σ = 0 y = 0 x = z Σ = 0 x = 0 y = z Σ = 0 x = y x = z y = z

ABD 100 y = 0 1

100 z = 0 1

100 y = z 1

010 x = 0 1

010 z = 0 1

010 x = z 1

001 x = 0 1

001 y = 0 1

001 x = y 1 1 −1

110 z = 0 1 1 −1

110 x = y 1 1 −1

110 Σ = 0 1 1 −1

101 y = 0 1

101 x = z −1 1 1

101 Σ = 0 1 −1 1

011 x = 0 1

011 y = z −1 1 1

011 Σ = 0 1 −1 1

111 x = y 1 1 −1

111 x = z 1 1 −1

111 y = z 1 1 −1

BEF 100 y = 0 −1 −1 −1 −1

100 z = 0 −1 −1 −1 −1

100 y = z −1 −1 −1 −1

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Table 11: The boundary homomorphism C3 → C2 for Γ2\(Z,Zs(2)); Part II

100 010 001 110 101 011 111

y = 0 z = 0 y = z x = 0 z = 0 x = z x = 0 y = 0 x = y z = 0 x = y Σ = 0 y = 0 x = z Σ = 0 x = 0 y = z Σ = 0 x = y x = z y = z

OAC 100 y = 0 −1 −1

100 z = 0 −1 −1

100 y = z −1 −1

101 y = 0 −1 −1

101 x = z −1 −1

101 Σ = 0 −1 −1

001 x = 0 −1 −1

111 x = z −1 −1

OAEF 100 y = 0 1 1

100 z = 0 1 1

100 y = z 1 1

101 y = 0 1 1

101 x = z 1 1

101 Σ = 0 1 1

001 x = 0 1 1

111 x = z 1 1

OCBF 010 x = 0 −1 −1

010 z = 0 −1 −1

010 x = z −1 −1

110 z = 0 −1 −1

110 x = y −1 −1

110 Σ = 0 −1 −1

100 y = 0 −1 −1

111 x = y −1 −1


3.2 Quotients of X∞,s(i) by Γi

We will derive Theorem 1.6 from Theorem 1.8 and from the following result.

Theorem 3.1 Let p be any prime. Then the reduced cohomology of thequotients of X∞,s(i) by the action of the respective groups is given as follows.

a) H∗(Γ0\X∞,s(0);Fp) = 0

b) H∗(Γ1\X∞,s(1);Fp) = Σ(Fp)2

c) H∗(Γ2\X∞,s(2);Fp) = Σ(Fp)6

Proof. We will compute the mod - p (co-)homology from the cell complexesof the homotopy equivalent spaces Γi\Zs(i). First of all we note that in all

cases we have H0(Γi\Zs(i);Fp) = 0 (e.g. because Z is connected and becauseof Theorem 1.8).

a) Because of Euler characteristic considerations it suffices in the case ofΓ0 to show that the boundary map ∂2 : C2 −→ C1 in the cellular complexC∗(Γ0\Zs(0)) is a monomorphism. This can be easily seen from figure 1.

b) In the case of Γ1 the Euler characteristic argument shows that is enough

to verify H2(Γ1\Zs(1);Fp) = 0. For this we need to show that the boundaryhomomorphism ∂2 is injective. This boundary homomorphism can be easilydetermined by the information provided in Section 2.6 and is described intable 12 below. Injectivity is now easily checked.

c) Finally we consider the case of Γ2. Again by the Euler characteristic

argument it suffices to show H2(Γ2\Zs(2);Fp) = 0. The boundary map ∂2 isnow described in tables 13 and 14 and again injectivity is easily checked. 2

3.3 Quotients of X∞ by Γi

In order to determine H∗(Γi\X∞;Fp) ∼= H∗(Γi\Z∞;Fp) it remains to computethe relevant connecting homomorphismsm in the long exact sequence for thehomology of the pair Γi\(Z,Zs(i)).

In case i = 0 we obtain clearly H∗(Γ0\Z;Fp) = 0 and in the other two casesone checks easily that under ∂2 : C2(Γi\Z) −→ C1(Γi\Z) the images of therelative cycles in (3.1) resp. (3.3) ff. are linearly independent in the quotientof C1(Γi\Zs) by the image of ∂2 : C2(Γi\Zs(i)) −→ C1(Γi\Zs(i)); in fact,to see this it is enough to determine the “OA” - part of the total boundaryof these relative cycles and compare with tables 12 resp. 13 and 14. Inother words, the corresponding connecting homomorphism in the long exactsequence is injective and then even an isomorphism because of dimensionreasons. Part a), b) and c) of Theorem 1.6 follow.

42 Hans-Werner Henn

Table 12: The boundary homomorphism C2 → C1 for Γ1\Zs(1)

OAC OAEF OCBF BEF

001 110 111 001 110 111 100 011 111 110 101 011

BD 100 1

011 −1 1 1

111 1 −1 1

AC 001 1

110 1

111 1

AE 001 1

110 1

111 1

OC 100 −1 1

110 −1 1

111 −1 1

OF 100 −1 −1

110 −1 −1

111 −1 −1

EF 110 1 1

101 1 1

100 1 1

BF 011 1 −1

110 −1 −1

100 1 1

OA 001 1 1

110 1 1

111 1 1

100

101

The

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Table 13: The boundary homomorphism C2 → C1 for Γ2\Zs(2); Part I

OAC OAEF OCBF BEF

001 111 110 001 111 110 100 111 011 101 110 011

x = y x = y x = y z = 0 Σ = 0 x = y x = y x = y z = 0 Σ = 0 y = z y = z y = z x = 0 Σ = 0 x = z Σ = 0 x = y Σ = 0 y = z Σ = 0

BD 100 y = 0 1

100 y = z 1

011 x = 0 −1 −1 1

011 y = z 1 −1 1

111 x = y 1 1 1

111 y = z −1 1 1

AC 001 x = y 1

111 x = y 1

110 x = y 1

110 z = 0 1

110 Σ = 0 1

AE 001 x = y 1

111 x = y 1

110 x = y 1

110 z = 0 1

110 Σ = 0 1

OC 100 y = z −1 1

111 x = y −1 1

110 x = y −1 1

110 z = 0 −1 1

110 Σ = 0 −1 1

OF 100 y = z −1 −1

111 x = y −1 −1

110 x = y −1 −1

110 z = 0 −1 −1

110 Σ = 0 −1 −1

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Table 14: The boundary homomorphism C2 → C1 for Γ2\Zs(2); Part II

OAC OAEF OCBF BEF

001 111 110 001 111 110 100 111 011 101 110 011

x = y x = y x = y z = 0 Σ = 0 x = y x = y x = y z = 0 Σ = 0 y = z y = z y = z x = 0 Σ = 0 x = z Σ = 0 x = y Σ = 0 y = z Σ = 0

EF 100 z = 0 1 1

110 x = y 1 1 1

110 Σ = 0 1 1

101 x = z 1 1

101 Σ = 0 1 1

BF 100 y = z 1 1

110 x = y −1 −1

110 Σ = 0 −1 −1

011 y = z 1 1 −1

011 Σ = 0 1 −1

OA 100 y = 0

100 z = 0

100 y = z

001 x = 0

001 x = y 1 1

101 y = 0

101 x = z

101 Σ = 0

110 x = y 1 1

110 z = 0 1 1

110 Σ = 0 1 1

111 x = y 1 1

111 x = z


3.4 Quotients by SL(3,Z[1/2])

As before we abbreviate SL(3,Z[1/2]) by Γ. In this section we are concernedwith the proof of part d) of Theorem 1.6 and Theorem 1.8, i.e. with thecomputation of the mod - p cohomology of Γ\X and the mod - 2 cohomologyof the pair (Γ\X ,Γ\Xs). For this we consider the Γ - equivariant projectionmap p : X −→ X2 and the spectral sequences (which arise from the skeletalfiltrations of the bases) of the following associated maps

pX : Γ\X −→ Γ\X2∼= ∆2 ,

p(X ,Xs) : Γ\(X ,Xs) −→ Γ\X2∼= ∆2 ,

and also that of

pXs : Γ\Xs −→ Γ\X2∼= ∆2 .

The fibres of these maps over the i - simplices in ∆2 are homeomorphicto the spaces Γi\X∞ resp. to Γi\(X∞,X∞,s(i)) resp. to Γi\X∞,s(i) (cf.Section 2.1). Therefore Theorem 3.1 and the already proven parts a), b)and c) of Theorem 1.6 and Theorem 1.8 immediately give the following E1

- terms for the cohomology spectral sequences converging to H∗(Γ\X ;Fp),H∗(Γ\(X ,Xs);Fp) resp. H∗(Γ\Xs;Fp).

