on the theory of obstructions

Annals of Mathematics

On the Theory of ObstructionsAuthor(s): J. H. C. WhiteheadSource: Annals of Mathematics, Second Series, Vol. 54, No. 1 (Jul., 1951), pp. 68-84Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1969311 .

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ANNALS OF MATHEMATICS Vol. 54, No. 1, July, 1951

ON THE THEORY OF OBSTRUCTIONS

BY J. H. C. WHITEHEAD

(Received August 1, 1950; Revised January 22, 1951) 1. Introduction

N. E. Steenrod has solved1 the homotopy classification problem for maps K -+ S', where n > 2 and K is an (n + 1)-dimensional polyhedron. His solu- tion is stated in terms of separation cochains and depends on a certain theorem concerning obstructions (cf. (6.1) below). The latter, likewise the homotopy classification theorem, has been extended by M. M\. Postnikov and also, for n = 2, by Hassler Whitney to the case of maps in an (n - 1)-connected space X, which is arbitrary except that 7r1(X) is finitely generated.2

The main purpose of this note is to show how the theory of composite chain systems ([11], ?4) and the secondary boundary operator can be used in proving the theorems of Postnikov and Whitney. Our conditions are more restrictive than theirs, since we eventually confine ourselves to finite complexes. This is because our definitions of the squaring operations (?5 below) do not apply to infinite complexes. However the same theorems can be proved, by an elabora- tion of our methods, for maps of a finite complex in an arbitrary (n - 1)-connected space. This was done in an earlier draft. But the simplification due to the relation (5.9) below, for which the image space is also required to be a finite complex, seems to justify the loss of generality. In revising the first draft I have been greatly helped by a series of discussions with Steenrod, who suggested the use of the difference homomorphism and the relation (5.9).

2. General remarks on obstructions Let K be a connected CW-complex,3 which may be infinite. We restate the

elements of obstruction theory in terms of the groups

Aq(K) = 7rq(KQ), Rq(K) = 7rq(K', K-l),

where q = 1, 2, * and R1(K) = A1(K). Let

Aq 4 Rq Aq,, 4 A-+ Al

be the sequence of boundary homomorphisms and injections, where

Aq = AJ(K), Rq = Rq(K).

Let Zr = dTl(o), where

dr = jrL/r: Rr -* Rri1 (r _ 2;j1 = 1),

1 See Reference [1]. The problem had previously been solved for n = 2 by Pontrjagin ([31).

2See [4], [51, [6], [7]. 3 See ?5 of [8]. A 0-cell e0 e KO is to be taken as base point for all the homotopy groups

which we associate with K. When discussing a map f: K -+ X we take fe0 to be the base point in X.

68



ON THE THEORY OF OBSTRUCTIONS 69

and let Hq = Aq/f1Rq+i and Hr = Zr/dRr+l. Then IIq and Hr may be identified with irq(K) and with the integral homology group Hr = Hr(K), based on finite chains, where K is the universal covering complex of K. Let

Cq = Cq(K) (q = O1, ...)

be a system of chain groups, with a boundary operator a, which are related to Rq, d as in ?8 of [9]. We take C2 to be R2 made Abelian and Cq = Rq, c q+l = dq+l if q > 2. The group Cq is a free Hi1-module, which may be interpreted as the group of finite q-chains' in K.

Let G be any (discrete) additive Abelian group, which admits H, as a group of operators. We define the (equivariant) cochain groups5

C'(G) = Cq(K; G) = Ophom (Cq, G) (q > 0),

meaning the group of operator homomorphisms Cq -+ G. Thus infinite cocycles are allowed but Cq(G), and the related cohomology group Hq(G), are discrete.

Let Dq = Hq(Kq, Kql). Then Dq may be interpreted as the ordinary group of (finite) q-chains, without operators, and Dq(G) = Hom (Dq, G) is the ordinary group of (q, G)-cochains. Let ,: Cq -* Dq be the natural homomorphism. Then /iCq = Dq and ,?-'(0) is generated by all elements of the form6 xc - -c, with x E Hi , c E Cq . If Hi operates simply on G (i.e. if xg = g for every x E Hi, g E G) then a cochain mapping, j4*:Dq(G) -> Cq(G), is defined by A*-q = -u, where

- E Dq(G). Moreover t(xc - c) = 0, if t - Cq(G), because t is an operator homomorphism. Therefore iV-1(0) = 0 and ,* has an inverse, which is given by (A*-li),c = (c. Therefore /L*:D (G) ~ Cq(G) and Hq(G) may be regarded as the ordinary (discrete) cohomology group with coefficients in G (Cf. (25.2) in [131).

