PICTURE GROUPS OF FINITE TYPE

AND COHOMOLOGY IN TYPE An

KIYOSHI IGUSA, KENT ORR, GORDANA TODOROV, AND JERZY WEYMAN

Abstract. For every quiver of finite type we define a finitely presented group called apicture group. We construct a finite CW complex which is shown in another paper [10] tobe a K(π, 1) for this picture group. In [5] another independent proof was given for this factin the special case of type An with straight orientation and we use this CW complex tocompute the integral cohomology of picture groups of type An with straight orientation. Itis free abelian in every degree with ranks given by the “ballot numbers”. We also computethe ring structure on the cohomology of these groups.

Contents

Introduction 21. Spherical semi-invariant picture L(Q) 41.1. Notation 41.2. Semi-invariants 62. Picture group G(Q) 73. Picture space X(Q) 103.1. Local properties of D(β) 103.2. Construction of the picture space 134. Homology of X(An) 154.1. Weight filtration of C∗(Q;Z) in general 164.2. Semi-simple categories in type A 174.3. Non-primitive weights 184.4. Primitive weights 195. Proof that X(An) is a K(π, 1) for π = G(An) 215.1. Filtration of X(An) 225.2. HNN extensions 236. Cup product structure 246.1. Dual blocks 246.2. Multiplication of dual blocks 26References 27

Date: September 25, 2014.2010 Mathematics Subject Classification. 16G20; 20F55.The first author is supported by NSA Grant #H98230-13-1-0247.The third author is supported by NSF Grant #DMS-1103813.

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Introduction

Suppose that Q is a modulated quiver of finite type with n vertices. Then there is an n−1dimensional “picture” L(Q) which is a finite n − 2 dimensional subcomplex of the sphereSn−1. The top dimensional simplices of this complex can be labeled with positive rootsof the root system of the quiver which we view as generators of the unipotent Chevalleygroup UQ(Z) associated to the root system. This is a nilpotent group with one generatorεβ(1), which we denote x(β), for every positive roots β.) On the codimension one simplices,Chevalley relations are displayed. For example, x(α), x(β) commute if α+β is not a root andis nonzero (See definition in Section 1 below.) Consequently, every region in the complementof L(Q) in Sn−1 can be labelled with an element of the nilpotent group N(Q) in such away that the group label in adjacent regions differ by right multiplication by the generatorlabelling the wall separating the regions.

However, this nilpotent group is not the minimal group supporting the picture L(Q).Although all positive roots occur as labels, only a subset of the Chevalley relations occur.We let G(Q) be the finitely presented group with only those generators and relations whichactually occur in the picture L(Q). This group depends on the orientation of the quiver Qbut we believe the cohomology of the group depends only on the underlying root system.We call this the picture group of Q.

As an example, consider the following picture.

This picture has 6 smooth curves without inflection points. These curves meet transverselyat 9 vertices. Such a picture determines a “picture group”, unique up to isomorphism, with6 generators and 9 relations as follows.

(1) Label each smooth curve with a different letter. These are the 6 generators, say,a, b, c for the circles and x, y, z for the arcs.

(2) At each vertex we obtain a relation by reading the labels of the arcs coming into thevertex counterclockwise starting at any point. Read the label as g, g−1 dependingon the curvature of the arc. For example, at the top vertex, we get the relation:

r1 = aba−1x−1b−1

since, going counterclockwise around this vertex we: enter circle a, enter circle b,exit circle a, exit the circle x for which there is only an arc in the picture and, finally,we exit circle b to return to the starting point.

Following the construction in [9] of the nilmanifold for a torsion-free nilpotent group, weview an n−1 dimensional picture as the attaching map for an n-cell in a finite CW complex.The minimal CW complex which supports the attachment of the single n-cell given by thespherical semi-invariant picture we call X(Q). In a later paper [10] we prove that this is an

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Eilenberg-MacLane space K(π, 1) with π1 = G(Q). In the present paper we prove this inthe special case when Q is An with straight orientation. And we focus on this case.

The group G(An) has generators xij for all 0 ≤ i < j ≤ n subject to the followingrelations.

(1) xij , xk` commute if either j < k or i < k < ` < j.(2) [xij , xjk] = xik for all i < j < k where [x, y] := y−1xyx−1.

These imply that G(An) is generated by the n elements xj−1,j for j = 1, · · · , n. The spaceX(An) ' K(G(An), 1) is a CW-complex with Narayama number N(n, k) of k-cells. Thecohomology group Hk(G(An)) is free abelian of rank given by “ballot numbers” b(n, n−2k).The ring structure is also easy to describe.

These spaces and groups have many convenient properties. For example, the groupsG(An) form both a directed system and an inverse system since X(An) is a retract ofX(An+1). Also X(An) × X(Am) is a retract of X(An+m+1). The filtration of X(An) bysubcomplexes X(Am) for m < n has a refinement by subcomplexes which are all K(π, 1)’s.These properties help to determine the ring structure of the cohomology of G(An).

Outline of paper:

In Section 1 we construct the spherical semi− invariant picture L(Q) for any val-

ued quiver Q of finite type. More precisely, we construct L(Λ) for any finite dimensionalhereditary algebra and show that it depends only on the underlying valued quiver of Λ.

In Section 2 we define the picture group G(Q) and show that the picture L(Q) is a

picture for the group G(Q) with its defining presentation. The picture group is the universalgroup with this property.

In Section 3 we construct the picture space X(Q) . This is a CW-complex with one

k-cell for every set of k hom-orthogonal roots α1, · · · , αk ∈ Φ+(Q).

In Section 4 we compute the integral cohomology of the space X(An) . It is generated

by indecomposable elements with square zero. There are (n − 2k + 2)Ck indecomposableelements of degree k for 2k − 1 ≤ n where Ck is the kth Catalan number.

In Section 5 we outline a proof that X(An) is a K(π, 1) and that, therefore, the cal-

culation in Section 4 computes the cohomology of these groups. The proof uses a filtrationof X(An) by subcomplexes Y0 ( Y1 ( · · · ( Yn where Y0 = X(An−1), Yn = X(An) andeach Ym is a K(π, 1). We give some examples to see what these spaces and groups are.Missing details in this proof can be found in another paper [10] which gives a categoricalapproach to the construction of the picture space X(Q) for any modulated quiver of finitetype and extended it to finite convex subsets of the Auslander-Reiten quiver of a hereditaryalgebra of possibly infinite type. The first author has written another topological approachto the same special case of An with straight orientation in [5]. Since these other papers givetwo independent detailed and rigorous proofs of the main theorem of Section 5, we presentonly an outline of our original idea which was the inspiration for these other papers. Thisshows that the cohomology of the space X(An) is the same as the cohomology of the groupG(An).

Finally, in Section 6 we determine the cup product structure of the cohomology ring

of the picture group G(An).3

1. Spherical semi-invariant picture L(Q)

We construct the spherical semi-invariant picture L(Λ) for any valued quiver Q of finitetype. This is a codimension one subcomplex of the n − 1 sphere with suitable simplicialdecomposition where n is the number of vertices of Q. This is defined in terms of therepresentations of a hereditary algebra Λ of finite type. However, it depends only on theunderlying valued quiver Q of Λ. So, we often denote it by L(Q) instead of L(Λ).

1.1. Notation. Suppose that Λ is a finite dimensional hereditary algebra over a field K.We assume that Λ has finite representation type. Here is a summary of well-known factsand our notation. See [7], [8] for more details. Also [3] is the classical reference for valuedquivers.

So, the quiver of Λ is a valued quiver which is a disjoint union of Dynkin quivers. Recallthat the quiver Q is a directed graph with one vertex for every (isomorphism class of)simple module Si, i = 1, · · · , n with one arrow i → j if Ext(Si, Sj) 6= 0. The quiver Q hasvaluation given by fi = dimK Fi where Fi = End(Si) at each vertex i and edge valuation(dij , dji) on any arrow i → j if dij = dimFj Ext(Si, Sj) and dji = dimFi Ext(Si, Sj) so thatdijfj = djifi.

Given any Λ-module M , the dimension vector dimM is the vector in Nn whose ith coor-dinate is dimFi HomΛ(Pi,M) where Pi is the projective cover of Si with endomorphism ringcanonically identified with Fi = End(Si). A virtual representation is a homomorphism be-tween projective modules p : P → P ′ (thought of as the presentation of a module M) withmorphisms given by homotopy classes of chain maps. Up to isomorphism, the indecom-posable virtual representations are presentations of indecomposable modules and shiftedindecomposable projective modules Pi[1] := (Pi → 0). The dimension vector of a virtualrepresentation P → P ′ is defined to be dimP ′ − dimP . Then the dimension vector of thepresentation of any module is equal to the dimension vector of the module.

The Euler matrix E is the n× n integer matrix with entries

Eij = dimK Hom(Si, Sj)− dimK Ext(Si, Sj)

Then, the Euler-Ringel form 〈·, ·〉 : Zn × Zn → Z, defined by 〈v, w〉 = vtEw, satisfies

〈dimM, dimN〉 = dimK Hom(M,N)− dimK Ext(M,N).

Let Φ+(Q) be the set of positive roots of Q. These are the dimension vectors of the in-decomposable Λ-modules. If πi = dimPi then we call −πi the negative projective roots.These are the dimension vectors of the virtual representations Pi[1] = (Pi → 0). We saythat β is an almost positive root if it is either a positive root or a negative projective root.Using the notation hom(α, β) = dimK Hom(Mα,Mβ) and ext(α, β) = dimK Ext(Mα,Mβ),we say that α, β are hom-orthogonal if hom(α, β) = 0 = hom(β, α) and ext-orthogonal ifext(α, β) = 0 = ext(β, α). We use the notation |P [1]| = P and | −β| = β. So, |Mβ| = M|β|.

The cluster complex Σ(Λ) of Λ is defined to be the n−1 dimensional simplicial complex

whose vertices are the almost projective roots of Q. The k-simplices of Σ(Λ) are k+1 tuplesof pairwise ext-orthogonal almost positive roots. Since Λ has finite type, Σ(Λ) is a finitecomplex whose geometric realization is the sphere Sn−1.

One definition of the picture L(Λ) ⊂ Sn−1 is that it is the geometric realization of then − 2 skeleton of Σ(Λ). This is a “picture” for the group G(Λ) as defined below. In thisdefinition we use normal orientations on n−3 dimensional (codimension 2 in Sn−1) simplices

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ρ of L. Such a normal orientation induces a cyclic ordering of the n−2 simplices of L whichcontain ρ. We use k = n− 1 in the definition.