-

t

0

1

2

3

6

s0 1 2

3 3 1

1

Es,t1 (Γ\X )

-

t

0

1

2

3

6

s0 1 2

6 6

1

Es,t1 (Γ\(X ,Xs))

-

t

0

1

2

3

6

s0 1 2

3 3 1

6 6

Es,t1 (Γ\Xs)

The numbers in these diagramms give the dimension of Es,t1 as an Fp - vector

space. Missing numbers are to be interpreted as 0. The differential d1 on theline t = 0 (in the first and the third case) is as in the case of the simplicialchains on ∆2, in particular we get in these cases E2

0,0∼= Fp and E2

s,0 = 0

if s > 0 . In particular, we immediately obtain H∗(Γ\X ;Fp) ∼= Σ5Fp, i.e.

46 Hans-Werner Henn

part d) of Theorem 1.6. We also see that H i(Γ\(X ,Xs);Fp) = 0 for i ≤ 2,independent of the precise behaviour of the spectral sequences.

What remains to be calculated is the differential d1,21 : E1,2

1 −→ E2,21 in the

second case and the differential d1,11 : E1,1

1 −→ E2,11 in the third case. The con-

necting homomorphisms of the long exact sequences associated to the pairsΓi\(X∞,X∞,s(i)) induce (by Section 3.3) isomorphisms between Ei,1

1 (Γ\Xs)and Ei,2

1 (Γ\(X ,Xs)) for i = 1, 2, hence it suffices to do the calculation in onecase. We will show that in the third case d1

1,1 is an isomorphism if p = 2 andthis will finish the proof of Theorem 1.8.

For this consider the mod - 2 cohomology spectral sequences (arising froma skeletal filtration of the base) of the maps

EΓ×Γ Xs −→ Γ\Xs

and

EΓ×Γ X −→ Γ\X .

(For a discussion of the existence of a cellular structure on these bases werefer to the remark at the end of this section.) The E1 - terms of both spectralsequences agree except on the line t = 0 because the mod - 2 cohomologyof a fibre outside of Γ\Xs vanishes in positive dimensions. Consequently theE2 - terms of both spectral sequences also agree except on the line t = 0 andthere we get Es,0

2∼= Hs(Γ\Xs;F2) resp. Es,0

2∼= Hs(Γ\X ;F2).

Claim 1: E0,12 = 0 in both spectral sequences.

Proof. We consider the spectral sequence converging to H∗(EΓ ×Γ X ;F2).As we have seen above the groups Ei,0

2∼= H i(Γ\X ,F2) are trivial for i = 1, 2.

Therefore we have E0,12 = 0 if and only if H1(EΓ ×Γ X ;F2) = 0. From

Theorem 1.5 we know that H1(EΓ ×Γ Xs;F2) = 0 and from the discussionabove we know that H1(EΓ×Γ (X ,Xs);F2) ∼= H1(Γ\(X ,Xs);F2) = 0. Thenthe long exact sequence of the pair EΓ×Γ(X ,Xs) shows H1(EΓ×ΓX ;F2) = 0and we are done. 2

Now consider the class v2 ∈ H2(EΓ ×Γ Xs;F2) which is pulled back fromthe second universal Stiefel Whitney class in H∗(BSL(3,R);F2) under the

induced map of the composition EΓ ×Γ Xs π−→BΓi−→BSL(3,R) (where π

is given by sending Xs to a point and i by the canonical inclusion Γ ↪→SL(3,R)).

Claim 2: In the spectral sequence converging to H∗(EΓ×ΓXs;F2) the classv2 is detected on E0,2

1 .


Proof. We have E0,21∼=⊕

eH2(Γe;F2) where e runs through a set of Γ -

orbits of 0 - dimensional cells in Xs and Γe denotes the isotropy group ofthe cell e. We may write e = (e2, e∞); if the X2 - component e2 is givenby the vertex l0 defined by the standard Z2 - lattice in Q2 then we haveΓe = Γe∞, the isotropy group of the cell e∞ ⊂ X∞,s with respect to theaction of SL(3,Z). For any such cell the class v2 restricts to the Stiefel-Whitney class of the representation of Γe arising from the embedding Γe ↪→SL(3,Z[1/2]) ↪→ SL(3,R). Now Γe contains at least one element of order 2and all such elements are conjugate in SL(3,R). It follows that v2 restrictsnon-trivially to any subgroup of order 2 in Γe and hence the claim is proved.2

Now assume that the differential d1,11 (Γ\Xs) is not an isomorphism. Then

H2(Γ\Xs;F2) 6= 0 and in the spectral sequence converging to H∗(EΓ ×Γ

Xs;F2) we have E2,02∼= H2(Γ\Xs;F2) 6= 0. Because of Claim 1 we conclude

that all of E2,02 survives to E∞, and because of Claim 2 we see that the

assumption implies that H2(EΓ ×Γ Xs;F2) is a vector space of dimensionbigger than 1 in contradiction to Theorem 1.5. This finishes the proof ofpart d) of Theorem 1.8. 2

Remark. a) Our approach to the computation ofH∗(Γ\(X ,Xs);F2) is ratherindirect and one might wonder why we did not analyze the differential

E1,21∼=

3⊕i=1

H2(Γ1\(X∞,X∞,s(1));F2) −→ H2(Γ2\(X∞,X∞,s(2));F2) ∼= E2,21

directly? The reason is that the three summands in the source (correspond-ing to the three Γ - orbits of 1 - dimensional cells in X2 resp. the three edgesin ∆2) are mapped differently under this differential; only on one summandis the map induced by the inclusion Γ2 ⊂ Γ1, on the other two summands itis induced by the inclusion of Γ2 into the isotropy groups of the edges {l0, l2}resp. {l1, l2} where as in Section 2.1 l0, l1, l2 are the classes of the Z2 - latticesL0 = 〈e1, e2, e3〉, L1 = 〈1

2e1, e2, e3〉 and L2 = 〈1

2e1,

12e2, e3〉 respectively. The

component of the differential corresponding to the inclusion Γ2 ⊂ Γ1 (corre-sponding to the edge {l0, l1}) is straightforward to determine: with respect toour cell structures on the spaces Γi\Z it is induced by a cellular map which isdetermined by the forgetful map S2 −→ S1. The component correspondingto the edge {l0, l2} can also be worked out on the level of the spaces Γi\Z. Incontrast the isotropy group H{l1,l2} of the edge {l1, l2} is not contained in Γ0,hence the deformation retraction X∞ −→ Z is not H{l1,l2} - equivariant andthis makes this component of the differential much harder to evaluate. Interms of integral lattices in R3 and the spacesWi/SO(3) this last component

48 Hans-Werner Henn

is induced by the map which associates to the triple (L0, L1, L2) of lattices(with L0 being well-rounded and m(L0) = 1) the pair (L1, L2). Because L1

need not be well-rounded one has to work out the effect of the deformationretraction L1/SO(3) −→ W1/SO(3) of Section 2.2 explicitly. Although onewould not expect that this could cause unsurmountable problems it wouldbe at the very least very laborious and the author found his initial attemptsto carry this out very frustrating.

b) We have tacitly used above that X is a Γ - equivariant CW - complexand we will use it again, namely in the final step of the proof of 1.4 whichcombines Proposition 1.9 and Theorem 1.5. It is quite likely that there issuch a stucture but we do not know of a suitable reference. However, it is easyto show that X has the equivariant homotopy type of a Γ - CW complex,and this will be enough: for example, we can do induction on the skeletaX k

2 of the evident Γ - equivariant cell structure of the simplicial complex X2

using that the preimages of the space X k2 and X k

2 −X k−12 with respect to the

projection map X −→ X2 are understood by Section 2.4.


4 The cohomology of SL(3,Z[1/2])

4.1 Mod - 2 cohomology

In this section we will complete the proof of Theorem 1.4. Because of Theo-rem 1.5 and Theorem 1.8 it is enough to prove Proposition 1.9, i.e. that theconnecting homomorphism

H4Γ(Xs;F2) −→ H5

Γ(X ,Xs;F2) ∼= F2

in the long exact sequence of the pair EΓ×Γ (X ,Xs) is an epimorphism, orequivalently that the natural map

H5Γ(X ,Xs;F2) −→ H5

Γ(X ;F2)

is trivial.For this consider the following commutative diagramm in which the hori-

zontal maps are induced by inclusions and the vertical maps by projections:

H5Γ(X ,Xs;F2) −→ H5

Γ(X ;F2)

∼=x q∗

xH5(Γ\(X ,Xs);F2)

∼=−→ H5(Γ\X ;F2)

The indicated arrows are isomorphisms because of Section 3.4 resp. becausethe isotropy groups in X − Xs are of odd order. Therefore we have to showthat the map q∗ : H5(Γ\X ;F2) −→ H5

Γ(X ;F2) is trivial.