By a pair, (K, L), we shall mean a connected CW-complex, K, together withl a sub-complex L C K, which may be vacuous. Let (eq) be the totality of oriented q-cells in K and let eq E Cq, likewise eq E Dq, denote the basis elements7 which correspond to the cell eq. Then Cq is freely generated by the set of elements (xeq) and Dq by (eq), for every eq E K, x Ec HI. The group Dq(L) = Hq(Lq, Lq-') is defined even if L is not connected. If eq E L we identify eq - Dq with eq E Dq(L). Thus DDq(L) is the subgroup of Dq , which consists of the q-chains carried by L. Let C (L) C Cq be the sub-group which is generated by the elements xeq for every x E HI , eq E L. Then the relative cochain groups,

Cq(K) L; G), Dq(K) L; G),

4Co and CG, when treated as groups of chains in K, are normalized as in ?12 of [9], Co being a free II,-module with a single basis element.

I See [13] and [161. 6 See (23.3) in [17]. 7 The chain eq is represented by a characteristic map for the cell eq, which, in the case

of e7 e Cq (q > o), is joined to the base point by a path in Krl (q > 1), or in a tree T, such that KOCTcKl, if q = 1 (see ?5 in [9]).



70 J. H. C. WHITEHEAD

are the sub-groups of C(G), DP(G) which annihilate C'(L), D'(L). It is obvious that AC (L) = Dq(L) and if Hll operates simply on G that

A*D'q(KI L; G) = Cq(Kj L; G).

Therefore in this case the group Hq(K, L; G), which is defined in terms of Cq(K, L; G), is the ordinary (relative) (q, G)-cohomology group of the pair (K, L). Eventually we shall only be concerned with coefficient groups with simple operators in H, and we shall use A* to make the identifications

Cq(KI L; G) = Dq(KI L; G) (q = 0,1, * ). Let r = Kr L, let q ? 2 and let f : Kq -- X be a map which has an ex-

tension f:Kq -+ X, where X is any space. Let H1 = ir,(X) and let

f:7rr(K ) -*1 (r > 1) be the homomorphism induced by f' (i.e. by f' ] Kq). Then H, operates on H11 through fi and fq,3:Cq+l -- H' is a cochain in C'+l(If) Since fjAq+i = 0 the cochain fL is a spherical annihilator and a fortiori a cocycle. It vanishes if, and only if, f' has an extension Kq+l -* X. Let f" :Kq -+ X be any other extension of f and let c(f', ff") e Cq (H') be the separation cochain determined by f', f". Since f', if" are defined in Lq+1 and since f' Lq = jf I Lq it follows that

fib1 f'lo E Zq+'(K, L; II')

c = c(f', If") e Gq(K, L; IIq) It is shown in Appendix B of [11] that fq - = cj. Since jf = a it follows that8

(2.1) fAll-CfABo = ca =C.

Therefore fM - f q3 in K- L. We write

wf= {fq3} I Hq+l(K, L; HI), where { } denotes the cohomology class of a given cocycle t. A given cochain c e Cq(K, L; Hl.) can be realized9 as c(f', f") by a suitable choice of the extension f". Therefore wf vanishes if, and only if, f has an extension Rq+l -+ X. Obviously wf depends only on the homotopy class, rel. eo, of f, where eo e K0 is the base point in K. The cocycle fq3 is called the (first) obstruction of f' and wf is the (second) obstruction of f.

Let 4: (K, L) -+ (P, Q) be a cellular map in a pair (P, Q). Let PT = PT v Q and let e p1 -+ X be a map which has an extension e' Pq - X. Thus we have

(2.2) Kq-l h pq e X,

where hy = oy (y E R"'). Let f = e'h. Then f has the extension e'h' :Kq -+X, where h':Kq - Pq is the map determined by 4. Let

O Cq+l (K) + Cq+l (P)

Aq(K) hit Aq(P) X q

8 Cf. (10.2) in [12]. 9 See (11.5) in [12], which may be applied without difficulty to CW-complexes.




be the homomorphisms induced by 4 and by e'. Then w1 = {eqh3} . Since ho = Aoi it follows thatl'

(2.3) Wf = (*We -

Throughout the rest of the paper Hi will operate simply on every coefficient group, G, in C'(G), C'(P; G) etc.

3. The secondary boundary operator

Let q ? 3 and let rq(K) = i7rq(K-l), where i: 7rq(K'-l) -* Aq is the injec- tion. Then it follows from the exactness of the homotopy sequence of K', K'-1 that rq = rq(K) is the kernel of j:Aq -+ Cq . Let s: Cq+1 -+ jAq be the homomorphism which is determined by

Fq

Cq+l A, -, Cq

3 jAq

0: Cq+i -+ Cq. Since Cq and hence jAq are free Abelian it follows that j has a right inverse X:jAq -+ Aq. That is to say," jX:jAq C Cq, whence jXa8 = a. Therefore

j - Xa8) = a - jXaO = 0,

whence (f - Xad)Cq,+ C rF Therefore a homomorphism, a: Cq+i rq, is defined by ta = - Xa8, where t: rq C Aq. We describe a as a secondary boundary homomorphism (of chains). In general X, and hence a, are not uniquely defined; nor are they operator homomorphisms with respect to the operators in Hl.

