Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs

Publ. Mat. 46 (2002), 97–118 STABLE RATIONAL COHOMOLOGY OF AUTOMORPHISM GROUPS OF FREE GROUPS AND THE INTEGRAL COHOMOLOGY OF MODULI SPACES OF GRAPHS Craig A. Jensen Abstract It is not known whether or not the stable rational cohomology groups ˜ H∗(Aut(F∞); Q) always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions). We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields that H5(ˆ Qm;Z), H6(ˆ Qm;Z), and H5(Qm;Z) never stabilize as m→∞, where the moduli spaces ˆ Qmand Qmare the quotients of the spines ˆ Xmand Xmof “outer space” and “auter space”, respectively, introduced in [3] by Culler and Vogtmann and [6]by Hatcher and Vogtmann. 1. Introduction Let Fndenote the free group on nletters and let Aut(Fn) and Out(Fn) denote the automorphism group and outer automorphism group, respectively, of Fn.In[5] Hatcher shows that the integral cohomology of the infinite symmetric group Σ∞is a direct summand of the integral cohomology of Aut(F∞). He mentions that it is unknown whether or not the complementary summand is zero and in particular whether or not ˜ H∗(Aut(F∞); Q) is always zero. In [6], Hatcher and Vogtmann again pose the question of whether or not the stable rational cohomology groups of Aut(Fn) and Out(Fn) all vanish, and show that it does 2000 Mathematics Subject Classification. Primary: 05C25, 20F32, 20J05; Secondary: 20F28, 55N91. Key words. Graphs, free groups, moduli spaces, outer space, auter space. 98 C. A. Jensen vanish in dimensions 1 through 6. A recent theorem of Madsen and Tillman gives (after inverting the prime 2) a product decomposition for the plus construction BΓ+of the classifying space for stable mapping class groups; however, it is currently unknown to what extent this enables one to answer the question posed by Hatcher and Vogtmann. Let ˆ Xmbe the spine of outer space (see Culler and Vogtmann in [3]) and let ˆ Qm=ˆ Xm/Out(Fm) be the corresponding moduli space of graphs. Similarly, let Xmbe the spine of auter space (see Hatcher and Vogtmann in [6]) and let Qm=Xm/Aut(Fm) be the corresponding moduli space of pointed graphs. In this paper, we show that Theorem 1. Let i∈{0,1}. For all positive integers k, either H4k+i(Out(F∞); Q)=0 or H4k+i+1(ˆ Qm;Z)never stabilizes as m→∞. Theorem 2. For all positive integers k, either H4k(Aut(F∞); Q)=0 or H4k+1(Qm;Z)never stabilizes as m→∞. From calculations in [7] that H4(Aut(F∞); Q)=H4(Out(F∞); Q)=H5(Aut(F∞); Q)=0, the above two theorems immediately show that Corollary 3. The cohomology groups H5(ˆ Qm;Z)and H6(ˆ Qm;Z)never stabilize as m→∞. Corollary 4. The cohomology group H5(Qm;Z)never stabilizes as m→∞. The two corollaries are true because as mincreases, torsion from increasingly higher primes is introduced in H5(ˆ Qm;Z), H6(ˆ Qm;Z), and H5(Qm;Z). There are natural inclusions QmQm+1, and it is known [6] that the induced map Hi(Qm+1;Q)→Hi(Qm;Q) is an isomorphism for m>3i/2. It is therefore important to keep in mind that the above two corollaries only hold with respect to integral cohomology. Integral Cohomology of Moduli Spaces of Graphs 99 A quick note about our notation is appropriate here. In general, groups without any additional structure will be written using multiplicative notation (e.g., Z/p ×Z/p ∼ =(Z/p)2) but modules like cohomology groups will be written using additive notation (e.g., Z/p⊕Z/p ∼ =2(Z/p)). In Section 2 we review the basics about outer and auter space, and in Section 3 we prove Theorem 1. Symmetry groups of graphs with 2p−1 holes are discussed in Section 4, which enables us to prove Theorem 2 in Section 5. This paper is based on a dissertation (see [8], [9]) written while the author was a student of Karen Vogtmann at Cornell, and the author would like to thank Prof. Vogtmann for her help and advice. The author would also like to thank Henry Glover for his helpful comments on this paper. 2. Basics about spectral sequences and Aut(F n ) Let Gbe a group acting cellularly on a finite dimensional CW-complex Xsuch that the stabilizer stabG(δ) of every cell δis finite and such that the quotient of Xby Gis finite. Further suppose that for every cell δof X, the group stabG(δ) fixes δpointwise. Let Mbe a G-module. Recall (see [2]) that the equivariant cohomology groups of the G-complex Xwith coefficients in Mare defined by H∗ G(X;M)=H∗(G;C∗(X;M)) and that if in addition Xis contractible (which will usually, but not always, be the case in this paper) then H∗ G(X;M)=H∗(G;M). In [2] a spectral sequence ˜ Er,s 1= [δ]∈∆r n Hs(stab(δ); M)⇒Hr+s G(X;M)(5) is defined, where [δ] ranges over the set ∆r nof orbits of r-simplices δ in X. If Mis Z/p or