This will be a consequence of the following two results.

Lemma 4.1 If the map q2∗ : H3(Γ2\X∞;F2) −→ H3

Γ2(X∞;F2) (induced by

projection) is trivial then so is q∗ : H5(Γ\X ;F2) −→ H5Γ(X ;F2).

Lemma 4.2 The map q2∗ : H3(Γ2\X∞;F2) −→ H3

Γ2(X∞;F2) is trivial.

Proof of Lemma 4.1 This follows immediately from naturality applied tothe following situation. If X 1 denotes the preimage of the 1-skeleton ∂∆2 of∆2 under the projection map X −→ X2 −→ Γ\X2

∼= ∆2 then consider thefollowing commutative diagram in which the vertical maps are induced byprojections and the horizontal maps by inclusions:

H5Γ(X ,X 1;F2) −→ H5

Γ(X ;F2)x xH5(Γ\(X ,X 1);F2) −→ H5(Γ\X ;F2) .

50 Hans-Werner Henn

It is clear from Section 3.4 that the horizontal arrow on the bottom line ofthe diagram is an isomorphism. By excision we see

H5(Γ\(X ,X 1);F2) ∼= H5((∆2, ∂∆2)× (Γ2\X∞);F2) ∼= Σ2H3(Γ2\X∞;F2)

and

H5Γ(X ,X 1;F2) ∼= H5

Γ2((∆2, ∂∆2)×X∞;F2) ∼= Σ2H3

Γ2(X∞;F2)

and the claim follows. 2

Proof of Lemma 4.2 This is more involved. We prefer to work in ho-mology and there we have to show that the map q2∗ : HΓ2

3 (X∞;F2) −→H3(Γ2\X∞;F2) is trivial, or equivalently that the non-trivial element (cf.Theorem 1.6) of H3(Γ2\X∞;F2) is not a permanent cycle in the homologyspectral sequence of the projection map. For this consider the short exactsequence of F2Γ2 - modules

0 −→ C3∂3−→ C2

∂2−→ I −→ 0 (4.1)

in which Ci denotes the i-th cellular chain group (with coefficients in F2)of the contractible Γ2 - space Z and I is the image of the boundary map∂2 : C2 −→ C1. Note that Ci can be identified with

⊕τ F2[Γ2/Γτ ] where τ

runs through a set of representatives of Γ2 - orbits of i - cells in Z and Γτ isthe isotropy subgroup of the chosen representative τ .

The E1 - term of the spectral sequence of the projection map is given asE1s,t∼= Ht(Γ2;Cs) and the differential d2

3,0 : E23,0 −→ E2

1,1 can be described asfollows: The group E2

3,0 is given as

E23,0∼= H3(Γ2\Z) ∼= Ker (H0(Γ2;C3) −→ H0(Γ2;C2))

while E21,1 is given as quotient

E21,1∼= Ker (H1(Γ2;C1) −→ H1(Γ2;C0))

Im (H1(Γ2;C2) −→ H1(Γ2;C1)).

All maps are, of course, induced by the differentials in the chain complexC∗. If z is an element in E2

3,0 ⊂ H0(Γ2, C3) then z = ∂y for some y ∈H1(Γ2; I) (with ∂ denoting the connecting homomorphism associated to theexact sequence (4.1)) and d2

3,0z is represented by i∗y ∈ H1(Γ2;C1) (with idenoting the inclusion of I into C1. In particular we see that the followingtwo conditions are equivalent:


1. d23,0 : E2

3,0∼= F2 −→ E2

1,1 is non-trivial.

2. Im (H1(Γ2; I)i∗−→ H1(Γ2;C1)) is strictly larger than Im (H1(Γ2;C2)

d12,1−→

H1(Γ2;C1)).

We will verify the second condition and this will complete the proof ofLemma 4.2. In fact, it suffices to verify the second condition after projectingoff to a suitable summand in H1(Γ2;C1) ∼=

⊕τ H1(Γ2;F2[Γ2/Γτ ]) where as

above τ runs through a set of representatives of Γ2 - orbits of 1 - cells inZ. We choose the 1 - dimensional cells τ1 resp. τ2 given by 1 · (AC, 001x =y) ⊂ Γ2 × D2/ ∼2 and 1 · (OA, 001x = y) ⊂ Γ2 × D2/ ∼2 (where 1 ∈ Γand our conventions for labelling the cells in Z ∼= Γ2 ×D2/ ∼2 are those ofSection 2.4.2). We will denote the corresponding projections by π1 and π2

respectively. Lemma 4.2 will then follow from the following two results. 2

Lemma 4.3 If y ∈ H1(Γ2;C2) is mapped non-trivially by the map H1(Γ2;C2)d1

2,1−→ H1(Γ2;C1)π1−→ H1(Γ2;F2[Γ2/Γτ1 ]) then y is also mapped non-trivially by

H1(Γ2;C2)d1

2,1−→ H1(Γ2;C1)π2−→ H1(Γ2;F2[Γ2/Γτ2 ]).

Lemma 4.4 There is an element u ∈ H1(Γ2; I) which maps non-trivially

by the map H1(Γ2; I)i∗−→ H1(Γ2;C1)

π1−→ H1(Γ2;F2[Γ2/Γτ1 ]) and trivially by

H1(Γ2; I)i∗−→ H1(Γ2;C1)

π2−→ H1(Γ2;F2[Γ2/Γτ2]).

Proof of Lemma 4.3. The differential d1s,∗ can be described as follows

(cf. chapter VII.8 of [B1]): via the identifications of Cs with⊕

σ F2[Γ2/Γσ]and of Cs−1 with

⊕τ F2[Γ2/Γτ ] the στ component of d1

s,∗ is induced by the

corresponding component F2[Γ2/Γσ]∂σ,τ−→ F2[Γ2/Γτ ] of the boundary map

Cs −→ Cs−1. By equivariance this component is determined by the imageof the coset 1 · Γσ ∈ F2[Γ2/Γσ], i.e. by understanding the incidence numbers[σ : gτ ] between the cell σ and all cells in the Γ2 - orbit of τ . We obtain∂σ,τ (1 · Γσ) =

∑gΓτ

[σ : gτ ]gΓτ where the sum is over the Γ2 - orbit of τ .

Because we project off via π1 and π2 and we are interested in homology indegree 1 only it suffices to consider “singular” 2 - dimensional cells σ ⊂ Zs(2)for which [σ : gτi] is non-trivial for some cell in the orbit of τi; in particularall cells of the form g · (ACD, s), g · (ABD, s) and g · (ABE, s) in Γ2\Z ∼=Γ2 × D2/ ∼2 are “non-singular” and can be ignored. By going through thediscussion of the relevant 2 - cells in Section 2.6 and using Theorem 2.7 we seethat we only get contributions to ∂σ,τi(1·Γσ) for σ = σ1 := 1·(OAC001, x = y)in the case of τ1, and σ = σ1 or σ = σ2 := 1 · (OAEF, 001x = y) in the case

52 Hans-Werner Henn

of τ2. Furthermore σ1 and σ2 are the only cells in their Γ2 - orbit for whichthe incidence numbers are non-trivial, namely equal to 1.

Therefore it suffices to consider the following situation in which we identifyH1(Γ2;F2[Γ2/Γσ]) with H1(Γσ;F2) for σ ∈ {σ1, σ2, τ1, τ2} and drop the coef-ficients from the notation; the maps i1 resp. i2 denote the inclusions of Γσ1

resp. Γσ2 into Γτ2 (cf. table 1 and table 2 for the isotropy groups and theirinclusions).

H1(Γσ1)⊕H1(Γσ2) ∼= H1(Z/2)⊕H1(Z/2) (a, b)y yH1(Γτ1)⊕H1(Γτ2) ∼= H1(Z/2)⊕H1(Z/2⊕ Z/2) (a, i1∗a+ i2∗b) .

The proof of the Lemma is now reduced to showing that a 6= 0 impliesi1∗a+ i2∗b 6= 0, and this is clear. 2

Proof of Lemma 4.4. Of course, the connecting homomorphism associatedto the exact sequence (4.1) has to send the element u in question to theelement in H3(Γ2\Z;F2) ⊂ H0(Γ2;C3) given by the cycle of (3.2):

[100y = 0] + [100z = 0] + [010x = 0] + [010z = 0] + [001x = 0] + [001y = 0] .