Now let ll, = 1 and let jAq be a direct summand of Cq, say Cq = jAq + T. Then X has the extension : Cq -* Aq, where AT = 0. The homomorphisms a, ta are (q + 1, Aq)-cocycles and A e C'(Aq). Also Xa, = pa. Therefore

(3.1) 0 = ta + Ads = ta + A = ta + 6A.

Since l-l(0) = 0 it follows that a e Z'+l(rq) and {I 1 = { ta I = I* {I a in consequence of (3.1).

The condition that jAq is a direct summand of Cq can be expressed invari- antly. For jAq C Zq and Zq is a direct summand12 of Cq . Hence it follows that jAq is a direct summand of Cq if, and only if, it is a direct summand of Zq.

10 We shall consistently use t*, 0* to denote the homomorphisms of Hr(P, Q; G), which are induced by a map t: (K, L) -> (P, Q) and by a homomorphism 0: G -* G'.

11 If Q c P, where P is any set, then f: Q C P will denote the identical map of Q into P. We shall also denote it byf: Q = P in case Q = P.

12 In the following argument we make repeated use of the fact that a sub-group, Go, is a direct summand of a free Abelian group, G, if, and only if, GIGo is free Abelian.




This is so if, and only if, Zq/jAq is free Abelian. Let

k:Aq- lq, I :Zq -+Hq

be the natural homorphisms. Then the natural homomorphism i:H, -+ H, is defined by jka = lja(a e Aq). Therefore

ljAq= Hq = Sq,

where Sq C Hq is the group of spherical homology classes. Also

l-'(0) = aCq+l C jAq.

Therefore Zq/jAq , Hq/Sq and jAq is a direct summand of Cq if, and only if, Hq/Sq is free Abelian. This last condition is a homotopy invariant of K.

Let K' be a simply connected complex, let A' = Aq(K'), Cq = Cq(K') etc., and let H/S' be free Abelian. Let a', 3', operating on C'+,, be the analogues of a, in (3.1). Let 0:K -> K' be a cellular map and let

os: C, r- C' f :Aq >A'

be the homomorphisms induced by 0. Since jf = 4) j it follows that frq C rq . Therefore a homomorphism g: rq -> r' is defined by t'g = ft, where t': r' C Aq . Since

0j*10} == = {f=3} =f*{t3} it follows that

tI**a'} = {*t'} = 0*'} = f*t* {a} = t*g*1a}.

Since j: A -> C' has a right inverse it follows that rF is a direct summand of A' . Therefore t', and hence t* , have left inverses and it follows that

(3.2) 4* = g* {a}

The homomorphism 4)* depends only on the homotopy class of 4) and so does g, in consequence of ??3, 11 in [11]. Therefore (3.2) is a naturality condition on la}, within the homotopy'3 category of simply connected complexes such that Hq/Sq is free Abelian.

Let X be a simply connected CW-complex such that Hq(X)/Sq(X) is free Abelian. Let (a) be a set of elements which generate Hq(X) and let Oa: &-+ Xq be a map which represents the element a, where (o?q+) is a set of oriented (q + 1)- simplexes. Assuming that the simplexes o-q+ are disjoint from each other and from X we form a CW-complex,

p =~~+1(e+ q+1 .1 (3.3) P = X <, (e ) a - aq

by identifying each point y E aq+l vwith Oay E Xq. Clearly 111(P) = 1 and

__ Hq(P) = 0, Hq(P) ~ Hq(X)/Sq(X). 13 I.e. the category in which the mappings are homotopy classes of maps of one complex

in another.




Therefore P satisfies our condition concerning Sq. Since Pq = X' we have

Aq(P) = Aq(X), r=P ,X = X (r q).

Let H]J = llq(X) and let k:Aq(X) --> H be the natural homomorphism. Let

t:rq(X) C Aq(X). Then

t = kc rq(X) Hq

is one of the (natural) homomorphisms in the (invariant) sequence :(X), which is defined in [11]. Let e:Pq-' = X. Then

(3.4) We = {kf3} E Hq+l(P, X; 1H4).