Definition 1.1. Suppose that G is a group given by generators and relations: G = 〈X |Y〉where each y ∈ Y is a word in X ∪ X−1 and k ≥ 2. Then a k-dimensional picture for Gis defined to be a k − 1 dimensional subcomplex L of a triangulated k sphere Sk togetherwith an orientation of the normal bundle in Sk of every k− 1 simplex and k− 2 simplex ofL and labels x(σ) ∈ X for each k − 1 simplex σ in L so that

(1) There exists a locally constant function g : Sk\L → G so that, for every k − 1simplex σ of L, g(τ ′) = g(τ)x(σ) when τ ′ is on the positive side of σ.

(2) For every k − 2 simplex ρ of L, the k − 1 simplices σi of L which contain ρ, can benumbered in agreement with the cyclic ordering given by the normal orientation ofρ in the sphere, so that∏

x(σi)εi ∈ Y ∪ {xx−1 |x ∈ X}

where εi = +1 if the positive side of σi faces σi+1 and εi = −1 if not. We use thenotation y(ρ) =

∏x(σi)

εi .

Let X0 ⊆ X be the set of all labels x(σ) for all k − 1 simplices σ of L and let Y0 ⊆ Y bethe set of all elements of Y which occur as labels y(ρ) for some k− 2 simplex ρ of L. Thenwe call G0 = 〈X0 | Y0〉 the group determined by L with its normal orientation and systemof labels x(σ).

Remark 1.2. For some applications ([6],[9]), it is useful to specify the starting point of theword y(ρ) in (2). Such a starting point is given by one of the k-dimensional simplicesof Sk which contains ρ. This is called the base point direction. Without this choice, therelation y(ρ) will only be well-defined up to cyclic orientation. Without the orientationof ρ, the relation y(ρ) would only be well-defined up to inversion. But, given the labelsx(σ) and normal orientations for the k − 1 simplices of L, the group G0 is still uniquelydetermined (thereby justifying the terminology). If L is a picture for G then there is acanonical homomorphism G0 → G induced by the inclusion X0 ↪→ X .

Let L be a one-dimensional subcomplex of any triangulation of the 2-sphere S2 which,when considered as a graph, contains no leaves. (Every vertex of L is adjacent to at leasttwo edges.) Choose a normal orientation of each edge and vertex in L. Let x : L1 � X byany surjective mapping of the set of edges of L to any finite set X . For each vertex v ∈ L0,let y(v) be the product of the labels x(ei)

εi on the edges adjacent to v starting with anyedge and going either clockwise or counterclockwise according to the orientation of v, withexponent εi = ±1 according to the orientation of ei. If the words y(v) are cyclically reducedand aperiodic then L is a picture for the group G0 = 〈x(e), e ∈ L1 | y(v), v ∈ L0〉 and G0 isthe group determined by L ⊂ S2. Figure 1 gives an example.

More generally we have the following.

Proposition 1.3. If L is any codimension one subcomplex of any triangulated k-sphere Sk

with normal orientations on k − 1 and k − 2 simplices and labels in a set X on the k − 1simplices, there is group G0, unique up to isomorphism, determined by L. Furthermore, ifL is a picture for another group G, there is a unique homomorphism G0 → G which takesthe generators of G0 to the corresponding generators of G.

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x

x

y x

z

z

y

z

y

Figure 1. This graph with indicated labels and normal orientation, givenby placing the labels on the positive side of each edge and taking positiveorientation at each vertex, determines the group

G0 =⟨x, y, z |xyz−1y−1, yzx−1z−1, zxy−1x−1

⟩which is the fundamental group of the complement of the trefoil knot.

The proof of Proposition 1.3 is given by Remark 1.2 above.We will use semi-invariants to provide a system of labels and normal orientations for

L(Λ) ⊆ Sn−1. This will simultaneously define a group G(Λ) and show that L(Λ) is an n−1dimensional picture for this group.

1.2. Semi-invariants. For every positive root β ∈ Φ+(Q), let Mβ be the unique indecom-posable Λ-module with dimension vector β. Then the (integral) support of β is defined tobe the set of all dimension vectors of all virtual representations V = (p : V1 → V0) so thatHom(V,Mβ) = 0 = Ext(V,Mβ) or, equivalently,

Hom(p,Mβ) : Hom(V0,Mβ) ∼= Hom(V1,Mβ)

is an isomorphism. The determinant of this linear map is a semi-invariant of weight β. (See[8].)

The real support of β, denoted D(β), is defined to be the closure in Rn of the set of allvectors in Qn an integer multiple of which lies in the integral support of β. The virtualstability theorem [8] states that

(1.1) D(β) = {v ∈ Rn | 〈v, β〉 = 0 and⟨v, β′

⟩≤ 0 ∀β′ ⊆ β}

where β′ ⊆ β means that Mβ contains a submodule isomorphic to Mβ′ . Formula (1.1)implies in particular that D(β) depends only on the valued quiver Q.

Note that D(β) is the closure of a convex open subset of the hyperplane

H(β) = {v ∈ Rn | 〈v, β〉 = 0}.

This hyperplane has a normal orientation. The positive side is given by

H+(β) = {v ∈ Rn | 〈v, β〉 ≥ 0}.

Thus, each D(β) is a normally oriented codimension one subspace of Rn.

Lemma 1.4. For every cluster T1, · · · , Tn in the cluster category of Λ there are uniqueroots γ1, · · · , γn ∈ Φ(Q) so that

〈dimTi, γj〉 = δij dimK End(Ti).

Furthermore, dimTi lies in D(|γj |) for all i 6= j.

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Proof. By an observation of Schofield, the modules |Ti| can be ordered to form an exceptionalsequence. (Put the shifted projectives last.) For each i there is a unique exceptional module

Mi so that (Mi, |T1|, |T2|, · · · , |Ti|, · · · , |Tn|) is an exceptional sequence. This implies thatHom(|Tj |,Mi) = 0 = Ext(|Tj |,Mi) for all j 6= i. We claim that

〈dimTi,dimMi〉 = ±dimK End(Ti)

Then γi = ±dimMi with the same sign as above is the solution of the equation.The claim follows from that fact that dimMi is equal to ±dimTi modulo the Z-span of

dimT1, · · · , dimTi−1. This follows easily from the fact that braid mutation of exceptionalsequences preserves the Z-span of the dimension vectors of the modules. �

Since D(|γi|) is convex it contains all nonnegative linear combinations of dimTj , j 6= i.

Theorem 1.5. Let D(Λ) ⊂ Rn be the union

D(Λ) =⋃

β∈Φ+(Q)

D(β)

Then L(Λ) = D(Λ) ∩ Sn−1 where Sn−1 is the unit sphere in Rn.

Proof. By the lemma, D(Λ) contains the k − 2 skeleton of Σ(Λ) and therefore, L(Λ) is asubset of D(Λ) ∩ Sn−1. Conversely, suppose that v ∈ D(Λ) ∩ Sn−1 and v /∈ L(Λ). SinceL(Λ) is a closed set and every point in D(Λ) is a limit of rational points, there is a rationalvector w ∈ D(Λ) so that w/||w|| is not in L(Λ). By definition of L(Λ) this implies that somepositive scalar multiple of w has the form mw =

∑ai dimTi for some cluster T1, · · · , Tn

where ai are positive integers. But,⊕T aii is the generic module of dimension vector mw.

So,∑ai dimTi ∈ D(β) implies dimTi ∈ D(β) for all i. But this is impossible since dimTi

are linearly independent and D(β) is a subset of a hyperplane through the origin. �

Since the subsets D(β) ∩ Sn−1 ⊆ L(Λ) are normally oriented and labeled with positiveroots β ∈ Φ+(Q) and depend only on Q, we get the following.

Corollary 1.6. L(Λ) ⊂ Sn−1 is an n − 1 dimensional picture for a group G(Λ) withgenerators x(β) for β ∈ Φ+(Q). Furthermore, L(Λ) together with its normal orientationand system of labels depends only on the underlying valued quiver Q of Λ.

Because of this we write L(Λ) = L(Q).Summary: Section 1 constructs the spherical semi-invariant picture L(Λ) = D(Λ)∩Sn−1

where D(Λ) is the union of domains D(β) of virtual semi-invariants of weight β. These setsare normally oriented and labelled β. So, they determine a group G(Q) which is universalwith the property that L(Λ) is a picture for G(Q).

2. Picture group G(Q)

We define the picture group G(Q) to be the group determined by the subcomplexL(Q) ⊆ Sn−1. By Proposition 1.3, L(Q) is a picture for the group G(Q) with its definingpresentation.

The generators of the picture group are, by definition, the labels of the walls in L(Q).Since we sometimes think of L(Q) as an n− 2 dimensional subcomplex of Sn−1 and some-times as an n − 1 dimensional subcomplex of Rn, we will refer to the codimension insteadof the dimension of its pieces. The walls are the codimension one sets. Since these walls

are D(β) for all positive roots β of Q, we have a generator x(β) for each β ∈ Φ+(Q).

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We now consider a codimension p ≥ 2 simplex ρ of L(Q) = L(Λ). In this section weare only interested in the case p = 2, but the general case is needed for the next section.By definition the vertices of ρ form a partial cluster T1, · · · , Tn−p in the cluster category ofΛ. By Lemma 1.4, the codimension 1 simplices of L(Q) which contain ρ are contained inD(βj), j > n− p, for some extension T1, · · · , Tn of the partial cluster to a full cluster whereβj = |γj | ∈ Φ+(Q) in the notation of the lemma. Furthermore, the condition Ti ∈ D(βj)for i = 1, 2, · · · , n − p is equivalent to the condition that Mβj lies in the right hom-ext

perpendicular category |T |⊥ of the underlying module |T | of T = T1 ⊕ · · · ⊕ Tn−p. Since T

has n − p components, |T |⊥ is wide subcategory of mod-Λ of rank p which is the modulecategory of a hereditary algebra with p nonisomorphic simple objects.

Let Mα1 , · · · ,Mαp be the simple objects of the wide subcategory |T |⊥. Since Λ has finitetype, we can number the roots so that ext(αi, αj) = 0, or equivalently, 〈αi, αj〉 = 0 for i < j.

All other objects of |T |⊥ have the form Mγ where γ =∑rjαj , rj ≥ 0, is a nonnegative

integer linear combination of the αj .

Lemma 2.1. For any positive root γ ∈ Φ+(Q), the following are equivalent.

(1) There is an n− 1 simplex σ in L(Q) so that ρ ⊂ σ ⊆ D(γ).(2) The indecomposable module Mγ lies in |T |⊥.

(3) The modules |Ti|, i = 1, · · · , n− p, lie in ⊥Mγ.(4) γ has the form γ =

∑riαi where ri ≥ 0.

We denote by Φ+(α∗) the set of all γ ∈ Φ+(Q) satisfying these equivalent conditions.