Consider now the following chain in C3 whose class in H0(Γ2;C3) agrees withthis cycle:

z : = [1 · (100y = 0)] + [1 · (100z = 0)] + [1 · (010x = 0)] +

+[1 · (010z = 0)] + [1 · (001x = 0)] + [1 · (001y = 0)] .

Let σ denote the 2 - dimensional cell 1·(ABD, 001x = 0) in Γ2×D2/ ∼2∼= Z.

This cell generates a free F2[Γ2] - submodule that we denote by F2[Γ2]〈σ〉.Let π3 : C2 −→ F2[Γ2]〈σ〉 denote the projection map. Then Section 2.4.2and inspection of table 5 in Section 2.6 yield π3∂3z = λσ where λ = (1 +gsg

ABgs−1) ∈ F2[Γ2], s ∈ S2

∼= Γ2\Γ0 is the element 001x = 0, gs ∈ Γ0 is afixed chosen coset representative of s and gAB ∈ Γ0 is as in Section 2.4. Notethat because of sgAB = s we have gsg

ABgs−1 ∈ Γ2, and in fact, by Section

2.4.2, the element gsgABgs

−1 is the unique non-trivial element in the isotropygroup of the cell 1 · (AB, 001x = 0). In Z the cell g · (AB, 001x = 0) getsidentified with the cell 1 · (AC, 001x = y) = τ1 if g is determined by theequation ggs = gs′g

AD, s′ ∈ S2 is the element 001x = y, gs′ is a fixed chosencoset representative of s′ and gAD is again as in Section 2.4.2. It follows thatthe assignment σ 7→ g−1τ1 induces an isomorphism F2[Γ2]〈σ〉/λF2[Γ2]〈σ〉 ∼=F2[Γ2/Γg−1τ1 ] of F2[Γ2] - modules.


Now let F2[Γ2]〈z〉 be the F2[Γ2] - submodule of C3 generated by the cycle z;obviously this is a free F2[Γ2] - module. Let C∂z

2 be the F2[Γ2] - submodule ofC2 which is generated by all cells appearing in ∂3z if this is written as linearcombination of cells; observe that C∂z

2 is a direct summand of C2 as a F2[Γ2]- module. Next let I∂z be the quotient of C∂z

2 by the F2[Γ2] - submodulegenerated by ∂3z. Then we get the following diagramm of exact sequencesof F2[Γ2] - modules:

0 −→ C3∂3−→ C2 −→ I −→ 0x i

x j

x0 −→ F2[Γ2]〈z〉 −→ C∂z

2 −→ I∂z −→ 0y π3

y π3

y0 −→ λF2[Γ2]〈σ〉 −→ F2[Γ2]〈σ〉 −→ F2[Γ2/Γg−1τ1 ] −→ 0

where the left hand vertical arrow in the upper half of the diagram is aninclusion, i is the inclusion of a direct summand, the upper left hand rectanglecommutes and induces j. The lower left hand vertical arrow is given by z 7→λσ, hence the lower left hand rectangle commutes by the formula for π3∂3(z)and induces the map π3. By going through table 5 in Section 2.6 one checks

that π3 agrees with the composition I∂zj−→ I

i−→ C1π1−→ F2[Γ2/Γg−1τ1 ]

where π1 denotes the composition of π1 with left multiplication by g−1.Now the upper half of the diagramm shows that there is an element u′ ∈

H1(Γ2; I∂z) such that ∂j∗u′ = z. (Use that z ∈ H0(Γ2;C3) comes from an

element, still denoted by z, in H0(Γ2;F2[Γ2]〈z〉) whose image in H0(Γ2;C∂z2 )

vanishes because i is split injective.) Pick any such u′ and let u := j∗u′.

Then the lower half of the diagramm shows that the connecting homomor-phism maps π3∗(u

′) to the non-trivial element in H0(Γ2;λF2[Γ2]〈σ〉) ∼= F2, inparticular π3∗(u

′) 6= 0 and hence π1∗i∗u 6= 0.

Finally, using Section 2.6 once more, it is straightforward to check thatπ2∂2 : C2 −→ F2[Γ2/Γτ2 ] vanishes on C∂z

2 and hence π2ij : I∂z −→ F2[Γ2/Γτ2 ]is the zero map and the second statement of the Lemma follows. 2

4.2 Mod - 3 cohomology

In this section we prove Theorem 1.10. We take advantage of our investi-gations in Sections 2.4, 2.5 and 2.6. In particular we will make use of thedescription of the Γj - space Z given by Theorem 2.6 resp. Theorem 2.7, i.e.

54 Hans-Werner Henn

we will identify Z with Γj×Dj/ ∼j and write A, B, . . . for the points of thisspace given by the class of (1, A), (1, B), . . ..

We break the proof into two parts.

Proof of Theorem 1.10 (a)-(c). Let Zs,3(j) denote the 3 - singular locus ofZ with respect to the action of Γj.

Part c) of the Theorem follows immediately from Section 2.6 because in

this case Zs,3(2) = ∅, and therefore we get H∗(Γ2;F3) ∼= H∗Γ2(Z;F3) ∼=

H∗(Γ2\Z;F3) ∼= Σ3F3 by Theorem 1.6.

In the cases of Γ0 and Γ1 we derive from Section 2.6 that the space Γj\Zs,3(j)consists of two components. In the case of Γ0 one of the components consistsof the image of the Γ0 - orbit of the 0 - cell A in Zs,3(0) (with isotropy groupisomorphic to D12). The other one consists of the image of the Γ0 - orbit ofthe subcomplex with the two 1 - simplices OC and OF in Zs,3(0); the Γ0 -orbits of the 1 - dimensional cells have isotropy isomorphic to S3 and theΓ0 - orbits of the three 0 - dimensional cells have isotropy group isomorphicto S4. In the case of Γ1 one component in Γ1\Zs,3(1) comes from the 0 -cell A001 (where we use the convention of Section 2.6 for labelling the cells),again with isotropy group isomorphic to D12; the other one comes from thesubcomplex with 1 - cells OC111 and OF111, with isotropy groups isomor-phic to S3 for the 1 - dimensional cells and the 0 - dimensional cell F111,and isomorphic to S4 for the 0 - dimensional cells O111 and C111.

Now we consider the spectral sequences associated to the maps

EΓj ×Γj Zs(j) −→ Γj\Zs,3(j), j = 0, 1 .

Because the inclusions ofS3 into S4 and of S3 into D12 induce isomorphismsin mod - 3 cohomology we find in both cases an isomorphism

H∗Γj(Zs,3(j);F3) ∼=2∏i=1

H∗(S3;F3) .

Furthermore it is clear thatH∗(Γj\Zs(j);F3) = F3⊕F3; from Theorem 1.6 we

know H∗(Γj\Z;F3) = 0 and therefore we conclude H∗(Γj\(Z,Zs,3(j));F3) =ΣF3. Finally the long exact sequence for the Borel cohomology of the pair(Z,Zs,3(j)) gives part a) and b). 2

To prove part (d) of Theorem 1.10 one could now consider the spectralsequence of the map EΓ×ΓX −→ ∆2 and we will actually make some use ofthis spectral sequence. However, both for the final description of the result


as well as for the proofs centralizers of elementary abelian 3 - subgroups turnout to be helpful again. In fact, we will combine information derived from theknowledge of these centralizers with information coming from the analysis ofthe spectral sequence of the map EΓ×ΓX −→ ∆2 . We start by analyzing theelementary abelian 3 - subgroups of GL(3,Z[1/2]). First it is clear that thereare no elementary abelian 3 - subgroups of rank 2 (isomorphic to Z/3×Z/3)because there are none in GL(3,R).

Proposition 4.5 In GL(3,Z[1/2]) there are precisely two conjugacy classesof subgroups isomorphic to Z/3.

Proof. These conjugacy classes are in one to one correspondence with theisomorphism classes of modules M over the group algebra Z[1/2][Z/3] whichare free of rank 3 as Z[1/2] - modules and on which Z/3 acts faithfully. Suchmodules are classified by the (obvious modification for the ring Z[1/2] of the)Diederichsen-Reiner Theorem (cf. Theorem (74.3) of [CR]); one of the twoclasses corresponds to the free Z[1/2][Z/3] - module F on one generator, theother one to T ⊕R, the direct sum of the trivial module T and the moduleR := Z[1/2][ζ3] where a fixed chosen generator of Z/3 acts by multiplicationwith ζ3, a fixed chosen third root of unity. 2

We pick a subgroup E1 corresponding to the module F and a subgroup E2

corresponding to T ⊕R. If E is a subgroup of a group G we write CG(E) forthe centralizer of E in G and NG(E) for the normalizer of E in G. The unitsin a ring R will be denoted by R×. We will now analyze the centralizers andnormalizers of Ei. We start with the case of E2.