Let (K, L) be a pair such that dim (K - L) < q + 1. That is to say, =q+= K, where K _ = K _ , L. Let f: K1-* X be a map which has an extension f': Kq > X. Since H1q(P) = 0 the map f' has an extension 4: K -> P. Let 41: (K, L) -> (P, X) be the map of the pair (K, L) which is determined by 4. That is to say 4ly = 4y for each point y e K. Then it follows from (2.3), with Q = X, h = f, e' = 1 in (2.2), and from (3.4) that

(3.5) Wf =

Let g: KRq -* X be a map which agrees with f in L and which has an extension g9:Kq -> X. Let 4': K -? P be an extension of g' and let 4ik: (K, L) -* (P, X) be the map which is determined by ik. Let

(0- ik)*:H[(P; G) -- Hr(K, L; G)

be the difference homomorphism, as defined in ?22 of [1], for any r ? 0 and any coefficient group G. Let

j* Hq+l (p p X; ') Hq +l (p; Hf)

be the homomorphism induced by j: P C (P, X). Then it is obvious from the definition of (4- ,f)* that

*~~~~~~')* 1*- 44' = (4+-O *

Therefore it follows from (3.5) that

wf - wg = (4 - ip)*j*{kI3} = (a - )*IkIP

where { kf I p is the cohomology class of kf3 in P, not in P - X. Since f3 ' 1a in P it follows that

{ko}p = {kLa}p = {la}p = 1*{a} ({la} = a}p).

Obviously ( - ,)* commutes with i*. Therefore

(3 6) Wf - W. = 4*( - 4)*Jal.




4. Characteristic classes

Let n > 2, let X be an (n - 1)-connected complex and let C' = Cq(X), H, = Hq(X) etc. Since X is (n - 1)-connected there are natural isomorphisms

I Hn Hn X H'(X; G) Hom(H', G),

where G is any coefficient group. We use these isomorphisms to make the identifications

(4.1) = Hn, H'(X; G) = Hom(H', G).

Since X is (n - 1)-connected we have jAq = Z' for q = n, n + 1. Let

C n z H n ils ~~~~~11

At k II

mean the same as in ?3, with q = n. Then k = lpj. Also a given homomorphism u: H' -> G is identified with

u = {ul} e H(X; G).

In particular let G = H' and let 1:H' = H'. Then 1 = {Iu}. Let f: Kn -> X be a cellular map and let

f xy Cn --> C'n ) fn: An A'n

be the homomorphisms induced by f, where Cn = Cn(K) etc. Then f, j = jfn and kfn: An > Hn is the homomorphism induced by f. Let

(4.2) c(f) = jif, Cn(K; H')

Then bc(f) = lufgl = luf,0j# = lujfn3 = kfn3. Therefore bc(f) is the first obstruction to the extension of f through Kn+l. Let f have an extension

: K n+--> X. Then bc(f) = 0 and we write

Vf = {c(f))} e H(K; H').

On replacing f by a map fi c f, such that f1K n- is a single point, vs is seen to be the characteristic cohomology class of f. It depends only on the homotopy class of f and the latter is determined by vs . Since 1 = {I u} and

f 1 = f lo: Cn -+C'n

where f is induced by f', it follows from (4.2) that

(4.3) Vf = f *(1)

Let (K, L) be a pair and let f', g' K+l -+ X be given cellular maps which coincide in L. Since X is (n - 1)-connected we have g' -' h', rel.L, where h' is




cellular and h' = f' in K7''. Let f = If' KI h = h' I K . Since f, h agree in K7- they determine separation cochains

a(f, h) e Cn (K, L; A n)X c(f, h) e Cn (K, L; Hn)X

where c(f, h) = ka(f, h). Also

- = ja(f, h), according to Appendix B in [11], where hg, Cn -+ Cn is induced by h. Therefore

(f' -9') = (f' -h')*()

= -h(fo- ho)}

= {lija(f, h) } = {ka(fh)}

(4.4) = {c(f, h)}.

Let g':K7+l > X be given and also an element u e H (K, L; Hn).

(4.5) There is a mapf' K7n+' _> X such thatf' I L = g' I L and (f' - g')*(l) = u.

Let c e Zn (K, L; 1I.) be a cocycle in the class u and let f: Kn -+ X be a map such that f I Rn-i = g | Kn-1 and c = c(f, g), where 9 = 9 I Kn. Since Oc = 0 and since g has the extension g' it follows from (2.1) above, with q = n and f', f" replaced by f, g, that f has an extension of f' :K7n+' - X. Therefore (4.5) follows from (4.4).

5. Squaring operations Let K be a finite, simplicial complex with ordered vertices and let L be a

sub-complex of K. Let cup-i products of cochains in K be defined as in [1] and let Cn (K, L) be the group of integral cochains, relative to L. Let cl, * *, cm be a cononical basis for Cn (K, L), numbered so that bcx = axc' for X 1, * , t and Acx = O if X > t, where ax > O. ax I x+i and(c, ,c) is part of a basis for C!'+'(K, L). Let G be a given coefficient group and let r(G) and y(g), [g, g'] e r1(G) (g, g' e G) be defined as in [111. We proceed to define maps

po: Cn(K, L; G) _> C2n+l(K, L; r (G))

pi:Cn (K, L; G) _> C2 n(K, L; r (G)),

on the understanding that Pi is only defined if n is even. Let Cn (K, L; B) be represented as the tensor product of B and Cn (K, L) and let 1 = g1 Ci + * + 9m Cm(9 E G) be a given element of Cn (K, L; G). Then

m PO= E 7(gx) + i ) +* C c (.1

( 5.1 )m pjyt E 'y (gx) * (CX ' CX + CX '_' 5CX) + HE 9IgX, Ha] - CX A_ en .