Proof. As explained above, Lemma 1.4 implies that (1) and (2) are equivalent. (2) and(3), which are clearly equivalent, imply (4) since Mαi are the unique simple objects in thecategory |T |⊥. Conversely, suppose that γ =

∑rjαj ∈ Φ+(Q). Then

〈dimTi, γ〉 =∑

rj 〈dimTi, αj〉 = 0

Since Λ has finite representation type, this implies that Mγ ∈ |T |⊥. So, all four statementare equivalent. �

Lemma 2.2. The interior of the simplex ρ with vertices dimTi lies in the interior of eachD(αj) but it lies on the boundary of D(γ) for any γ =

∑rjαj which is not one of the αj.

Proof. Take any fixed v ∈ int ρ. Then, v lies in D(γ) if and only if ρ ⊆ D(γ). This happensif and only if γ =

∑rjαj for some rj ≥ 0. It follows that v is not contained in D(β) if β

is a subroot of any αj . By the virtual stability theorem (1.1), this implies that 〈v, β〉 < 0.Since this is an open condition, we have 〈w, β〉 < 0 for all w in some neighborhood of v.Therefore, v lies in the interior of each D(αj).

Any nontrivial linear combination γ =∑rjαj , will contain some αj as a subroot. Also,

αj , γ will be linearly independent. So, the hyperplanes H(αj), H(γ) intersect transverselyalong a codimension 2 subspace which contains the simplex ρ. Since 〈v, αj〉 ≤ 0 for allv ∈ D(γ), the set D(γ) is restricted to the negative side of H(αj). So, ρ lies on theboundary of D(γ) as claimed. �

Finally, we need to characterize which pairs of roots α, β ∈ Φ+(Λ) arise in the way thatwe described.

Lemma 2.3. Suppose that α1, · · · , αp ∈ Φ+(Λ). Then Mαj are the simple objects of a widesubcategory of mod-Λ of rank p if and only if they are pairwise hom-orthogonal.

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Proof. We prove only the sufficiency of this condition is clearly necessary. Since Λ has finitetype, we can number the roots so tha ext(αi, αj) = 0 for i < j. Then, reversing the order

gives an exceptional sequence M = (Mαp , · · · ,Mα1) making A = (⊥M)⊥ into a rank p widesubcategory with complete exceptional sequence M . Since the αj are hom-orthogonal, Mαj

are the simple objects of A. �

We will use the notation Ab(α∗) = (⊥M)⊥ for this wide subcategory. By Lemma 2.1,Φ+(α∗) is the set of dimension vectors of indecomposable objects of Ab(α∗). We call Ab(α∗)the wide subcategory spanned by α∗ since “generated” is not the right word.

Theorem 2.4. If Q is a valued Dynkin quiver, the picture group G(Q) determined by thespherical semi-invariant picture L(Q) has the following presentation.

(1) G(Q) has one generator x(β) for every positive root β ∈ Φ+(Q).(2) For each pair (α, β) of hom-orthogonal roots in Φ+(Q) so that ext(α, β) = 0, we

have the relation:

(2.1) x(α)x(β) =∏

x(riα+ siβi)

where the product is over all positive roots of the form riα+ siβ in increasing orderof the ratio ri/si (going from 0/1 to 1/0).

Proof. Each codimension one face simplex of L(Q) lies in D(β) for some positive roots βand is labeled x(β). By Lemma 2.2, the relation which occurs around a codimension twosimplex ρ of L(Q) is a word in x(rα + sβ) in which the letters x(α), x(β) occur twice andthe other letters occur once. In the semi-simple case where Mα,Mβ do not extend eachother, the only D(γ) containing ρ are D(α), D(β) which meet transversely with ρ in theirintersection. So, the relation around ρ is x(α)x(β) = x(β)x(α) in this case.

If Ext(Mβ,Mα) 6= 0 then there are extensions Mγ where γ = rα + sβ. (Example 2.5below gives a case by case description.) Figure 2 shows where D(rα + sβ) occur. Theyare oriented counterclockwise as shown in the figure and the slope of the positive normaldirection is proportional to r/s. Therefore, the sets D(rα+sβ) are in cyclic order accordingto this slope and we get the relation (2.1) above. �

D(β)

D(β)

D(α)D(α)

D(rα+ sβ)

ρ

Figure 2. Image of L(Q) under the projection Rn → R2 given by v 7→(〈v, β〉 , 〈v, α〉). By definition, D(α), D(β) map to the x and y axes. Inthe non-semisimple case, 〈α, β〉 = 0 and 〈β, α〉 < 0, all sets D(rα + sβ) forr, s > 0 map to the fourth quadrant as shown.

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Example 2.5. There are only six types of relations (2.1) which occur in the presentationgiven in the theorem. This is because the wide category (⊥M)⊥ is equivalent to the modulecategory of a hereditary algebra of finite type with two vertices. And there are only fourpossibilities as listed below. (But Cases (3) and (4) have two subcases depending on whetherthe arrow points towards the short root or the long root. So, the total is six.)

(1) A1 × A1. This corresponds to the case when the modules Mα,Mβ do not extendeach other and the wide category that they generate is semi-simple. So, Φ+(α, β) ={α, β} and the relation is:

x(α)x(β) = x(β)x(α).

(2) A2. Here Ext(Mβ,Mα) is one dimensional over both Fβ = End(Mβ) and Fα =End(Mα). The wide category has 3 indecomposable objects forming an exact se-quence Mα →Mα+β →Mβ and G(Q) has relation:

x(α)x(β) = x(β)x(α+ β)x(α).

(3) B2∼= C2. In this case, either Ext(Mβ,Mα) is 1-dimensional over Fβ and 2-dimensional

over Fα or vise versa. In the first case, where β is the long root, we have Φ+(α, β) ={α, β, α+ β, 2α+ β} and the relation is

x(α)x(β) = x(β)x(α+ β)x(2α+ β)x(α).

(4) G2. Here Ext(Mβ,Mα) is 1-dimensional over Fβ and 3-dimensional over Fα or viseversa. There are six positive roots giving the relation:

x(α)x(β) = x(β)x(α+ β)x(3α+ 2β)x(2α+ β)x(3α+ β)x(α).

In all cases there are irreducible morphism between the corresponding modules in the op-posite order than how they appear in the relations. For example, in Case (4) there areirreducible morphisms

Mα →M3α+β →M2α+β →M3α+2β →Mα+β →Mβ.

If we compare these relations with the Chevalley relations for the generators of themaximal unipotent subgroup UQ of the algebraic group of the underlying Dynkin diagramof Q, we see that there is an epimorphism G(Q) � UQ(Z). (Send x(β) to εβ(1) in thenotation of [11].)

Summary: In Section 2 (Theorem 2.4) we gave a presentation of the picture G(Q) whichis determined by the labeled picture L(Q) constructed in Section 1.

3. Picture space X(Q)

In Section 3 we will construct the picture space X(Λ) assuming that Λ is a hereditaryalgebra of finite representation type. This will be a finite CW-complex together with asystem of closed codimension-one subsets J(β) ⊂ X(Q) for all β ∈ Φ+(Q). Since X(Λ) willdepend only on the underlying valued quiver Q, we will write X(Q) = X(Λ).

3.1. Local properties of D(β). The construction of the space X(Q) depends on the localproperty of the sets D(β) as given in Proposition 3.2 below for β in a wide subcategoryAb(α∗) spanned by a pairwise hom-orthogonal set of roots α∗. Roughly speaking, it saysthat the intersection pattern of these sets depends only on the valued quiver of Ab(α∗). Thisis the quiver with one vertex for each αj with valuation fj = hom(αj , αj) and an arrowi→ j whenever ext(αi, αj) 6= 0 with valuation (dij , dji) so that dijfj = djifi = ext(αi, αj).

10

Let Q(α∗) denote this valued quiver. Then Q(α∗) depends only on the numbers 〈αi, αj〉since this is equal to fj when i = j and −ext(αi, αj) when i 6= j.

Let α∗ = {α1, · · · , αp} ⊆ Φ+(Q) be any set of hom-orthogonal roots and let T =

{T1, · · · , Tn−p} be any partial cluster in the cluster category of Λ so that Ab(α∗) = |T |⊥.

For example, take the components of any tilting object of ⊥Ab(α∗). Let Rα∗ denote thep-plane spanned by the vectors α1, · · · , αp. Then, it follows from Lemma 1.4 that the spanof the roots dimTi is equal to

⊥Rα∗ = {w ∈ Rn | 〈w,αj〉 = 0 for all j}.

Let πα∗ : Rn → Rα∗ be the projection along ⊥Rα∗. Then, for every x ∈ Rn, πα∗(x) ∈ Rα∗is uniquely determined by the condition 〈x, αj〉 = 〈πα∗(x), αj〉 for all j.

Lemma 3.1. For β∗ a hom-orthogonal set of roots in Φ+(α∗) we have πβ∗x = πβ∗πα∗(x).I.e., πβ∗ : Rn → Rβ∗ factors uniquely through πα∗ : Rn → Rα∗.

Proof. Since Rβ∗ ⊆ Rα∗, ⊥Rα∗ = kerπα∗ ⊆ ⊥Rβ∗ = kerπβ∗ . The lemma follows. �

Let ρ be the n−p−1 simplex in the cluster complex Σ(Λ) spanned by the almost positiveroots dimTi. Then ρ ⊂ ⊥Rα∗. Let v be any point in the interior of ρ. (I.e., v =

∑vi dimTi

where vi > 0 for all i.)

Proposition 3.2. For each β ∈ Φ+(Q) let D(β, ρ) be the set of all vectors x ∈ Rn so that,for all sufficiently small ε > 0, v + εx ∈ D(β). Then D(β, ρ) is nonempty if and only ifβ ∈ Φ+(α∗) and D(β, ρ) is the set of all x ∈ Rn satisfying the condition:

(3.1) 〈x, β〉 = 0 and⟨x, β′

⟩≤ 0 for all subroots β′ of β which lie in Φ+(α∗).

Remark 3.3. Let Dα∗(β) ⊂ Rα∗ be the set of all x ∈ Rα∗ satisfying (3.1). This propositionsays that D(β, ρ) is the inverse image of Dα∗(β) under πα∗ : Rn → Rα∗.

Suppose that τ is a simplex containing ρ and let Mβj be the simple objects in the rightperpendicular category of |T ′| where T ′ is the direct sum of all vertices Ti of τ . ThenΦ+(β∗) ⊆ Φ+(α∗). If β ∈ Φ+(β∗) then D(β, τ) is a nonempty subset of D(β, ρ) and D(β, τ)is the inverse image under πβ∗ : Rn → Rβ∗ of the subset Dβ∗(β) ⊆ Rβ∗. By Lemma 3.1,πβ∗ factors through Rα∗. So, Dα∗(β) ⊆ Rα∗ contains the inverse image of Dβ∗(β) under theinduced map Rα∗ → Rβ∗ .