Proposition 4.6 The centralizer CGL(3,Z[1/2])(E2) is isomorphic to Z/3 ×Z[1/2]× × Z[1/2]×.

Proof. The centralizer is isomorphic to the group of automorphisms ofthe corresponding Z[1/2][Z/3] - module, i.e. to the group of units in itsendomorphism ring. After tensoring with Q both F and R ⊕ T becomeisomorphic; both R⊗Q and T ⊗Q are irreducible, in particular there are noZ[1/2][Z/3] - module maps between R and T .

Therefore we obtain CGL(3,Z[1/2])(E2) ∼= Z[1/2]× × (Z[1/2][ζ3])× and it iseasy to check (say by using the norm map from the cyclotomic extensionQ[ζ3] down to Q) that the map

Z/3× Z[1/2]× → (Z[1/2][ζ3])×

(a, b) 7→ ζa3 b

is an isomorphism. 2

56 Hans-Werner Henn

Remark. The norm can also be used to show that Z[ζ3] is a Euclideanring and therefore a principal ideal domain. This simplifies the proof andstatement of the Diederichsen-Reiner Theorem for modules over Z[Z/3] andZ[1/2][Z/3].

Corollary 4.7 E2 is contained in SL(3,Z[1/2]) and there is a unique con-jugacy class of elementary abelian 3 - subgroups in SL(3,Z[1/2]) which mapsto the GL(3,Z[1/2]) - conjugacy class of E2. Furthermore

CSL(3,Z[1/2])(E2) ∼= Z/3× Z[1/2]× ,

NSL(3,Z[1/2])(E2) ∼= CSL(3,Z[1/2])(E2)o Z/2

and the isomorphism can be chosen such that the conjugation action of Z/2on Z/3 is non-trivial while it is trivial on Z[1/2]×.

Proof. It is clear that E2 is contained in SL(3,Z[1/2]) and also thatCSL(3,Z[1/2])(E2) is isomorphic to Z/3 × Z[1/2]×. Furthermore, if σ denotesthe Galois automorphism of Z[1/2][ζ3] then σ⊕(−id) normalizes E2 and thisshows NSL(3,Z[1/2])(E2) ∼= CSL(3,Z[1/2](E2) o Z/2 with the conjugation actionas claimed.

Now assume E′ is another subgroup of SL(3,Z[1/2]) which becomes conju-gate in GL(3,Z[1/2]) to E2, say be an element g. Now the determinant fromGL(3,Z[1/2]) to Z[1/2]× remains onto when restricted to CGL(3,Z[1/2])(E2),hence we can write g = g1g2 with g1 ∈ CGL(3,Z[1/2])(E2) and g2 ∈ SL(3,Z[1/2])and this implies that E and E′ are already conjugate in SL(3,Z[1/2]). 2

Proposition 4.8 The centralizer CGL(3,Z[1/2])(E1) is isomorphic to Z/3 ×Z[1/2]× × Z.

Proof. The module F contains the direct sum of the submodules Ker(g−1)(generated as abelian group by 1 + g + g2) and Ker(1 + g + g2) (generatedby 1 − g and 1 − g2) with quotient isomorphic to Z/3 (observe that 3 =(1 + g + g2) + (1 − g) + (1 − g2)). These submodules are isomorphic to Tresp. R and are preserved by any automorphism of F . In other words we geta homomorphism

Aut(F ) −→ Aut(R⊕ T ) ∼= Z/3× Z[1/2]× × Z[1/2]× .

This is obviously injective and we claim that its image is isomorphic to Z/3×Z[1/2]× ×Z. To see this note that the subgroup Z/3 is clearly in the image;scalar automorphisms show that the diagonal of Z[1/2]××Z[1/2]× is also in


the image. Therefore it suffices to determine which of the automorphismsαε,n : R ⊕ T → R ⊕ T , (r, t) 7→ (r, ε2nt) (with ε ∈ {0, 1} and n ∈ Z) extendsto one of F . Because of 3 = (1 + g + g2) + (1 − g) + (1 − g2) an extensionexists iff ε2n(1 + g + g2) + (1− g) + (1− g2) = (g + g2)(ε2n − 1) + (ε2n + 2)is divisible by 3. This happens iff ε2n− 1 is divisible by 3. In other words, nmay be chosen arbitrarily but ε is then determined by n. 2

Corollary 4.9 E1 is contained in SL(3,Z[1/2]) and there is a unique con-jugacy class of elementary abelian 3 - subgroups in SL(3,Z[1/2]) which mapsto the GL(3,Z[1/2]) - conjugacy class of E1. Furthermore

CSL(3,Z[1/2])(E1) ∼= Z/3× Z ,

NSL(3,Z[1/2])(E1) ∼= CSL(3,Z[1/2])(E1)o Z/2and the isomorphism can be chosen such that the conjugation action of Z/2on Z/3 is non-trivial while it is trivial on Z.

Proof. The proof is analogous to that of Proposition 4.7. One only hasto check that the restriction of the determinant to CGL(3,Z[1/2])(E1) remainsonto and that the automorphism −σ ⊕ (-id) of R ⊕ T (with σ the Galoisautomorphism of Z[1/2][ζ3]) extends to an automorphism of F . 2

We can now use the “centralizer spectral sequence”

Es,t2∼= lims

EHt(CΓ(E);F3) =⇒ Hs+t

Γ (Xs,3;F3)

of [H1] to compute H∗Γ(Xs,3;F3) where as before Γ = SL(3,Z[1/2]), X is thespace X∞×X2, but now Xs,3 denotes the 3 - singular locus of X with respectto the action of Γ; the limit is here taken over the category of elementaryabelian 3 - subgroups of Γ. Because the 3 - rank of Γ is equal to 1, thespectral sequence degenerates into an isomorphism

H∗Γ(Xs,3;F3) ∼=∏(E)

(H∗(CΓ(E);F3))NΓ(E)/CΓ(E) ∼=∏(E)

H∗(NΓ(E);F3)

where the product is indexed by conjugacy classes of elementary abelian 3 -subgroups of Γ (see 3.3.1 of [H1]). In our case there are two conjugacy classeswhose normalizers are isomorphic to S3×Z×Z/2 resp. S3×Z resp., so weobtain the following result.

Proposition 4.10

H∗Γ(Xs,3;F3) ∼=2∏i=1

H∗(S3 × Z;F3)

58 Hans-Werner Henn

We now turn towards the proof of part (d) of Theorem 1.10. By Proposition4.10 it suffices to prove the following result.

Proposition 4.11 a) H∗Γ(X ,Xs,3;F3) ∼= ΣF3 ⊕ Σ2(F3)2 ⊕ Σ5(F3).

b) The boundary map

H∗Γ(Xs,3;F3) −→ H∗+1Γ (X ,Xs,3;F3)

is surjective. Its kernel in degree 4 is of dimension 3 and is generated by theimage of the Bockstein of H3 and one further element which has non-trivialrestriction to the two factors in Proposition 4.10.

The proof of Proposition 4.11 depends on another result whose proof wegive at the end of this section.

Proposition 4.12 The restriction map H∗(Γ1;F3) −→ H∗(Γ2;F3) is onto,and with respect to the isomorphism H3(Γ1;F3) ∼= H3(S3;F3) ⊕H3(S3;F3)of part (b) of Theorem 1.10 the kernel in degree 3 restricts non-trivially toboth factors.

Proof of Proposition 4.11. We consider the E1 - term of the spectral se-quence of the map

EΓ×Γ (X ,Xs,3)→ ∆2 ∼= Γ\X2 .

By the proof of part (a) - (c) of Theorem 1.10 we get E0,11∼= E1,1

1∼= (F3)3,

E2,01∼= E2,3

1∼= F3 and Es,t

1 = 0 in all other cases. In particular, we see thatH3

Γ(X ,Xs,3;F3) = H4Γ(X ,Xs,3;F3) = 0.

Now one could try to directly compute the differentials in order to prove(a). This can presumably be done directly, but we proceed in a different waywhich at the same time turns out to be quite useful for the proof of part (b).

We use the spectral sequence of the map

EΓ×Γ X → ∆2 ∼= Γ\X2 .