1X=1 X<JS




The map pi carries cocyles and coboundaries into cocyles and coboundaries; also the induced map

(5.2) p H n(K L; G) -? H2n-i+l (K, L; r(G))

is natural. More precisely, let 4 be a map of (K, L) in a pair (K', L'), let h be a homomorphism of G into a coefficient group G' and let h: r(G) -> rJ(G') be the homomorphism induced by h. Then

(5.3) OPi = p0*, hopi = Pjh*

These assertions follow from arguments which are similar to those used in ?16 of [11] to prove them in case i = 1, L = 0. The map (5.2) is the Pontrjagin square if i = 1, with n even, and we describe it as the Postnikov square ([4]) if i = 0. As in [11] it follows that, if u, v E Hn(K, L; G), then

WU(u + v) = pu + Piv + u k_ V,

where u By v is defined in terms of the pairing1 g *t = [g, g']. Also p1mu = m2Plu, whence 2P1u = u d, u.

Let t in (5.1) be a cocyle. Then oxgx = 0, whence

2ox7y(gx) = xfgx), gX] = [uxg, gX] = 0.

Therefore 2pof = 0. Since

'Y(g + g') = y(g) + y(g') + [g, g']

a similar argument shows that

po(t + a') = po4 + Pot'

if t is a cocycle and {' is any (n, G)-cochain. Therefore po is a natural homomorphism such that

2poHn(K, L; G) = 0.

As an example of a non-trivial Postnikov square let K be a 3-dimensional lens space whose fundamental group, H1, is of even order m. Let L = 0 and let G = H1,. Then r(G) is cyclic of order 2m. It may be verified, by means of Poincar6 duality and intersections, that, if u and v generate H'(K; G) and H3(K, r(G)), then Pou = mv.

Let (K, L) be arbitrary, let M be a sub-complex of L and let

(5.4) a:Hr(L, M; B) -> Hi+'(K, L; B)

be the coboundary operator in the cohomology sequence of the triple (K, L, M), with any coefficient group B. I say that, if n is even, then

(5.5) =po = pluH nH8-1(LI lJl; G) _+ H2n(K, L; r1(G)).

14 Since [g, g] = 2-y(g) this is twice the ordinary pairing if G is cyclic infinite.




This is proved in ?8 below. Let n be arbitrary, let 0 < i < n and let k = n - i. Let G2 = G/2G and let

e G2 be the coset wihich contains a given element g e G. Let cl X cm e C'(K, L) be the same as before and let

sk:Cn(K, L; G) -_ Cn+k(K, L; G2)

be the map which is given by

(5.6) 8 k = 91 C Hi c, + + gm'Cm _Ji Cm,

where g = U cl + ... + gm cm. The map Sk can be treated in the same way as pi but we shall use a different method, which is suggested by a remark in [5]. Since 2G2 = 0 the group G2 may be regarded as a vector space over the field of integral residue classes mod. 2. Let (e,) be a basis for G2 and let a commu- tative bilinear multiplication in G2 be defined by e2 = e,,, eve, = 0 if vt V. Then 92 = 9 for any 9 e G2 . Let a pairing (G, G) --+2 be defined by (g, g') = Let t in (5.6) be a cocycle. Then agx = 0, whence #x = 0 if skx is odd. Since

; Ci, Cs Cis c ex (mod. 2) if cx, c, are cocycles, mod. 2, it follows that t ',j t skS, where t < is defined in terms of the above pairing (G. G) G2 . Therefore Sk induces the homomorphism

k = Sq:kHHn(K, L; G) Hn+k(K, L; G2),

where Sqk means the same as Sqn-k in [1]. If we change from (e,) to another basis for G2, but keep the basis (cx) fixed, we see that {k is independent of the choice of the basis (e,). If we keep (e,) fixed and change to another canonical basis for Cn(K, L) we see that Bk is independent of the choice of the canonical basis (cx). Hence it follows that {k is natural and, from (9.6) in [1], that

(5.7) cak = (kia

where 5 means the same as in (5.4). Since pi, k are natural they induce natural maps of Cech cohomology groups,

which are defined by finite coverings of arbitrary pairs of spaces R and S C R (cf. [11], ?16). Moreover Pi, {Sk satisfy (5.5) and (5.7) when thus defined for the pairs (R, S) and (S, T) in a triple (R, S, T) (T C S). In particular Pi, Ok are defined in this way for a pair of cell complexes (K, L), where K is finite but not necessarily simplicial. All our complexes will now be finite cell complexes.