Proof. Since D(β) is a closed set, the condition v + εx ∈ D(β) for small ε implies thatv ∈ D(β). This implies that ρ ⊂ D(β) which holds if and only if β ∈ Φ+(α∗) by Lemma2.1. So, D(β, ρ) is nonempty only in this case.

We show that the condition (3.1) is necessary. If v+ εx ∈ D(β) then 〈v + εx, β〉 = 0 and〈v + εx, β′〉 ≤ 0 for all β′ ⊆ β. Since 〈v, β〉 = 0 and ε > 0, these conditions imply (3.1).Conversely, suppose that (3.1) holds. Then 〈v + εx, β〉 = 〈v, β〉 + ε 〈x, β〉 = 0. Let γ ⊂ βbe any subroot. If γ ∈ Φ+(α∗) then (3.1) implies that 〈v + εx, γ〉 ≤ 0 for all ε > 0. Ifγ /∈ Φ+(α∗) then we know that v /∈ D(γ). Therefore 〈v, γ〉 < 0. (Otherwise, 〈v, γ〉 = 0 and〈v, γ′〉 ≤ 0 for all γ′ ⊆ γ ⊂ β making v ∈ D(γ).) So, 〈v + εx, γ〉 < 0 for sufficiently small εwhich implies that v + εx lies in D(β) for sufficiently small positive ε as claimed. �

Recall that the cluster complex Σ(Q) is a simplicial complex whose geometric realizationis |Σ(Q)| ∼= Sn−1. Since Sn−1 is a manifold, the dual cell decomposition is an n − 1dimensional CW complex. We attach a single n cell to this dual cell complex to get an ndimensional CW complex which we denote by E(Q). Then |E(Q)| ∼= Dn. Proposition 3.2

11

gives us an equivalence between certain p-cells in this cell decomposition. By identifyingthese equivalent cells we will obtain the picture space X(Q).

We recall that the dual cell E(ρ) to the n− p− 1 simplex ρ in |Σ(Q)| is the p-cell whichis the union of all simplices τ in the first barycentric subdivision of |Σ(Q)| so that τ ∩ ρ isthe barycenter of ρ. This implies that the other vertices of τ are barycenters of simplices σwhich contain ρ. For every β ∈ Φ+(Q) let J(β, ρ) be the subcomplex of E(ρ) consisting ofall simplices τ whose vertices are barycenters of simplicies σi which are contained in D(β).J(β, ρ) is nonempty if and only if β ∈ Φ+(α∗).

We use the general fact that E(ρ) is simplicially isomorphic to the cone on the firstbarycentric subdivision of the link of ρ in Σ(Q). This is the simplicial complex whosevertices are all almost positive roots γ which are ext-orthogonal to dimT1, · · · ,dimTn−p.The vertices γi span a simplex in Lk(ρ) if and only if they are ext-orthogonal.

Corollary 3.4. There is a simplicial isomorphism ϕρ : Lk(ρ) ∼= Σ(Q(α∗)) which is uniquelydetermined by the property that, for every vertex γ of Lk(ρ), ϕρ(γ) is a positive scalarmultiple of πα∗(γ) ∈ Rα∗.

Remark 3.5. (a) By Proposition 3.2 this implies that a vertex γ of Lk(ρ) lies in D(β) forsome β ∈ Φ+(α∗) if and only if ϕρ(γ) lies in Dα∗(β).

(b) We also obtain as a consequence the following naturality condition on ϕρ. Supposethat τ is a simplex in Lk(ρ) and σ = ρ ∗ τ is the smallest simplex in Σ(Q) containing ρand τ . Then Lk(σ) ⊆ Lk(ρ). Take ϕσ : Lk(σ) ∼= Σ(Q(β∗)) where β∗ gives the simpleobjects in |σ|⊥. Let τ ′ = ϕρ(τ) and let Lk′(τ ′) be the link of τ ′ in Σ(Q(α∗)). Then, β∗ alsogives the simple objects in the right perpendicular category of |τ ′| in Ab(α∗). So, we gettwo isomorphisms Lk(σ) ∼= Σ(Q(β∗)). We need to know that they agree. Equivalently, thefollowing diagram commutes.

Lk(σ)

ϕρ|Lk(τ)��

⊂ //

ϕτ

yy

Lk(ρ)

��Σ(Q(β∗)) Lk′(σ)

ϕτ ′oo ⊂ // Σ(Q(α∗))

It suffices to show that the triangle commutes. But this follows from Corollary 3.4 abovesince each vertex γ of Lk(σ) maps by ϕτ to the unique vertex of Σ(Q(β∗)) which is propor-tional to πβ∗(γ) which comes from πα∗(γ) ∝ ϕρ(γ) by Remark 3.3.

Proof. By Lemma 1.4, each vertex γ in Lk(ρ) is contained in D(β) for p− 1 linearly inde-pendent vectors β ∈ Φ+(α∗). This implies that v + ε(γ − v) lies in each of these D(β) forall small ε > 0 where v ∈ intρ. So, γ − v ∈ D(β, ρ) = π−1

α∗Dα∗(β). Since πα∗(v) = 0, thisimplies that πα∗(γ) lies in each Dα∗(β). So, it is a scalar multiple of an almost positive rootin Φ(α∗) which we define to be ϕρ(γ).

To see that ϕρ takes simplices to simplices, take a maximal simplex in Lk(ρ) spannedby p vertices γ1, · · · , γp. Then, for sufficiently small εj > 0, we have that v +

∑εj(γj − v)

does not lie in D(β) for any β since it lies in the interior of a top dimensional simplex ofthe cluster complex. This condition characterizes which sets of vertices in Lk(ρ) form asimplex.

By the proposition, this implies that, when rj > 0,∑rjϕρ(γj) does not lie in Dα∗(β)

for any β ∈ Φ+(α∗). This is equivalent to the condition that ϕρ(γj) are ext-orthogonal andtherefore form a simplex in Σ(Q(α∗)). So, ϕρ is a simplicial isomorphism. �

12

Corollary 3.6. Let ρ, ρ′ be two n−p−1 simplices spanned by the dimension vectors of twopartial clusters T = {T1, · · · , Tn−p} and T ′ = {T ′1, · · · , T ′n−p} in the cluster category of Λ

so that Ab(α∗) = |T |⊥ = |T ′|⊥. Then there is a simplicial isomorphism ψρ : E(ρ) ∼= E(ρ′)which sends J(β, ρ) onto J(β, ρ′). Furthermore, if ρ ⊆ σ = ρ ∗ τ so that E(σ) ⊂ E(ρ), thenthe isomorphism ψρ restricts to the isomorphism ψσ : E(σ) ∼= E(σ′) where σ′ = ρ′ ∗ τ . Wealso note that J(β, σ) = J(β, ρ) ∩ E(σ).

Proof. We use the general fact that E(ρ) is the cone on the first barycentric subdivisionof the link Lk(ρ) of ρ in Σ(Q). By Corollary 3.4, Lk(ρ), Lk(ρ′) are both isomorphic toΣ(Q(α∗)). So, Lk(ρ) ∼= Lk(ρ′) and, therefore, E(ρ) ∼= E(ρ′). We refer to the elements ofE(ρ) corresponding to vertices of Lk(ρ) as the corners of the cell E(ρ). (The vertex γ ofLk(ρ) corresponds to the barycenter of the n− p simplex ρ ∗ γ.)

By Remark 3.5(a), the set J(β, ρ) ⊆ E(ρ) is the cone on the inverse image of thesubsets Dα∗(β) ⊆ Σ(Q(α∗)) under the isomorphism ∂E(ρ) ∼= Lk(ρ) ∼= Σ(Q(α∗)). So,ψρ : E(ρ)→ E(ρ′) must send J(β, ρ) to J(β, ρ′).

By Remark 3.5(b), the subset Lk(ρ ∗ τ) ⊆ Lk(ρ) maps to Lk(ρ′ ∗ τ) under ϕ−1ρ′ ϕρ and

the induced map is equal to ϕ−1ρ′∗τϕρ∗τ . The two maps agree on where they send each vertex

of Lk(ρ ∗ τ). Therefore they agree on where they send each corner of E(ρ ∗ τ) under themap ψρ∗τ . So, ψρ∗τ agrees with ψρ. �

3.2. Construction of the picture space.

Definition 3.7. The picture space X(Q) is defined to be CW complex obtained from E(Q)by identifying p cells E(ρ) ∼= E(ρ′) using the simplicial isomorphisms given by the corollaryabove. The compatibility of the map ψρ with ψσ that we just proved implies that theidentifications on the p-cells agrees with the identifications on lower cells. So, X(Q) is awell defined CW complex constructed one cell at a time by induction on dimension of cells.

Theorem 3.8. X(Q) is an n-dimensional CW complex with one cell of dimension k forevery set α∗ = {α1, · · · , αk} of pairwise hom-orthogonal roots in Φ+(Q). Denote it ekα∗.

Then the closure of the cell ekα∗ is a subcomplex of X(Q) isomorphic to X(Q(α∗)). Inparticular, X(Q) is the closure of the single cell enε1,··· ,εn where εi are the simple root in

Φ+(Q). The cell epβ∗ is in the closure of ekα∗ if and only if β∗ ⊆ Φ+(α∗).

Proof. The cell ekα∗ is the one obtained by identifying all E(ρ) where Ab(α∗) = |ρ|⊥, equiva-

lently ρ is a cluster inside the cluster category of ⊥Ab(α∗). So, the cells of X(Q) are indexedby all such sets α∗ which are the spanning sets of wide subcategories Ab(α∗). When ρ ⊂ σthen σ = ρ∗ τ and E(σ) ⊂ E(ρ). So, |σ|⊥ = Ab(β∗) ⊆ Ab(α∗). And conversely (|σ|⊥ ⊆ |ρ|⊥iff ρ ⊆ σ). But E(ρ) is the cone on ∂E(ρ) ∼= sdLk(ρ) ∼= sdΣ(Q(α∗)) by Corollary 3.4. And,X(Q(α∗)) is obtained from sdΣ(Q(α∗)) by the same recipe as X(Q):

X(Q(α∗)) =∐τ

Eα∗(τ)/ ∼

where Eα∗(τ) is the cell in sdΣ(Q(α∗)) dual to τ . The subscript indicates that we areworking in the quiver Q(α∗). In the larger quiver, we have Eα∗(τ) = E(ρ ∗ τ). Thus thecells of X(Q(α∗)) correspond to those cells of X(Q) which are identified with E(ρ ∗ τ) forvarious τ . These are exactly the simplices which contain ρ and the cells are identified inthe same way in both cell complexes by Remark 3.5(b). �

13

We will examine in more detail the structure of the space X(Q) and J(β) ⊂ X(Q) onecell at a time by induction on dimension. We use the notation X(Q)k for the k-skeleton ofX(Q) and J(β)k−1 = J(β) ∩X(Q)k.