By Theorem 1.10 its E1 - term is given by

Es,∗1∼=

3∏i=1

H∗(Γs;F3) if s = 0, 1 and E2,∗1∼= H∗(Γ2;F3) ∼= F3 ⊕ Σ3F3 .

By Proposition 4.10 we already know H∗(Γ;F3) in large dimensions. Thistogether with the multiplicative structure of the spectral sequence forces the


behaviour of d1 : E0,∗1 −→ E1,∗

1 and gives for all ∗ > 0 with ∗ ≡ 3, 4 mod 4that the kernel and cokernel of the map

d1 : (F3)6 ∼= E0,∗1 −→ E1,∗

1∼= (F3)6

is of dimension 2. By Proposition 4.12 the restriction map H∗(Γ1;F3) −→H∗(Γ2;F3) is onto and therefore d1 : E1,∗

1 −→ E2,∗1 is onto as well. In par-

ticular we find that the spectral sequence collapses at E2 and with someextra effort one could probably also determine the multiplicative structure.Here we need only the additive result in dimensions up to 5 where we findH0(Γ;F3) ∼= F3, H1(Γ;F3) = H2(Γ;F3) = 0, H3(Γ;F3) ∼= H5(Γ;F3) = (F3)2,H4(Γ;F3) ∼= (F3)3.

Part (a) and the surjectivity in part (b) of the proposition follow nowimmediately from the long exact sequence of the pair (X ,Xs,3) together withthe knowledge that H3

Γ(X ,Xs,3;F3) = H4Γ(X ,Xs,3;F3) = 0. It is also clear

that the kernel in degree 4 is of dimension 3 and contains the image of theBockstein of H3. The following proposition finishes the proof. 2

Proposition 4.13 The restriction maps

H∗(Γ;F3) −→ H∗(NΓ(Ei);F3) ∼= H∗(S3 × Z);F3)

are surjective for i = 1, 2 except in degree 1.

Proof of Proposition 4.13. We abbreviate NΓ(Ei) by Ni. By Smith theorythe space XEi is mod 3 - acyclic, so we try to understand the Ni - space XEi

and for this we consider the canonical Ni - equivariant map XEi −→ XEi2

induced by the Γ - equivariant projection X −→ X2. The quotient of X2 bythe action of Γ is ∆2. It is an elementary exercise to verify that the quotientof XEi

2 by Ni is the 1 - skeleton ∂∆2 of ∆2, and furthermore that the isotropygroup of the j - simplices in ∂∆2 are isomorphic to Ni ∩ Γj for j = 0, 1.

Now we compare the mod - p cohomology spectral sequences of the maps

EΓ×Γ X → ∆2 ∼= Γ\X2

andEΓ×Ni XEi → ∂∆2 ∼= Ni\XEi

2 .

As observed before the first spectral spectral sequence has as E1 - terms

Es,∗1∼= (H∗(Γs;F3))⊕3 if s = 0, 1 and E2,∗

1∼= H∗(Γ2;F3) ∼= F3 ⊕ Σ3F3 ,

while the second has

E1

s,∗ ∼= (H∗(Γs ∩Ni;F3))⊕3 if s = 0, 1 and E1

2,∗= 0 .

60 Hans-Werner Henn

The map on E1 - terms is induced by the restriction maps H∗(Γs;F3) −→H∗(Γs ∩ Ni;F3) for s = 0, 1. The groups Γs ∩ Ni are easily identified withS3 (for E1) resp. S3 × Z/2 (for E2). The map on Es,∗

1 corresponds fors = 0, 1 to the projection onto the i-th factor, i = 1, 2 (with respect to theproduct decomposition of the source, cf. Theorem 1.10(a+b)). Now we useProposition 4.12 to finish the proof. 2

Finally we turn towards the proof of Proposition 4.12.

Proof of Proposition 4.12. We dualize and work in homology. Furthermorewe use Theorem 1.10(a+b) to identify H∗(Γ0;F3) with H∗(Γ1;F3) via themap induced by inclusion. Therefore we may consider the homomorphismH∗(Γ2;F3) −→ H∗(Γ0;F3) again induced by inclusion. By Shapiro’s lemmathis homomorphism can be identified with the map

H∗(Γ0;F3[Γ0/Γ2]) −→ H∗(Γ0;F3)

induced by the canonical map of Γ0 - modules F3[Γ0/Γ2] −→ F3. We denotethe kernel of this map by K. The following lemma is the main step in theproof.

Lemma 4.14 H2(Γ0;K) ∼= F3.

We continue with the proof of Proposition 4.12. The lemma immediatelyimplies the first part of 4.12. For the second part we consider the two non-conjugate elementary abelian 3 - subgroups E1 and E2 of Γ0; they are the3 - Sylow subgroups of the two S3’s which detect H∗(Γ0;F3). The proofof the proposition will be complete once we have seen that the inclusionsof both S3’s into Γ0 induce isomorphisms H2(S3;K) −→ H2(Γ0;K). (Wenote that F3[Γ0/Γ2] is projective as F3[S3] - module, and hence H2(S3;K) ∼=H3(S3;F3) ∼= F3.) In fact, this follows at once from the following observa-tion: via mod - 2 reduction both S3’s map monomorphically to SL(3,F2)and there they agree with the normalizer of a 3 - Sylow subgroup; thereforethe composition

H2(S3;K) −→ H2(Γ0;K) −→ H2(SL(3,F2);K)

(the second arrow being induced by mod - 2 reduction) is an isomorphism.2

We turn towards the proof of Lemma 4.14. We could explicitly work outa projective resolution of the trivial F3[Γ0] - module F3 from the resolutionprovided by the cellular chains of Z, and then compute H2(Γ0;K) from thisprojective resolution. As this would be quite involved we construct just asmuch of this resolution as necessary.


Proof of Lemma 4.14. We consider the mod - 3 cellular chain complex of Zand break it apart into the following exact sequences of Γ0 - modules wherei1δ2 = ∂2 and i0δ1 = ∂1:

0 −→ C3∂3−→ C2

δ2−→ I2 −→ 0 (4.2)

0 −→ I2i1−→ C1

δ1−→ I1 −→ 0 (4.3)

0 −→ I1i0−→ C0

ε−→ F3 −→ 0 (4.4)

The lemma will follow from the long exact sequence in TorΓ0∗ (−;K) associ-

ated to the exact sequence (4.4) and the following claims. (Here and in thesequel we abbreviate TorF3[Γ0]

∗ (−;K) by TorΓ0∗ (−;K).)

Claim 1: TorΓ02 (C0;K) ∼= (F3)4.

Claim 2: TorΓ01 (I1;K) = 0 and TorΓ0

2 (I1;K) ∼= (F3)3.

Claim 3: The map TorΓ02 (I1;K) −→ TorΓ0

2 (C0;K) induced by i0 is injec-tive.

Proof of Claim 1. The Γ0 - modules Ci are direct sums⊕

σ F3[Γ0/Γσ] whereσ runs through the set of Γ0 - orbits of i - dimensional cells in Z and Γσ isthe isotropy group of a chosen representative of the orbit σ. By Sections 2.4and 2.5 we have 5 orbits of 0 - cells in Z corresponding to the vertices A,O, C, F and D in Γ0\Z, with respective isotropy groups D12 (for A), S4

(for O, C and F ) and D8 (for D). By Shapiro’s lemma we have thereforeisomorphisms

TorΓ02 (C0;K) ∼= H2(D12, K)⊕ (H2(S4;K))⊕3 ⊕H2(D8, K) .

The contribution coming from D8 is trivial because the order of D8 is primeto 3. Furthermore Γ2 has no 3 - torsion, hence the 3 - Sylow subgroup of allthe other finite subgroups acts freely on F3[Γ0/Γ2] and hence this module isprojective when restricted to the other finite subgroups. As a consequence weobtain H2(D12;K) ∼= H3(D12;F3) ∼= F3 and H2(S4;K) ∼= H3(S4;F3) ∼= F3

and the claim follows.2

Proof of Claim 2. Here we use the exact sequences (4.2) and (4.3). Just asabove we find

TorΓ0i (Cs;K) ∼=

⊕σ

Hi+1(Γσ;F3) (4.5)

62 Hans-Werner Henn

if s = 0, 1, 2, 3 and all i > 0. If i = 0, we observe that for all σ we haveH1(Γσ;F3) = 0, and hence the functors TorΓ0

0 (Cs;−) carry the short exactsequence

0 −→ K −→ F3[Γ0/Γ2] −→ F3 −→ 0

into short exact sequences. Hence we have a short exact sequence of com-plexes

0 −→ TorΓ00 (C∗;K) −→ TorΓ0

0 (C∗;F3[Γ2/Γ0]) −→ TorΓ00 (C∗;F3) −→ 0 (4.6)

for which the homology is known in the middle and on the right by Theorem1.6.