A (finite) cell-complex, K, is of the same homotopy type as some finite simplicial complex K' ([8], p. 239). A homotopy equivalence 4: K -+ K' induces isomorphisms

?*: Hq(K; G) Hq (K ), G) :Hq(K'; G) Hq (K; G), where H (X; G) is the Cech (q, G)-cohomology group of the space covered by X (q > 0, X = K or K'). Let

W: H (K';a G) C(K'; G)




be the natural isomorphism. Then we identify H'(K; G) with H'(K; G) by means of the isomorphism 0 = * 0'4*-'. It is easily verified that 0 is independent"' of the choice of K' or of the homotophy equivalence 4. Similarly we identify H'(K, L; G) with the corresponding Cech cohomology group of a pair (K, L).

Let X , Jb: K -* X be maps in a complex X, which coincide in the sub-complex L C K. Let d be the difference operator

d = (O - 4p)*:Hr(X; B) -) Hr(K, L; B),

with any coefficient group B. Then

(5.8) dAk = ekd

by (22.11) in [1]. Let u e H'(X; G), where n is even. Then I say that

(5.9) dpl u = p1 du + du <, 4,*u.

For it is obvious that j*d = - 4b*, where j: K C (K, L). Therefore

j*dplu = - -pl -A*piu plo*u- p*u

= pi(j*du + 4b*u) -pi n

= pij*du + (j*du) , *u

= j*(pidu + du i, #*u).

Therefore (5.9) is valid if j*-'(O) = 0. If j*-' (0) $ 0 we imbed K in a complex P, which has a sub-complex Q, such that K _, Q = P, K r- Q = L and each of K, Q is a retract16 of P. Let g: P -* K and h: P Q be retractions and let

. . . 6 IT(P, Q; B) H'(P; B) _ H'r (Q; B) **

be the cohomology sequence of P, Q. Since hi = 1, where i: Q C P, we have i*h* = 1. Therefore i* is onto, 6 = 0 and j*-'(O) = 0.

Since OIL = 4pIL the maps X, i,& can be factored into

K ) P 4',- X,

15 This is a special case of the following theorem on natural equivalences between functors ([141). Let S, T: ?I -e 0 be functors from a category 2l to a category A. Let 2l' C 2l be the sub-category, which consists of all the mappings between a sub-set of the objects in A, such that each object in 21 in equivalent to an object in Al'. Let S' = S2l1', T' = T12S' and let S', T' be related by a natural equivalence 71':S' -* T'. Then 1' can be "extended", in a unique way, to a natural equivalence q:S -* T.

16 For example we can define P as K U Q, where Q is a copy of K such that (K - L) n Q = 0 and each point in L coincides with the corresponding point in Q. E. H. Spanier pointed out to me that (O - 4p) * can be defined in terms of the Eilenberg-Steenrod axioms ([15]), without reference to cochains, by means of this construction. This observation suggested the above proof of (5.9).




where O:K C PI 0'0= X,0 *O = + and O'q = V'q = =gq if q eQ. Let

d'= (o' - v1/)*: Hr(X; B) Hr(p, Q; B).

Then it follows from (22.5) in [1] that

d = ()'O - V'o)* = 0*d'

where O1: (K, L) C (P, Q). Since j*-'(O) = 0 in (P, Q) we have

d p u = 0*d' pi u = O* (pi d'u + d'u <, p'*u)

- PlO d'u + (O*d'u) <, O*Vj*u = P1du + du \, 4P*u

and (5.9) is established. Let n > 2 and let X be an (n - l)-connected complex. We make the nat-

ural identifications (4.1) and also'7

(5.10) r = r(H2) or H' /2H',

according as n = 2 or n > 2, where rP+i = rn+1(X). Since X is (n - l)-connected every (n + 1)-cycle is spherical. Let

a I eH n+2(X ; Pr'1 {ot}6H (x~n+)

mean the same as in ?3, with q = n + 1. I say that

(5.11) { a } = pi(l) or 62(j),

according as n = 2 or n > 2. For let n = 2 and let :X- Y be a map in a simply connected complex Y. Let the identifications similar to (4.1) and (5.10) be made in Y and let

ly:H n(Y) = H n(Y), r: P3(Y)

be the identity and the homomorphism induced by 4. Then an argument similar to the one at the bottom of p. 94 in [11], in which only the top part of the dia- gram on that page is used, shows that

0*p3(ly) = 94p(l).

Hence, and from (3.2) above, (5.11), with n = 2, follows from (14.1) in [10] and an argument similar to the one on p. 95 of [11]. A similar argument estab- lishes (5.11) if n > 2 (see [19]).