The cell complex X(Q) has a single 0-cell (vertex) e0. It have a 1-cell e1β for every

positive root β ∈ Φ+(Q). The endpoints of each 1-cell are attached to the unique 0-cell.This gives a 1-dimensional CW complex X(Q)1 whose fundamental group is the free groupwith generators x(β) where β ∈ Φ+(Q). The generator x(β) is represented by the cell e1

β.

Before attaching more cells to X(Q)1, we give a recursive description of the sets J(β)k

in terms of the attaching maps of the cells.

Proposition 3.9. The 0-dimensional subset J(β)0 ⊂ X(Q)1 consists of the center pointof the cell e1

β. Given J(β)k−1 ⊂ X(β)k for k ≥ 1 and attaching maps ηi : Sk = ∂Dk+1 →X(Q)k, the set J(β)k is the union of J(β)k−1 and certain subsets of each cell ek+1

i as

follows. For each k+1 cells, J(β)k∩ek+1i is the cone of the inverse image of J(β)k−1 under

the attaching map ηi : Sk → X(Q)k assuming the image of ηi meets J(β)k−1. Otherwise

J(β)k ∩ ek+1i is empty.

Continuing with the construction of X(Q), we take one 2-cell e2α,β for every unordered

pair of hom-orthogonal roots α, β. This 2-cell is attached to the 1-skeleton X(Q)1 usingthe relation (2.1) corresponding to the pair {α, β}. In Example 2.5, Case (1), this gives atorus S1 × S1 with J(α) = S1 × ∗ and J(β) = ∗ × S1. In Case (2) we get a torus with oneboundary component given by the 1-cell e1

α+β. Cases (3) and (4) also give closed subsetsof tori. To be more precise, we define each 2-cell to be a convex polygon with m+ 2 sideswhere m is the number of elements of Φ+(α, β) (m = 2, 3, 4, 6 in Cases (1), (2), (3), (4),respectively). The attaching map sends these m + 2 sides to the 1-cells corresponding tothe letters in the relation (2.1).

For all k, the cell complex X(Q) will have one k-cell ekα∗ for every unordered set of kpairwise hom-orthogonal roots α∗ = {α1, α2, · · · , αk}.

Example 3.10. Consider the quiver A2 : 1← 2.

(1) Σ(A2) is a pentagon with five vertices and 5 edges. These vertices are the almostpositive roots α, β, γ,−α,−β connected in a cycle. The simple roots are α, γ.

(2) The following figure represents the cone on the first barycentric subdivision of Σ(A2).Except for the cone point ∅, each point bρ is the barycenter of a simplex ρ in Σ(A2).The point bρ is labeled by the set of vertices of ρ.

{β, γ} γ {−α, γ}

β

{α, β} ∅ −α

α

{−β, α} −β {−α,−β}14

(3) The entire object is E(∅) and the sides are E(α), E(β), E(γ), E(−α), E(−β). Thevertices are E(ρi) where ρi are the 2-simplices of Σ(A2).

(4) Since |ρi|⊥ = 0, these vertices are all identified to one point e0 in X(A2). The 1-cellsE(α), E(−α) are identified since |α|⊥ = | − α|⊥ = Ab(γ) and E(β), E(−β) are alsoidentified since β⊥ = Ab(α). Finally, γ⊥ = Ab(β).

(5) J(α) is the cone on the two points β,−β, J(β) is the cone on the point γ and J(γ)is the cone on the points α,−α.

Summary: In Section 3 we constructed the picture space X(Q). It is a quotient spaceof the cone on the first barycentric subdivision of the cluster complex Σ(Q) under certainidentifications. The space C(sdΣ(Q)) is decomposed as a union of cells E(ρ) for all simplicesρ in Σ(Q) (plus one big cell E(∅) := C(sdΣ(Q))) and E(ρ) is identified with E(ρ′) if andonly if the right perpendicular categories of |ρ|, |ρ′| agree.

4. Homology of X(An)

In this section we compute the homology of X(An) with straight orientation. The firststeps in the computation work in general in the sense that the cellular chain complex ofX(Q) always has a weight filtration. In the case Q = An with straight orientation, thehomology of the associated graded complex is easy to compute and equal to the homologyof the actual complex by an easy argument. We will see that the integral homology ofX(An) has no torsion. So the cohomology is also free abelian with the same rank. The cupproduct structure will be determined at the end.

Recall that the k-cells of X(Q) are indexed by all sets of k pairwise hom-orthogonalpositive roots βi of Q. Let 〈β1, · · · , βk〉 ∈ Ck(Q,Z) denote the corresponding free generatorof the cellular chain complex C∗(Q,Z) of X(Q). We understand the order of the βi to begiven up to even permutation. Under an odd permutation, the sign changes. Thus:⟨

βσ(1), · · · , βσ(k)

⟩= sgn(σ) 〈β1, · · · , βk〉

This generator has degree k. We often call this generator (with either sign) a cell and denoteit by β∗. We define the weight of the cell 〈β1, · · · , βk〉 to be the vector

∑βi ∈ Nn. Given

two weights w,w′ we say that w ≤ w′ if wi ≤ w′i for i = 1, · · · , n and w < wi if w ≤ w′ andw 6= w′. Note that if w < w′ then w comes before w′ in lexicographic order.

Given β∗ = 〈β1, · · · , βk〉, recall that Ab(β∗) is the abelian category spanned by the rootsβi and Φ+(β∗) is the set of all positive roots which can be written as nonnegative integerlinear combinations of the roots βi.

For each α ∈ Φ+(β∗), recall that Dβ∗(α) is the set of all∑viβi ∈ Rβ∗ satisfying the

stability conditions (3.1). Then, by Theorem 1.5, the union of the Dβ∗(α) intersected with

the unit sphere Sk−1 in Rβ∗ is the spherical semi-invariant picture for the hereditary abeliancategory Ab(β∗).

Lemma 4.1. Let β∗ = 〈β1, · · · , βk〉 be a set of hom-orthogonal positive roots. Then there isa 1-1 correspondence between positive roots γ ∈ Φ+(β∗) and hom-orthogonal subsets α∗ =〈α1, · · · , αk−1〉 of Φ+(β∗). The correspondence is given by γ⊥ = span(α∗). Furthermore, γis in the interior of Dβ∗(αi) for all i. Finally, wt(α∗) ≥ wt(β∗) if and only if Mγ is not aprojective object in the abelian category Ab(β∗).

Proof. The formula γ⊥ = span(α∗) gives the 1-1 correspondence. Assume for simplicity ofnotation that k = n and βi are simple roots. Then wt(β∗) = (1, 1, · · · , 1) and wt(α∗) ≥

15

wt(β∗) iff and only is∑αj is sincere, i.e., there is no index i so that the ith coordinate of

each αj is zero. But, if this happens then the i-th projective root πi is left perpendicularto all αj which implies γ = πi. So, the last statement holds. The statement that γ lies inthe interior of each Dβ∗(αi) was already shown in Lemma 2.2. �

4.1. Weight filtration of C∗(Q;Z) in general.

Proposition 4.2. The boundary of β∗ = 〈β1, · · · , βk〉 in the cellular chain complex C∗(Q;Z)is the sum of terms α∗ = 〈α1, · · · , αk−1〉 listed below, with coefficient ε(β∗, α∗) = ±1.

(1) One of the roots αi is equal to the sum of two of the roots βj and the remaining α’sare equal to the remaining β’s. In this case, α∗, β∗ have the same weight.

(2) αi are hom-orthogonal roots in Φ+(β∗) and∑αi >

∑βj.

Furthermore, ε(β∗, α∗) is the sign of the change of basis matrix from the basis 〈β1, · · · , βk〉to the basis 〈α1, · · · , αk−1, γ〉, each ordered up to even permutation, where γ is the uniquepositive root so the α∗ = γ⊥.

Proof. (1) is the only way that the k − 1 positive roots can add up to∑βi. (2) is the

only way that∑αi >

∑βi. The only remaining cases are

⟨β1, · · · , βi, · · · , βk

⟩which are

right perpendicular to projective roots πi. But these terms occur twice as summands of dβ∗with opposite sign corresponding to the vertices πi and −πi in the spherical semi-invariantpicture for β∗. So, they cancel. The signs comes from the definition of induced orientationon the boundary of a disk. The plane Rα∗ plays the role of the tangent plane to the unitsphere in Rβ∗ at the vector γ. The induced orientation is ε(β∗, α∗). The two cancelling

terms have signs given by the bases⟨β1, · · · , βi, · · · , βk, πi

⟩and

⟨β1, · · · , βi, · · · , βk,−πi

⟩.

Which are opposite. �

Corollary 4.3. dβ∗ = 0 if and only if Ab(β∗) is semi-simple. Equivalently, the sum of anytwo of the roots βi is not a root.

We call β∗ a semi-simple cell or semi-simple set if these conditions hold.

Proof. Under this condition, there are no α∗ as described in the Proposition. �

Corollary 4.4. The chain complex C∗(Q;Z) is filtered by weight in the sense that theadditive subgroup generated by the cells of weight ≥ w form a subcomplex C∗(Q;Z)w. �

Definition 4.5. We will say that β∗ = 〈β1, · · · , βk〉 is a resolution of α∗ = 〈α1, · · · , α`〉and the α∗ is a degeneration of β∗ if k > ` and there is an epimorphism p : {1, · · · , k} �{1, · · · , `} so that each αj is the sum of all βi for i ∈ p−1(j). If β∗ has no degenerationswe call it semi-simple. (This is equivalent to Ab(β∗) being semi-simple.) If β∗ has noresolutions, we call it primitive.

Example 4.6. In type A5 with straight orientation, 〈β04, β45, β12, β23〉 is a resolution of〈β05, β13〉 since β05 = β04 +β45 and β13 = β12 +β23. The later is semi-simple and the formeris primitive. We use the notation

βij = εi+1 + εi+2 + · · ·+ εj

where εj is the jth simple root of the root system An.

Note that, for any β∗ which is not semi-simple, the degeneration of β∗ with minimaldegree is necessarily semi-simple.

16

4.2. Semi-simple categories in type A. Suppose now that Q = Q(An) with straightorientation:

1← 2← · · · ← n

The positive roots are labeled βij where 0 ≤ i < j ≤ n as described in the example above.