From the exact sequence (4.2), formula (4.5) and our analysis of the cellstructures and their symmetries in Sections 2.4, 2.5 and 2.6 we deduce thatTorΓ0

i (I2;K) = 0 if i > 1. For i = 1 it is isomorphic to the homologyin dimension 3 of the complex TorΓ0

0 (C∗;K); this in turn is isomorphic tothe homology in dimension 3 of the complex TorΓ0

0 (C∗;F3[Γ2/Γ0]), i.e. toH3(Γ2\X∞;F3) ∼= F3 by Theorem 1.6. For i = 0 we obtain

TorΓ00 (I2;K) ∼= Coker(TorΓ0

0 (C3;K)∂3−→ TorΓ0

0 (C2;K)) .

Now we can compute TorΓ0i (I1;K) for i = 1, 2 from the long exact sequence

in Tor which is associated to the exact sequence (4.3). Using once more ouranalysis in section 2.4, 2.5 and 2.6 we see that TorΓ0

1 (C1;K) = 0 and weobtain a short exact sequence

0 −→ (F3)2 ∼= TorΓ02 (C1;K) −→ TorΓ0

2 (I1;K) −→ TorΓ01 (I2;K) ∼= F3 −→ 0

(4.7)where the contribution to TorΓ0

2 (C1;K) comes from the two Γ0 - orbits of 1 -dimensional cells corresponding to OC and OF with symmetry group isomor-phic toS3 in both cases. For TorΓ0

1 (I1;K) we use again that TorΓ01 (C1;K) = 0

and that the map TorΓ00 (I2;K) −→ TorΓ0

0 (C1;K) is injective (because thecomplex TorΓ0

0 (C∗;K) has no homology in degree 2 by Theorem 1.6). 2

Proof of Claim 3. We proceed in several steps. In a first step we reducethe evaluation of

i0∗ : TorΓ02 (I1;K) −→ TorΓ0

2 (C0;K) ∼= H2(D12, K)⊕ (H2(S4;K))⊕3

to the study of a particular chain map. In a second step we descroibe thischain map and in a final step we finish the computation of i0∗.


First step. We consider the restriction of the map i0∗ to the subgroupTorΓ0

2 (C1;K) ∼= (H2(S3;K))⊕2 ∼= (F3)2 (cf. (4.7)). This restriction is in-duced by injections of the isotropy groups (isomorphic to S3) of the edgesOC and OF into the isotropy groups (isomorphic to S4) of the verticesO, C and F . Each of these injections induces an isomorphism of coho-mology in H2(−;K) ∼= H3(−;F3), hence these injections map (H2(S3;K))⊕2

monomorphically to the summand (H2(S4;K))⊕3 of TorΓ02 (C0;K). It suffices

therefore to show that the composition of i0 : I1 −→ C0 with the projectionπ : C0 −→ F3[Γ0/D12] induces a non-trivial map in TorΓ0

2 (−;K). To thisend we should construct F3[Γ0] - projective resolutions P∗ of I1 and Q∗ ofF3[Γ0/D12] and lift πi0 to a chain map α : P∗ −→ Q∗.

In the case of Q∗ we start with the minimal projective resolution Q′∗ for F3

as a F3[D12] - module and take Q∗ = F3[Γ0]⊗F3[D12] Q′∗. In the case of I1 we

obtain a projective resolution P∗ by taking first projective resolutions R2∗ ofI2 and R1∗ of C1; then we lift i1 : I2 −→ C1 to a chain map i1 : R2∗ −→ R1∗and obtain a double complex R∗∗ whose total complex gives the desiredprojective resolution P∗. More concretely, we can take the exact sequence0 −→ C3 −→ C2 −→ I2 −→ 0 as a projective resolution R2∗ of I2. For C1

we use the direct sum decomposition C1∼=⊕

σ F3[Γ0/Γσ] and, if Rσ1 ∗ is a

minimal projective resolution of the trivial F3[Γσ] - module F3, then we takeR1∗ =

⊕σ F3[Γ0]⊗F3[Γσ] R

σ1 ∗. In these terms, the projective resolution P∗ of

I1 looks as follows:

· · · −→ R14

∂P3−→ R13

∂P2−→ R12 ⊕ C3

∂P1−→ R11 ⊕ C2

∂P0−→ R10

with ∂P3 = ∂R13 , ∂P2 =

(∂R1

2

0

), ∂P1 =

(∂R1

1 i11

0 −∂R20

)and ∂P0 = (∂R1

0 i10).

We denote the direct summands of C2 resp. R1 corresponding to the 2 -dimensional cells BEF and OCBF resp. the 1 - dimensional faces of these2 - dimensional cells by C2 resp. R1. It is clear that the lift i1 of i1 can bechosen in such a way that i10(C2) ⊂ R10 so that

· · · −→ R14∂P3−→ R13

∂P2−→ R12∂P1−→ R11 ⊕ C2

∂P0−→ R10

is a subcomplex of P∗. Furthermore the lift α of πi0 can be chosen suchthat this subcomplex maps trivially to Q∗. Therefore, if we denote the directsummands of C2 resp. C1 correponding to the other 2 - dimensional resp. 1 -dimensional cells by C2 resp. C1 and if we use that C1 is projective (because

all isotropy groups are of even order), i.e. R10 = C1 and R1∗ = 0 for ∗ > 0,

we obtain a factorization of α to a map α of complexes P∗ −→ Q∗ as follows

64 Hans-Werner Henn

(with i10 being induced by ∂C1 ) :

0 −→ C3

−∂R20−→ C2

i10−→ C1

α2

y α1

y α0

yQ3

∂Q3−→ Q2

∂Q2−→ Q1

∂Q1−→ Q0

Second step. Now we construct α in detail. First we observe that the com-plexes P∗ resp. Q are induced from F3[D12] - complexes P ′∗ resp. Q′∗ and thatα can also be constructed as a map induced from a chain map α′ of D12 -chain complexes. So it is enough to construct α′.

We denote the natural S3 - modules F3[S3/S2] by S and the tensor prod-uct of S with the non-trivial one-dimensional S3 - module by S(−1). Withrespect to the action of the 3 - Sylow subgroup of S3 the two module struc-tures agree, so if we denote a fixed generator of this 3 - Sylow subgroup byt, we have well defined linear maps S(−1) −→ S and S(−1) −→ S whichdeserve to be labelled t2− t. We consider all these modules as D12 - modulesvia the natural homomorphism D12 −→ S3 and leave it to the reader toverify that the minimal resolution Q′∗ of the trivial F3[D12] - module F3 isperiodic of order 4 and can be described as follows:

· · ·S(−1)t2−t−→ S

1+t+t2−→ St2−t−→ S(−1)

1+t+t2−→ S(−1)t2−t−→ S .

Now we define maps α′i for i = 1, 2, 3 and leave it to the reader to verify thatthey fit together to give a D12 - equivariant chain map α′ : P ′∗ −→ Q′∗ asdesired.

As before we denote the isotropy groups of the cells OA, AB, . . . by ΓOA,ΓAB, . . .. Then the map

α′0 : F3[D12/ΓOA]⊕ F3[D12/ΓAB]⊕ F3[D12/ΓAD] −→ S

is given as follows (we choose the letter e as a generic letter for the generatorsof the various modules while gAB and gAD are the elements in ΓA = D12 whichhave been introduced in section 2.4):

α′0(eOA) = eS, α′0(eAB) = −gADeS, α′0(eAD) = −gABeS .

This is D12 - equivariant if the subgroup S2 which occurs in the definition ofthe module S is chosen as the subgroup generated by the image of gABgADgAB

with respect to the projection D12 −→ S3; for t we take the image of gADgAB.The map

α′1 : F3[D12/ΓOAC]⊕ F3[D12/ΓOAF ]⊕ F3[D12/ΓABD] −→ S(−1)


may then be given by

α′1(eOAC) = 0, α′1(eOAF ) = 0, α′1(eABD) = −eS(−1) .

Finally we have

α′2 : F3[D12] −→ S(−1), α′2(e) = eS(−1) .