6. The relative extension theorem Let (K, L) be a pair, with dim (K - L) < n + 2. Let X be an (n - 1)-con-

nected complex and let us make the identifications (4.1) and (5.10). Let f, 9: Kn -- X be maps which coincide in L and have extensions

17 See ?13 in [111 and the concluding remarks in ?14 in [111. If n = 2 this identification depends on a choice of one of the two generators of 7r3(S2).




f', g'79 :Kn~ -X. Since X is finite the group I' = H' is finitely generated. So therefore is r', obviously if n > 2 and by ?8 in [11] if n = 2. Therefore tr , C II'+, is finitely generated and so is H'+,. Since II'+, is an extension of tri 1 by H'+, it follows that I1' is finitely generated."8 Let P be given by (3.3), with q = n + 1, where (a) is a finite set of generators for H'+,. Then P is a finite complex. Let A, 4V:K -* P be extensions of f', g' and let d = (4) *. It follows from (3.6) and (5.11) that

w- w = 4*d{a}

= i*dpl(l) or 4M(1),

according as n = 2 or n > 2. Therefore it follows from (5.8), (5.9) and (4.3) that

(6.1) wf - Wg = i*(pju + u , vg) or idu,

where u = d(l) e H'(K, L; HI). It follows from (4.4) that u = {c(f, h)}, where h g 9, rel. L, and h IK n-i = f I gn-l.

Let e = f I L. Following Paul Olum ([18]) we use

On+2(e) C Hn+2(K, L; HI,)

to denote the totality of elements w1, for every extension, f': Kn+l > X, of e. Assume that e has an extension g" K -* X and in (6.1) let g = g' I KR. Then Wg = 0 and since

u = (4 - ,6)*(1) = (f' - ')*()

it follows from (4.5) that:

(6.2) On+,(e) consists of the elements t* (pi u + u v_ vg) or t*l2u, according as

n = 2 or n > 2, for every u e H (K, L;I n).

7. Homotopy classification Let (P, Q) be a given pair and let dim (P - Q) < n + 1. Let K = I X P

and let

L = (O X P) <, (I X Q) By (1 X P), M = (O X P) (I X Q).

Let 0: (P, Q) -* (L, M) be given by Op = (1, p) (p e P). Then, obviously,

If(P, Q; G) ? Hr(L, M; G) - 1r+ (K, L; G) are isomorphisms onto'9 for every r > 0.

Let X mean the same as in ?6 and let 0o, 41: P - X be maps which coincide

18 Let (gi), (hi) be sets of generators for itr+1 and H'+, . Then II'+, is obviously generated by (gi), (pi), where pi e Tln+1 is a representative of hi.

19 Concerning 3, this follows from the exactness of the cohomology sequence of the triple (K, L, M) and the fact that M is a deformation retract of K, whence Hr(K, M; G) = 0.




in Q and which are n-homotopic, rel. Q. That is to say there is a map tKn+l X, such that20

(7.1) f'(O, P) = (POP, f'(1, p) = ()ep, f'(t q) = 4)oq, for all points p e P, q E Q. Following Olum ([18]) we write

(7.2) Qnf+1 ((o 4)>1) = *6-1lOn+2(f L)

Thus On+1(0) , 41) consists of the elements

0*6-wf eHn+l(P, Q:,' +) (f = f Kn) for every f' which satisfies (7.1). It follows from his arguments, translated into geometric terms, that

a) Qn+1(4 , 4)) is a sub-group of H n'(P, Q; H,:+l)

b) On+l(po, 41) is a coset of On+1(0) , 400).

Obviously 40 c 41 , rel. Q, if, and only if, Wf = 0 for some f which satisfies (7.1). This is so if, and only if, Qf+1(Oo , 41) contains the zero element, which means that

On l((po , 41) = O? 1(4)o , 4?).

If ()i C-- 02:P -* X, rel. Q, it follows from (18.1), (18.2) and (16.5b) in [18], with n replaced by n + 1, that On+l(0p , 4)1) = On+1(40, , 2). Since 40, 41 are n-homotopic, rel. Q, we may therefore assume, after a preliminary homotopy of 41 , rel. Q, if necessary, that 4),11hn = 001Pn. Let this be so and let g':K '+' -- X be given by

(7.3) 9'(0, p) = 9op, g'(l, p) = 0lp, g'(t, i3) - for all points p e P, _ f in. Then2'

i {C(o I (pi) =*61 Qfon+l(o , )

where g = g'lKn and c()o , 4)i) is the separation cocycle of 4), , 4)1 . Therefore 4o - 4)1 if, and only if,

{cC()o, 4)1) I ?E Onli(0? , 4)).

We proceed to determine on+] (0) , 40). Let 0o = 41 and let f', 9' mean the same as in (7.1), (7.3). Then g' has the

extension g" K -* X, where g" (t, p) = 0op for every p e P. Therefore it follows from (7.2) and (6.2) that O"+'(4) , 0o) consists of the elements

(7.4) 0*-1i* (plu + u , vg) or 0*6 i*2U

for every u E H (K, L; lln).

20 Rn+1 == (0 X p) U (I X Pn) U (1 X p). 21 The choice of sign in ( Ic(oo,+,) }, which depends on the conventions used, does not

affect the following argument.