Lemma 4.7. Two roots βij , βk` are hom-orthogonal if and only if one of the following twoconditions holds.

(1) i < k < ` < j or k < i < j < `.(2) j ≤ k or ` ≤ i.

Furthermore, βij , βk` are hom-ext-orthogonal if and only if, in addition, equality cannot holdin (2). �

We say that the closed intervals [i, j], [k, `] are noncrossing if they are either disjoint orone is contained in the interior of the other. The lemma says that βij , βk` are hom-ext-orthogonal if and only if the intervals [i, j], [k, `] are noncrossing

Let α∗ = 〈α1, · · · , αk〉 be a semi-simple cell. Then each root αs is equal to βij for some0 ≤ i < j ≤ n. And the closed intervals [i, j] are pairwise noncrossing. We define the blocksof α∗ to be the maximal intervals Bs = βisjs , s = 1, · · · ,m among these α’s. We arrangethese so that

0 ≤ i1 < j1 < i2 < j2 < · · · < im < jm ≤ n .Then the weight w of β∗ has the property that wt 6= 0 if and only if t lies in one of theintervals [is + 1, js]. Furthermore, wt = 1 if t = is + 1 or t = js for some s. Thus the blocksBs are completely determined by the weight w of the semi-simple cell α∗.

We say that a weight w ∈ Nn is admissible if it is the weight of some hom-orthogonalset β∗. This includes w = 0 which is the weight of the empty hom-orthogonal set.

Lemma 4.8. A weight w ∈ Nn is admissible if and only if |wi−wi+1| ≤ 1 for all 0 ≤ i ≤ nwith the convention that w0 = 0 = wn+1.

Proof. The condition is clearly necessary. For example if wi+1 ≥ wi + 2 then any β∗ withweight w will have two objects βij and βik one of which is a subroot of the other and aretherefore not hom-orthogonal.

To see that the condition is sufficient, express w as the sum of the smallest possiblenumber of roots. This is the number of indices i with the property that wi < wi+1. This isalso the number of indices j so that wj > wj+1. If we view each i as an open parenthesis“(” and each j as a closed parenthesis “)” then we see that they are paired. Then thecollection of roots βij for such pairs i, j forms a hom-orthogonal set of roots with totalweight

∑βij = w. So, w is admissible. �

The pairing between i and j used in the above proof can be described as follows. Givenj, the corresponding i is the largest integer i < j so that wi < wj . Since the degree of thiscell 〈βij〉 is minimal, it is semi-simple. The following lemma shows that there are no othersemi-simple objects with the same weight.

Lemma 4.9. For every admissible weight w ∈ Nn there is a unique semi-simple set α∗ withweight w.

Proof. As observed above, the blocks of α∗ are uniquely determined by w =∑αi. They

are B = βij where [i + 1, j] is a maximal connected component of the support of w. Eachof these blocks is a element of the cell α∗. When we remove these blocks, we are left with a

17

semi-simple cell which is uniquely determined by its weight w′ by induction on w. But w′

is uniquely determined by w. So, α∗ is uniquely determined by w. �

An admissible weight w will be called primitive if the unique semi-simple set with weightw is primitive. This is equivalent to the statement that there exists a unique hom-orthogonalset with weight w. We will show that the primitive semi-simple sets freely generate thehomology of X(An). Then we will enumerate all such hom-orthogonal sets and obtain“ballot numbers.”

Proposition 4.10. An admissible weight w is primitive if and only if wi 6= wi+1 for all iexcept in the case wi = 0 = wi+1.

Proof. Suppose that two consecutive coordinates of w are equal and positive: wa = wa+1 >0. Then we will show that w is not primitive. Let β∗ be the unique semi-simple cell of weightw. Then we first note that, for any βij in β∗, i, j are not equal to a by the description ofthe set of i, j given in the proof of Lemma 4.8.

Consider all objects βij in β∗ so that i < a < j and let βpq be the one with minimallength. Then a resolution of β∗ is given by replacing βpq with βpa and βaq. To see that thisworks, note that every other object of β∗ either contains βpq in its interior (and thereforecontains βpa and βaq in its interior) or lies in the interior of βpq, i.e., has the form βij wherep < i < j < q. Minimality of [p, q] implies that either j < a in which case βij lies in theinterior of βpa or a < i in which case βij lies in the interior of βaq.

Conversely, suppose that w is not primitive. Let β∗ be the minimal hom-orthogonal setwith weight w with, say, k elements. Let α∗ be a resolution of β∗ with k + 1 elements.Take the object of β∗ which resolves into an extension of two objects of α∗, say βij is inβ∗ and βia, βaj are in α∗. Then the index a cannot occur as a first or second index ofany other objects of β∗. For example if βpa is another object in β∗ then βpa, βia are nothom-orthogonal. Therefore wa = wa+1 ≥ 1. �

Corollary 4.11. Suppose that w is a primitive block and β∗ = 〈β1, · · · , βk〉 is the uniquesemi-simple set with weight w. Let βij be the longest root in β∗, so that wp 6= 0 only fori < p ≤ j. Then j − i = 2k − 1.

Proof. The number of roots k in a semi-simple set with weight w is equal to the sum∑|wi+1 − wi| since each root contributes 2 to this sum. When these numbers are equal to

1 then the sum is equal to the number of terms which is j − i+ 1. �

4.3. Non-primitive weights. Suppose that w is a non-primitive weight. Then we willshow that the weight subquotient complex of C∗(An) given by

C∗(An)(w) := C∗(An)w/∑w′>w

C∗(An)w′

has zero homology. We do this using the resolution set of w which we define to be the setof all integers i so that wi = wi+1. By Proposition 4.10, this set is nonempty.

Suppose that β∗ is the unique semi-simple set with weight w. Then the resolutionsof β∗ are in bijection with nonempty subsets S of the resolution set. The resolution β(S)corresponding to S is given recursively as follows. If S is empty, then β(S) = β∗. Otherwise,let s0 be the largest element of S and let S0 = S\s0. Then β(S) is given by replacing theobject βij of β(S0) of smallest length so that i < a < j and replacing it with the two objectsβis0 , βs0j . In the reverse direction, a resolution α∗ of β∗ determines resolution set S of allindices s which occur as both a first and last index of an object of α∗.

18

By Proposition 4.2, the boundary of β(s1, · · · , sr) is equal to the sum

d(β(s1, · · · , sr)) =r∑i=1

±β(s1, · · · , si, · · · , sr)

or r terms, each with coefficient plus or minus 1.

Lemma 4.12. For any non-primitive weight w, the subquotient complex C∗(An)(w) is con-tractible.

Proof. Since w is non-primitive, the resolution set S of w is nonempty. Choose one elements0 ∈ S. Then a chain contraction h of C∗(An)(w) is given by

h(β(T )) =

{0 if s0 ∈ T±β(T ∪ {s0}) otherwise

where the sign is front of β(T∪{s0}) is the same as the sign of β(T ) as a term in dβ(T∪{s0}).This will insure that β(T ) occurs with coefficient 1 in (dh + hd)(β(T )). The other termscancel as a result of the fact that d2β(T ∪ {s0}) = 0. �

This proves the following theorem.

Theorem 4.13. The homology of the associated graded complex⊕

w C∗(An)(w) is freelygenerated by the primitive semi-simple sets β∗. �

Corollary 4.14. The homology of the space X(An) is freely generated by the primitivesemi-simple sets β∗.

Proof. Semi-simple (ss) sets are cycles in C∗(An). By the theorem, the primitive ss cellsβ∗ generate the homology of the chain complex. It remains to show that no integer linearcombination of such cycles is a boundary.

Suppose not. Let z be an integer linear combination of primitive ss cells of degree kwhich is the boundary of a k + 1 chain z = dc. Let w be a weight which is minimal inlexicographic order so that cw 6= 0 where cw is the component of c of weight w. Choose cso that this minimal weight w is maximal in lexicographic order. Then w is non-primitivesince, otherwise, dcw = 0 and c can be replaced with c − cw contradicting the maximalityof the minimal weight w. This implies that zw = 0. So, the image of cw in C∗(An)(w) is acycle and therefore a boundary. Say, cw = dx in C∗(An)(w). In the chain complex C∗(An),the boundary of x may have higher weight terms. So, c− dx has no terms of weight w buthas new higher weight terms. This contradicts the maximality of w in all cases. So, weconclude that z is not a boundary and no linear combination of primitive semi-simple cellsis a boundary.

Equivalently, the homology of C∗(An) is isomorphic to the homology of the associatedgraded chain complex. �

It remains to determine the list of all primitive weights.

4.4. Primitive weights. By Corollary 4.14, Hk(X(An);Z) is free abelian for every n, k.Let r(n, k) denote its rank. Then r(n, k) is the number of primitive semi-simple weightsw ∈ Nn of degree k. We show that these numbers are equal to the “ballot numbers” byshowing that they satisfy the same recursion.

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By Lemma 4.8 and Proposition 4.10, the primitive weights w ∈ Nn of degree k areall sequences of nonnegative integers w = (w1, · · · , wn) satisfying the following conditionswhere w0 = 0 = wn+1 by convention.

(1) wi+1 = wi + 1 or wi+1 = max(wi − 1, 0) for all i = 0, · · · , n.(2) There are exactly k values of i for which wi+1 = wi + 1.

We recall that a block of this weight w is a maximal sequence of consecutive nonnegativecoordinates. By condition (1) each block has odd length. For example

w = (12123210012101)

has three blocks B07, B9,12, B13,14 of lengths 7, 3, 1 and degrees 4, 2, 1 respectively. Thereis only one possible block of length 1 and of length 3 which are as given in the example.However, there are two possible blocks of length 5: 12321 and 12121. And there are 5possible blocks of weight 7:

1234321, 1232321, 1212321, 1232121, 1212121

Also, a block of length 2j + 1 has degree j + 1.

Lemma 4.15. The number of possible blocks of length 2j+1 is given by the Catalan number

Cj =1

j + 1

(2j

j

)The proof is left to the reader.

Lemma 4.16. The ranks r(n, k) are uniquely determined by the following recursion: r(n, 0) =1 for all n ≥ 0 and for k > 0 we have:

r(n, k) =

{0 if n ≤ 2k − 2

r(n− 1, k) +∑

1≤j≤k r(n− 2j, k − j)Cj−1 otherwise

where, for convenience of notation, we use the convention that r(−1, 0) = 1.

Proof. Since X(An) is connected, we have r(n, 0) = 1 for n ≥ 0. The convention r(−1, 0) =1 is used to define ther term r(n − 2j, k − j) when n = 2k − 1 and j = k. To get fromw0 = 0 to wn+1 = 0 with k steps up and k steps down we must have at least n + 1 ≥ 2k.So, r(n, k) = 0 when n+ 1 < 2k.