Third step. We are now ready to finish the calculation of i0∗. The followingelement of K ⊂ F3[Γ0/Γ2] (cf. formula (3.2) in section 3.1)

[100y = 0]− [100z = 0]− [010x = 0] + [010z = 0] + [001x = 0]− [001y = 0]

represents a class in H2(P ′∗ ⊗F3[D12] K) and it suffices to show that its imagevia

α′2 ⊗ idK : K ∼= F3[D12]⊗F3[D12] K −→ S(−1)⊗F3[D12] K

is non-trivial in H2(Q′∗ ⊗F3[D12] K), i.e. is not in the image of

S ⊗F3[D12] Kt2−t−→ S(−1)⊗F3[D12] K .

We leave this verification to the patient reader with the hint that the calu-lation can be significantly simplified by making use of the decomposition ofF3[Γ0/Γ2] as a F3[D12] - module (cf. section 2.6). 2

4.3 Higher torsion in the integral cohomology

It is clear from Corollary 1.7 that the p - torsion in H∗(SL(3,Z[1/2]);Z)is trivial for primes p > 3. Furthermore the mod - 3 Bockstein spectralsequence and Theorem 1.10 shows that the 3 - torsion is all of order 3 and iseasily understood from the results in the last section. Therefore we restrictattention to higher 2 - torsion.

For this consider the mod - 2 Bockstein spectral sequence for SL(3,Z[1/2]):We know from Theorem 1.4 that H∗(SL(3,Z[1/2]);F2) maps injectively ontothe subalgebra of H∗(SD3;F2) ∼= F2[x, y] ⊗ E(f, g) generated by v2 = x2 +xy+y2, v3 = x2y+xy2, d3 = x2g+y2f and d5 = x4g+y4f (cf. [H1]). Thereforewe have Sq1v2 = v3 while Sq1 is zero on the other algebra generators andthus we see that the E2 - term of this spectral sequence is isomorphic toF2[v2

2]⊗E(d3, d5). The crucial point is now which order Bockstein of d3 killsv2

2.To settle this we consider the mod - 2 Bockstein spectral sequence for

GL(2,Z[1/2]): In this case H∗(GL(2,Z[1/2]);F2) maps injectively onto thesubalgebra of H∗(D2;F2) ∼= F2[x, y] ⊗ E(f, g) generated by w1 = x + y,

66 Hans-Werner Henn

w2 = xy, e1 = e + f and e3 = x2g + y2f [H1] and the E2 - term identifieswith F2[w2

2] ⊗ E(e1, e3). The restriction map from H∗(SL(3,Z[1/2]);F2) toH∗(GL(2,Z[1/2]);F2) maps d3 to e3 and v2 to w2 +w2

1, hence v23 to w2

2 +w41

which in the E2 - term is identified with w22. Therefore it suffices to determine

which higher order Bockstein of e3 kills w22.

Now we compare the mod - 2 Bockstein spectral sequence for GL(2,Z[1/2])with that of SL(2,Z[1/2]); we recall that H∗(SL(3,Z[1/2]);F2) ∼= F2[w2] ⊗E(e3) (cf. [Mi]). The notation suggests the behaviour of the restriction map,i.e. the elements w2 and e3 of H∗(GL(2,Z[1/2]);F2) map to the elementsin H∗(SL(2,Z[1/2]);F2) with the same name. Furthermore, the elementw2 comes from an integral class in H∗(SL(2,Z[1/2]);Z) (namely the firstChern class c1), hence Sq1 acts trivially on it. Therefore, the E2 - termin the case of SL(2,Z[1/2]) is isomorphic to F2[w2] ⊗ E(e3) and hence itis enough to determine which higher order Bockstein of e3 kills w2

2 in theBockstein spectral sequence for SL(2,Z[1/2]), or equivalently which is theadditive order of the second power of the integral lift c1 of w2. This canbe checked to be of order 8, e.g. by playing off the mod - 2 cohomologycomputation [Mi] against an integral cohomology computation based on theamalgam description SL(2,Z[1/2]) ∼= SL(2,Z) ∗∆ SL(2,Z) [Se]. Here ∆ isthe subgroup of SL(2,Z) consisting of all matrices which are upper triangularmodulo 2.

We summarize our discussion in the following result.

Proposition 4.15 The higher 2 - torsion in H∗(SL(3,Z[1/2]);Z) is all oforder 8 and is represented in the mod - 2 Bockstein spectral sequence by theclasses v2

2n and d5v22n (n > 0); the classes 1 and d5 represent classes of

infinite order. 2

The integral cohomology of SL(3,Z[1/2]) can now be easily written downexplicitly. We leave the details to the interested reader.


5 The cohomology of GL(3,Z[1/2])

Let GL±(3,Z[1/2]) be the preimage of the subgroup {±1} of (Z[1/2])× underthe determinant GL(3,Z[1/2]) −→ (Z[1/2])×. The group GL±(3,Z[1/2])splits as SL(3,Z[1/2])× Z/2, so we understand its mod p - cohomology byTheorem 1.4, Corollary 1.7 and Theorem 1.10. We will work out the mod -p cohomology spectral sequences of the group extension

1 −→ GL±(3,Z[1/2]) −→ GL(3,Z[1/2]) −→ Z −→ 1 (5.1)

where the homomorphism from GL(3,Z[1/2]) to Z is the determinant fol-lowed by the quotient map (Z[1/2])× −→ (Z[1/2])×/{±1} ∼= Z.

Note that the matrix 2 · id is central in GL(3,Z[1/2]), hence it acts triviallyon GL±(3,Z[1/2]) by conjugation. Its determinant is 8 = 23 which corre-sponds to the element 3 in Z under the determinant map. It follows that theconjugation action of Z on H∗(GL±(3,Z[1/2]);Fp) factors through an actionof Z/3.

The case p > 3. The conjugation action of Z/3 on H5(GL±(3,Z[1/2]);Fp)∼= Fp comes from one on integral cohomology, hence it is necessarily trivial.Furthermore the spectral sequence necessarily collapses at E2 and we obtainthe following result.

Proposition 5.1 Assume p > 3. Then there is an isomorphism of algebras

H∗(GL(3,Z[1/2]);Fp) ∼= H∗(SL(3,Z[1/2]);Fp)⊗H∗((Z[1/2])×;Fp) .

We have chosen Z[1/2]× as second factor in order to get “symmetric state-ments” for the different primes.

The case p = 2. Again we look at the spectral sequence of the group exten-sion (5.1). As in the case of primes p > 3 we claim that the conjugation actionof Z/3 on H∗(GL±(3,Z[1/2]);F2) ∼= H∗(SL(3,Z[1/2]);F2) ⊗H∗(Z/2;F2) istrivial.

First we note that this action leaves the two factors H∗(SL(3,Z[1/2]);F2)and H∗(Z/2;F2) invariant and is clearly trivial on the second factor. Bydimensional reasons it is clear that the action is trivial on v2, and because ofSq1v2 = v3 it is also trivial on v3. Now we know that the action of Z/3 onH3(SL(3,Z[1/2];F2) ∼= (F2)2 has an invariant subspace (namely the subspacegenerated by v3) and this forces it to be also trivial on d3. Next the formulaSq2d3 = d5 and multiplicativity of the action shows that Z/3 acts triviallyas claimed. We obtain E2

∼= H∗(SL(3,Z[1/2]);F2) ⊗ H∗((Z[1/2])×;F2) asalgebras. By Theorem 1.1 E2 consists of permanent cycles, i.e. the spectralsequence collapses and we have finally proved Theorem 1.3.

68 Hans-Werner Henn

The case p = 3. Once more we look at the spectral sequence of thegroup extension (5.1). Using the restriction map to the cohomology of thecentralizers CSL(3,Z[1/2])(Ei) together with the description of these groupsas provided by Section 4.2, it is easy to see that the action of Z/3 on

H∗(GL±(3,Z[1/2]);F3) is trivial. So as before we obtain an isomorphismE2∼= H∗(SL(3,Z[1/2]);F3)⊗H∗((Z[1/2])×;F3) as algebras, i.e. there is no

room for differentials and the spectral sequence collapses. By using the re-striction maps to the centralizers CGL(3,Z[1/2])(Ei) we see that the E2 - termgives also the algebra structure. We state the result of our discussion in thefollowing result.

Proposition 5.2 There is an isomorphism of algebras

H∗(GL(3,Z[1/2]);F3) ∼= H∗(SL(3,Z[1/2]);F3)⊗H∗((Z[1/2])×;F3) .


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Hans-Werner HennMathematisches Institut der UniversitatIm Neuenheimer Feld 288D-69120 HeidelbergGermany

Current address:Departement de MathematiqueUniversite Louis Pasteur7, rue Rene DescartesF-67084 StrasbourgFrance

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