Let v e H'(P; I ') be the characteristic cohomology class of 0o and let

H (K; HIn) - H (L; II) ---* Hn(P; H )

be the homomorphisms induced by i:L C K and by 6o :P -+ L, where 8op = Op = (1, p). Obviously v = 0 *i*v, . Let u e Hn(K, L; I ) be given and let

y = a-'u e H '(L, M; IlI), z = G*y e H n(P, Q; II ). Then plu = P16y = APoy, by (5.5), and

u Vg = (by) \, 'g = 5(y \, i*vg)

0*(y i*Vg) = (W*y) \, (O*i*vg) = z \y V,

according to (3.2) and (3.4) in [2]. Therefore

0*6 (plu + U \ Vg) = 0*(Poy + y i*Vg)

= Poz + z x, v (n = 2).

Also 0*6-16u2 = 620*y = 2z (n > 2) in consequence of (5.7). Obviously 0*6c1 commutes with i* Therefore it follows from (7.4) that

(7.5) On+l(0fr age) = tipH n-(P Q; I, ')

where p:H n1 (p, Q; HI,) - H n+(P, Q; rP+1) is the homomorphism given by

(7.6) pz = poz + z , v or 62ZI

according as n = 2 or n > 2.

8. Proof of (5.5): apo = p1b Let u, v e H n-(L, M; G) Then it follows from (3.2) in [2] that bu \, 5v = 0

Therefore plb(u + v) = plbu + plb, whence p1b as well as 6po, is a homomorphism. Therefore it is sufficient to prove that

(8.1) 6 KPo(q b) '~- P16&K(q b) in K -L,

where by is the coboundary operator in Y(Y = K or L), b is a single element in a canonical basis for Cn-1(L, M) and U .b = 0.

After a preliminary subdivision, if necessary, we assume that no simplex in K - L has all its vertices in L. Let the vertices in K be ordered so that each vertex in K - L precedes each vertex in L. Then

(8.2) a Phi c = 0 if a e C'(L), c e Cq (K L),

as on p. 304 of [1], where C'(L) is identified with the sub-group of C'(K) which annihilates all p-chains in K - L. Let Aa = Ka - hLa for any a e C'(L). Then Aa e C'+'(K, L) and it follows from (8.2) that

hLa \ a = O, a) \ OKAa = x, a a = O.




Therefore

(8.3) a ,a Aa = 6Ka K2 Aa = 5K(a Aa) = 0.

Let g* b mean the same as in (8.1) and let22 6Lb 0 (r), rg = 0. Then b 5K5Lb = 0, by (8.2). Therefore

(8.4) SK(b '1 6Lb) = 5Kb \, SLb Ab a5Lb

Also 6Lb \ Ab = 0, 6Lb -1 KAb = 0, by (8.2). Therefore, on expanding 6K(bLb _,1Ab) by means of (5.1) in [1], we have

Ab \ aLb =SKaLb \j Ab (2T) (8.5)

-KAb \ Ab. Also 8KAb = - 6KLb =- 0 (T). Therefore

6K(6KAb \, Ab) -KAb ,j Ab + Ab \l SKAb (2r, r2)

Moreover SKAb \2Ab E C2n-1(K, L). Hence, and because 2ry(g) = (r2g() = 0, it follows from (8.4) and (8.5) that

6KpO(g. b) = y(*68K(b \. h5b) (8.6) ~y(g) Ab l 6KAb in K-L.

Since rg = 0 we have 8K(g * b) = g9bKb = g * Ab. Since Ab E Cn(K, L) and 5KAb 0(r), and because of (8.3), a calculation similar to the one whicl follows (16.19) in [11] shows that

p, (g *Ab) - -y (g) *Ab \,JlbKAb in K - L.

Therefore (8.1), and hence (5.5), follow from (8.6).

MAGDALEN COLLEGE OXFORD

REFERENCES

1. N. E. STEENROD, Products of cocycles and extensions of mappings, Ann. of Math., 48 (1947), 290-320.

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hedron into a simply connected polyhedron of arbitrary dimension, C. R. (Doklady) Akad. Sci. URSS (N.S.), 64 (1949), 461-2.

5. M. M. POSTNIKOV, Classification of continuous mappings of an (n + 1)-dimensional complex in a connected topological space, aspherical in dimensions less than n, C. R. (Doklady) Akad. Sci. URSS (N.S.), 71 (1950), 1027-8.

6. HASSLER WHITNEY, Classification of the mappings of a 3-complex into a simply connected space, Ann. of Math., 50 (1949), 270-84.

22 cl= C2 (r) means that cl = c: + rca, where ca, C2, ca are integral cochains.




7. HASSLER WHITNEY, An extension theorem for mappings into simply connected spaces, ibid., 285-96.

8. J. H. C. WHITEHEAD, Combinatorial homotopy 1, Bull. Amer. Math. Soc., 55 (1949), 213-45.

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London (Series A), 202 (1950), 253-63.



on the theory of obstructions

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