Now consider all primitive semi-simple weight w with n, k ≥ 1. There are two cases.Case 1: wn = 0. In that case (w1, · · · , wn−1) is a primitive semi-simple weight of degree

k. So, there are r(n− 1, k) weights in this case.Case 2: wn = 1. Let 2j − 1 be the length of the last block of w. Then wn−2j = 0 and

w′ = (w1, · · · , wn−2j−1) is a primitive semi-simple weight of degree k − j. Since there areCj−1 possibilities for the last block of w and there are r(n − 2j, k − j) possibilities for w′

we have Cj−1r(n− 2j, k − j) possibilities for w in this case. This proves the recursion. �

Definition 4.17. The ballot number b(k, j) is defined to be the number of ways in whichk “yes” votes and j “no” votes can be cast in an ordered sequence in such a way that thenumber of “yes” votes is always greater than or equal to the number of “no” votes. Inparticular b(k, j) = 0 if j > k.

Since the count starts at (0, 0) and votes are cast one at a time by assumption, we havethe following recussion: b(k, j) = 0 unless k ≥ j ≥ 0, b(0, 0) = 1 and

b(k, j) = b(k − 1, j) + b(k, j − 1)20

for k ≥ 1. It is a well-known property of Catalan numbers that b(k, k) = Ck. An extensionof this observation is the following recursion.

Lemma 4.18. For m ≥ k ≥ 1 we have

b(m, k) = b(m− 1, k) +

k∑j=1

b(m− j, k − j)Cj−1

Proof. There are two cases.Case 1: The last vote cast was “yes”. There are b(m− 1, k) ways this could happen.Case 2: The last vote cast was “no”. Consider the difference m − k ≥ 0 between

the number of “yes” votes and the number of “no” votes. In case two this number wasm − k + 1 ≥ 1 before the last vote. Since this difference starts and ends at a smallernumber, this difference must have been equal to m− k at some earlier point. Let j > 0 beminimal so that the last 2j votes were tied j in favor and j against. There are Cj−1 waysthese last 2j votes could have been cast since the last vote was “no” and the first must havebeen “yes”. So, there are Cj−1b(m− j, k − j) ways that this could happen.

Adding up all possible cases, we get the stated recursion. �

Comparing the two lemmas, we get:

r(n, k) = b(n− k + 1, k)

which proves the following theorem.

Theorem 4.19. For n, k ≥ 0 the integral homology group Hk(X(An);Z) of the CW-complexX(An) for the quiver of type An with straight orientation is free abelian with rank equal tothe ballot number b(n− k + 1, k). �

Since b(m, k) = 0 for k > m, b(k, k) = Ck−1 and b(n, k) 6= 0 for k ≤ m we get thefollowing.

Corollary 4.20. Hk(X(An);Z) = 0 for k > n+12 and is nonzero for 0 ≤ k ≤ n+1

2 . Fur-thermore, Hk(X(A2k−1)) = Ck. �

Note that, for any Dynkin quiver Q with n vertices, X(Q) is an n-dimensional CWcomplex. So, we always have: Hk(X(Q)) = 0 for k > n.

Summary: In Section 4, we show that the homology of the space X(An) is freely gen-erated by “primitive semi-simple weights”. These are disjoint unions of “blocks”. Blocksare enumerated using Catalan numbers and the primitive weights are enumerated by ballotnumbers.

5. Proof that X(An) is a K(π, 1) for π = G(An)

Finally, in Section 5 we outline a proof that X(An) is a K(π, 1) and that, therefore,the calculation in Section 4 computes the cohomology of these groups. The proof is byinduction on n and uses a filtration of X(An) by subcomplexes Y0 ( Y1 ( · · · ( Yn whereY0 = X(An−1), Yn = X(An) and each Ym is a K(π, 1). We give some examples to seewhat these spaces and groups are. Missing details in this proof can be found in anotherpaper [10] where this theorem is extended to the case of a convex subset of the preprojectivecomponent of the Auslander-Reiten quiver of a hereditary algebra of not necessarily finitetype. The special case of An with straight orientation, as considered here, is also redone

21

combinatorially in [5] where the result is slightly strengthened: The space X(An) is locallyCAT(0).

We will first go over the base cases n ≤ 2. Then, for n ≥ 3, we use that fact that G(An)is an iterated HNN extension of G(An−1) to construct a sequence of spaces Ym going fromY0 = X(An−1) to Yn = X(An) and show by induction on m that each Ym is a K(π, 1).

5.1. Filtration of X(An). For n = 0, the space X(A0) is a single point which is a K(π, 1)with trivial group π.

For n = 1, X(A1) has C2 = 2 cells: one 0-cell and one 1-cell. So, it is the circleS1 = K(Z, 1).

For n = 2, X(A2) has C3 = 5 cells: one 0-cell, three 1-cells giving generators a, b, x forthe fundamental group and one 2-cell e(a, b) whose boundary gives the relation

∂e(a, b) = aba−1x−1b−1

In other words, the 2-cell e(a, b) is a pentagon (the associahedron) with one side identifiedwith the 1-cell x and the other four sides pasted twice to a and b. So, X(A2) is homeomorphicto a torus with an open 2-disk removed. This is homotopy equivalent to S1 ∨ S1 which is aK(π, 1) with π being the free group with two generators F2.

Suppose by induction that n ≥ 3 and X(Ak) is a K(π, 1) for k < n. Then we consideranother filtration of X(An) with subcomplexes Ym for 0 ≤ m ≤ n give as follows.

First, note that X(An−1) is the union of all cells in X(An) whose last block is Bij wherei < j < n. Let Ym be the union of X(An−1) and all cells β∗ so that, for any object of thehom-orthogonal set β∗ of the form βkn, we have k < m. Then

Y0 = X(An−1) ⊆ Y1 ⊆ Y2 ⊆ · · · ⊆ Yn = X(An)

So, Y0 is a K(π, 1) and we need to show that Yn is a K(π, 1). We may assume by inductionon 1 ≤ m ≤ n that Ym−1 is a K(π, 1). To show that Ym is a K(π, 1) we first observe thatπ1(Ym) is an HNN extension of π1(Ym−1). Then we will show that Ym is a “graph of groups”[4], p.91, and therefore a K(π, 1). We will give examples, definitions and proofs.

Example 5.1. Let n = 3 and m = 0. Then Y0 = X(A2) and

Y1 = Y0 ∪ e(β03) ∪ e(β03, β12)

Y0 ' S1 ∨ S1 and we are attaching one more 1-cell giving a new generator x(β03) for thefundamental group and one new 2-cell giving the commutation relation [x(β12), x(β03] = 1.So, Y1 ' S1 ∨ (S1 × S1) which is a K(π, 1) with π the free product of Z with Z × Z. Forn = 3,m = 1 we have:

Y2 = Y1 ∪ e(β13) ∪ e(β01, β13)

As in the case of Y1, Y2 is obtained by attaching a new 1-cell e(βmn) = e(β13) giving thenew generator x(β13) in π1Y2 and a new 2-cell giving the relation [x(β01), x(β13)] = x(β03).In other words:

x(β13)−1x(β01)x(β13) = x(β03)x(β01)

For n = 5,m = 2, Y3 is obtained from Y2 by attaching 10 new cells:

(5.1) Y3 = Y2 ∪ e(β25) ∪ e(β01, β25) ∪ e(β12, β25) ∪ e(β02, β25) ∪ e(β01, β12, β25)

∪e(β25, β34) ∪ e(β01, β25, β34) ∪ e(β12, β25, β34) ∪ e(β02, β25, β34) ∪ e(β01, β12, β25, β34)

We will use this to explain the general construction.

22

5.2. HNN extensions.

Definition 5.2. Suppose that H,G are groups and ϕ,ψ : H → G are injective homo-morphisms. Then HNN-extension N(H,G,ϕ, ψ) is defined to be the quotient of the freeproduct G ∗ Z with new free generator t modulo the relation

tϕ(h)t−1 = ψ(h)

for all h ∈ H.

Let Gn,m = π1(Ym) where Ym is defined above. Since Ym is a subcomplex of X(An), thegenerators and relations for Gn,m are the subsets of the generators and relations for G(An):

Lemma 5.3. Gn,m = π1(Ym) has generators and relations given by the 1-cells and 2-cellsof Ym as follows.

(1) Generators: x(βij) where 0 ≤ i < j ≤ n excluding x(βin) for i ≥ m.(2) Relations among these generators are:

(a) x(βij), x(βk`) commute if i, j, k, ` are distinct and noncrossing and(b) [x(βij), x(βjk)] = x(βik) if 0 ≤ i < j < k ≤ n and where j < m if k = n.

Examination of this list shows that Gn,m+1 = π1(Ym+1) has only one new generator notin Gn,m, namely x(βmn) and the only new relations are commutation relations with thisnew generators which can be expressed as follows.

(a) x(βij), x(βmn) commute if either 0 ≤ i < j < m or m < i < j < n.(b) x(βmn)−1x(βim)x(βmn) = x(βin)x(βim)

Equivalently, we have:

Proposition 5.4. Gn,m+1 is an HNN-extension of Gn,m with

(1) t = x(βmn)−1

(2) H = Hm = G(Am)×G(An−m−2)(3) The monomorphism ϕ : Hm → Gn,m is given on generators by

ϕ(x(βij), x(βk`)) = x(βij)x(βk+m+1,`+m+1) ∈ G(An−1) ⊆ Gn,m(4) The monomorphism ψ : Hm → Gn,m is given on generators by ψ(x(βij), x(βk`)) =

x(βij)x(βk+m+1,`+m+1) (same as ϕ) if j < m and

ψ(x(βim), 1) = x(βin)x(βim)

which lies in Gn,m since i < m.

Proof. We will check that ψ is a group homomorphism by showing that it preserves relations.We check only one relation: [x(βij), x(βjm)] = x(βim):

[ψx(βij), ψx(βjm)] = [x(βij), x(βjn)x(βjm)]

= x(βjm)−1x(βjn)−1x(βij)x(βjn)x(βjm)x(βij)−1

= x(βjm)−1x(βin)x(βij)x(βjm)x(βij)−1

= x(βin)[x(βij), x(βjm)] = x(βin)x(βim)

The other relations are easy to check. �

We use the following well-known theorem. (See [4] for an elementary exposition andproof.)

23

Theorem 5.5. Suppose that X,Y are base pointed spaces so that X is a K(H, 1) and Yis a K(G, 1). Let f, g : X → Y be a pointed continuous mapping which induces the mapsϕ,ψ : π1(X) = H → G = π1(Y ) respectively. Let Z = X × I

∐Y/ ∼ be the quotient space

of X × I∐Y modulo the identifications (x, 0) ∼ f(x) ∈ Y and (x, 1) ∼ g(x) ∈ Y for all

x ∈ X. Then Z is a K(π, 1) with π = N(H,G,ϕ, ψ).

We claim that, using X = X(Am)×X(An−m−2), Y = Ym with f, g : X → Y appropri-ately chosen maps, we get Z homeomorphic to Ym+1 proving that Ym+1 is a K(π, 1). See[10] for a detailed proof of a more general theorem proved using the same outline.

Theorem 5.6. The picture space X(An) is a K(π, 1) where π is the picture group G(An)with generators xij for all 0 ≤ i < j ≤ n and relations where we use the notation [x, y] :=y−1xyx−1.

(1) [xij , xjk] = xik for all i < j < k(2) [xij , xk`] = 1 if the closed intervals [i, j], [k, `] are either disjoint or one is contained

in the interior of the other.

We call [i, j] the extended support of xij . The actual support is [i + 1, j]. We say thattwo closed intervals are noncrossing if they are either disjoint or one is in the interior of theother. So (2) says that generators commute if their extended supports are noncrossing.

Proof. The presentation follows from Lemma 4.7 and (1) and (2) in Example 2.5. We haveswitched to the simplified notation

xij = x(βij).

�

Summary: Section 5 outlines the proof that X(An) is a K(π, 1) for π = G(An) and,therefore, the homology of X(An) is the homology of the picture group G(An).

6. Cup product structure

We now determine the ring structure on the cohomology H∗(X(An);Z). We use thefact that X(An) is a K(π, 1) for the group G(An) proved in detail in [10], [5] and outlinedin the last section. So, we deal with the cohomology of the group G(An) instead of thespace X(An). Since the homology is freely generated by the set of primitive weights w,the cohomology is also freely generated as an additive group by the dual elements w∗. Wewill show that, as a ring, the cohomology is generated by the duals w∗ to weights w havingonly one block. We call such generators dual blocks. First, we need an intrinsic (instead ofcomputational) description of the dual blocks with “full support.”

6.1. Dual blocks. Each block is supported on a subspace G(Qij) ⊆ G(An) which we willshow is a retract. For every 0 ≤ i < j ≤ n let Qij be the full subquiver of the quiver Q(An)with vertices i+ 1, · · · , j:

Qij : i+ 1← i+ 2← · · · ← j

This is the smallest subquiver which supports the root βij . Let G(Qij) be the correspondingsubgroup of G(An) which is generated by xpq = x(βpq) where i ≤ p < q ≤ j.

The CW complex X(An) contains a subcomplex X(Qij) which is the union of all cells

ekα∗ where each element of α∗ is a root βpq for i ≤ p < q ≤ j. The inductive proof thatX(An) is a K(π, 1) shows that X(Qij) is a K(π, 1) with π = G(Qij) and the inclusionX(Qij) ⊆ X(An) induces the inclusion G(Qij) ⊆ G(An).

24

Lemma 6.1. Let rij : G(An)→ G(Aj−i) be the homomorphism given on generators by

rij(xpq) =

{xp−i,q−i if i ≤ p < q ≤ j1 otherwise

Then the composition G(Qij)→ G(An)→ G(Aj−i) is an isomorphism.

Proof. The map rij is a homomorphism since it respects the relations of G(An). For ex-

ample, the relation [xij , xjk] = xik in G(An) becomes the relation xijx−1ij = 1 in G(Aj−i).

On G(Qij), the mapping rij sends relations among xpq to corresponding relations amongxp−i,q−i. So, it induces an isomorphism G(Qij) ∼= G(Aj−i). �

Proposition 6.2. If 0 ≤ p1 < q1 < · · · < pk < qk ≤ n then the subgroups G(Qpi,qi) ofG(An) commute with each other. So, we have inclusion and projection morphisms∏

G(Qpi,qi) ↪→ G(An)

∏rpi,qi−−−−−→

∏G(Aqi−pi)

Whose composition is an isomorphism.

Proof. Generators of G(Qis,js) have extended supports in [is, js] which are disjoint andtherefore commute. �

Proposition 6.3. Suppose that m = 2k+1 is odd. Then the cokernel of the homomorphismin integral homology induced by inclusion:

Hp(G(Q1,m))⊕m−1⊕j=2

Hp(G(Q0,j−1)×G(Qj,m))⊕Hp(G(Q0,m−1))→ Hp(G(Am))

is free of rank Ck = 1k+1

(2kk

)if p = k + 1 and is 0 otherwise.

Proof. The image of this homomorphism contains all primitive semisimple weights whichdo not have full support. So, the cokernel is freely generated by primitive blocks w withfull support, i.e., so that wi 6= 0 for 1 ≤ i ≤ n. These exist only when m is odd. They havedegree k + 1 and there are a Catalan number Ck of these by Lemma 4.15. �

Let K(G(Am)) ⊆ Hk+1(G(Am);Z) denote the kernel of the homomorphism

Hk+1(G(Am))→ Hk+1(G(Q1,m))⊕m−1⊕j=2

Hk+1(G(Q0,j−1)×G(Qj,m))⊕Hk+1(G(Q0,m−1))

induced by the inclusions of these subgroups into G(Am) where m = 2k + 1. By theproposition above, K(G(Am)) is free abelian of rank Ck. (Whereas Hk+1(G(Am)) is freeabelian of rank Ck+1.)

Definition 6.4. Choose a fixed basis for K(G(Am)). We define the dual blocks with fullsupport in Hk+1(G(Am);Z) to be any element of this basis. For n ≥ m, the images of theseCk generators under the split monomorphisms

Hk+1(G(Am))r∗i,i+m−−−−→ Hk+1(G(An))

induced by the retractions ri,i+m : G(An) � G(Am) for 0 ≤ i ≤ n −m will be called the

dual blocks in Hk+1(G(An)) with extended support [i, i+m]. The image of K(G(Am)) underr∗i,i+m will be denoted K(Qi,i+m).

25

Proposition 6.5. The cohomology of G(An) has a direct summand which is freely generatedby the dual blocks. I.e.,

⊕iK(Qi,i+m) is a direct summand of Hk+1(G(An)) for m = 2k+1.

Proof. Restricting to one degree, say k+ 1, we are talking about dual blocks with extendedsupport [pi, qi] where qi − pi = 2k + 1 for each i. Given any additive linear combination ofsuch dual blocks, consider the projection homomorphism

Hk+1(G(An)) � Hk+1(G(Qpi,qi))

induced by the inclusion G(Qpi,qi) ↪→ G(An). By definition of dual blocks, this homomor-phism is zero on dual blocks of degree k + 1 with extended support not equal to [pi, qi]since such extended supports will not be contained in [pi, qi]. Also, by definition of dualblocks, the dual blocks with extended support [pi, qi] map to distinct free generators of asplit summand of Hk+1(G(Qpi,qi)). The Proposition follows. �

6.2. Multiplication of dual blocks.

Lemma 6.6. If w∗1, w∗2 ∈ H∗(G(An)) are dual blocks whose extended supports intersect then

their cup product is zero: w1 ∪ w2 = 0.

Proof. This is just dimension counting. Suppose that w∗i has extended support [pi, qi] withqi − pi = 2ki + 1. Then, by definition, w∗i ∈ Hki+1(G(An)) is the pull back of a dual blockof full support in Hki+1(G(A2ki+1)) under the homomorphism

rpi,qi : G(An)→ G(A2ki+1)

Therefore, their cup product is the pull back of the cross product of these cohomologyclasses under the homomorphism

(rp1,q1 , rp2,q2) : G(An)→ G(A2k1+1)×G(A2k2+1)

Let a = min(p1, p2) and b = max(q1, q2). Then the homomorphism (rp1,q1 , rp2,q2) above

factors through G(Ab−a) which has cohomological dimension b b−a+12 c by Corollary 4.20.

If the extended supports of w∗1, w∗2 meet then we will have b − a ≤ 2k1 + 2k2 + 2 making

b b−a+12 c ≤ k1 + k2 + 1 which is less than the degree of w∗1 ∪ w∗2. So, this cup product must

be zero. �

Lemma 6.7. If w∗1, · · · , w∗k ∈ H∗(G(An)) are dual blocks with disjoint extended supportsthen their cup product is nonzero.

Proof. This follows immediately from Proposition 6.2. �

Theorem 6.8. The integral cohomology of the picture group G(An) is generated, as a ring,by the set of dual blocks w∗i . The cup product of these dual blocks is nonzero if and only iftheir extended supports are disjoint. Furthermore, as an additive group, H∗(G(An);Z) isfreely generated by the nonzero cup products of the dual blocks.

Proof. A simple counting argument shows that the number of nonzero cup products of dualblocks is equal to the total rank of the cohomology of G(An). Therefore, it suffices toshow that these cup products are linearly independent and generate a direct summand ofH∗(G(An)).

We work in a fixed degree, say d. Consider an arbitrary integer linear combination ofdegree d cup products of dual blocks. Take one of these expressions: w∗1∪w∗2∪· · ·∪w∗k where

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w∗i is a dual block with extended support [pi, qi] so that k is maximal. Then∑

(qi − pi) =2d− k is minimal. This implies that, the restriction homomorphism

Hd(G(An))→ Hd

(k∏i=1

G(Qpi,qi)

)is zero on all cup products in the given linear combination which have different extendedsupport. The reason is that any other extended support will contain a point not in the set⋃

[pi, qi] and this implies that that term will vanish under the restriction map and thereforethe cup product will also go to zero. The set of all cup products of dual blocks with thesame extended support as the given one freely generate the subgroup

(6.1) K(Qp1,q1)⊗K(Qp2,q2)⊗ · · · ⊗K(Qpk,qk)

which is a direct summand of Hd(∏k

i=1G(Qpi,qi))

and thus of Hp(G(An)) since each

K(Qpi,qi) is a direct summand of H∗(G(Qpi,qi)). Proceeding by downward induction on

k, we obtain a complete direct sum decomposition of Hd(G(An)) as claimed. �

Remark 6.9. We observe that this proof shows that H∗(G(An)) is a direct sum of subgroupsof the form (6.1).

When Q is a quiver of type An, the group G(Q) depends on the orientation of Q. Thisis a computer calculation using GAP which was carried out by D.Ruberman. Although thegroup depends on the orientation, we believe that the homology of the groups is independentof the orientation. According to He Wang, the difference lies in the Massey product structureof the cohomology. These are questions for further research.

Summary: Section 6 determines the cup product structure of the integral cohomologyof G(An).

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Department of Mathematics, Brandeis University, Waltham, MA 02454E-mail address: igusa@brandeis.edu

Department of Mathematics, Northeastern University, Boston, MA 02115E-mail address: g.todorov@neu.edu

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