Z(p)then a nice property should be noted about the spectral sequence (5). This property will greatly reduce the calculations we need to go through, and in general will make concrete computations possible. Since each group stab(δ) is finite, a standard restrictiontransfer argument in group cohomology yields that |stab(δ)|annihilates Hs(stab(δ); M) for all s>0. (For examples of these sorts of arguments see [2].) Since all primes not equal to pare divisible in Z/p or Z(p), this in turn shows that the p-part of |stab(δ)|annihilates Hs(stab(δ); M) 100 C. A. Jensen for s>0. In particular, if pdoes not divide some |stab(δ)|, then this [δ] does not contribute anything to the spectral sequence (5) except in the horizontal row s= 0. It follows that if our coefficients are Z/p or Z(p)then we are mainly just concerned with the simplices δwhich have “p-symmetry”. We now specialize to the cases where Gis Out(Fn) or Aut(Fn) and X is either the spine ˆ Xnof “outer space” or the spine Xnof “auter space”. Hatcher and Vogtmann’s definition of auter space closely follows Culler and Vogtmann’s (prior) definition of outer space, except that the graphs arising have basepoints. We review some basic properties and definitions of auter space below, where we concentrate on auter space because that is where most of the calculations in this paper will take place. Most of these facts can be found in [3], [6], [12], and [13]. Consider the automorphism group Aut(Fn) of a free group Fnof rank n(where nwill be 2p−1 for most of our work). Let (Rn,v 0)bethe n-leafed rose, a wedge of ncircles. We say a basepointed graph (G, x0) is admissible if it has no free edges, all vertices except the basepoint have valence at least three, and there is a basepoint-preserving continuous map φ:Rn→Gwhich induces an isomorphism on π1. The triple (φ, G, x0) is called a marked graph. Two marked graphs (φi,G i,x i) for i=0,1 are equivalent if there is a homeomorphism α:(G0,x 0)→ (G1,x 1) such that (α◦φ0)#=(φ1)#:π1(Rn,v 0)→π1(G1,x 1). Define a partial order on the set of all equivalence classes of marked graphs by setting (φ0,G 0,x 0)≤(φ1,G 1,x 1)ifG1contains a forest (a disjoint union of trees in G1which contains all of the vertices of G1) such that collapsing each tree in the forest to a point yields G0, where the collapse is compatible with the maps φ0and φ1. From [5] and [6] we have that Aut(Fn) acts with finite stabilizers on a contractible space Xn. The space Xnis the geometric realization of the poset of marked graphs that we defined above. Let Qnbe the quotient of Xnby Aut(Fn). Note that the CW-complex Qnis not necessarily a simplicial complex. Since Aut(Fn) has a torsion free subgroup of finite index [5] and it acts on the contractible, finite dimensional space Xn with finite stabilizers and finite quotient, Aut(Fn) has finite vcd. Let pbe an odd prime number, and let Z(p)be the localization of Z at the prime ideal (p). Then we can apply the spectral sequence (5) to get ˜ Er,s 1= [δ]∈∆r n Hs(stab(δ); Z(p))⇒Hr+s(Aut(Fn); Z(p))(6) where [δ] ranges over the set ∆r nof orbits of r-simplices δin Xn. Integral Cohomology of Moduli Spaces of Graphs 101 The spectral sequence (6) requires as input the stabilizers stabAut(Fn) (δ) of simplices δin Xn. Smillie and Vogtmann [12] examined the structure of these stabilizers in detail, and we list their results here. Consider a given r-simplex (φr,G r,x r)>···>(φ1,G 1,x 1)>(φ0,G 0,x 0) with corresponding forest collapses (Hr⊆Gr),...,(H2⊆G2),(H1⊆G1). For each i∈0,1,...,r, let Fibe the inverse image under the map Gr→···→Gi+1 →Gi of forest collapses, of the forest Hi. That is, we have Fr⊆···⊆F2⊆F1⊆Gr. It is shown in [12] that the stabilizer of the simplex under consideration is isomorphic to the group Aut(Gr,F 1,...,F r,x r) of basepointed automorphisms of the graph Grthat respect each of the forests Fi. For example, the stabilizer of a point (φ, G, x0)inXnis isomorphic to Aut(G, x0). 3. Graphs without basepoints Theorem 1 is a direct consequence of the stability theorems in [5] and the spectral sequence calculations in [4]. Proof of Theorem 1: From [5], H4k+i(Out(F∞); Q)=H4k+i(Aut(F∞); Q) and if m≥4k2+10k+1+i2/4+2ik +5i/2, then the standard map H4k+i(Aut(Fm); Z)→H4k+i(Out(Fm); Z) is an isomorphism. Observe that H4k+i(Out(F4k2+10k+1+i2/4+2ik+5i/2); Z)=H4k+i(Out(F∞); Z) is a finitely generated abelian group. If it contains a torsion free summand isomorphic to Z, then we are done and H4k+i(Out(F∞); Q)= 0. Otherwise, choose a prime qsuch that q+1≥4k2+10k+1+i2/4+2ik +5i/2 and so that for all primes p≥qthere is no p-torsion in H4k+i(Out(F∞); Z).We will show that H4k+i+1(ˆ Qp+1;Z) has p-torsion for infinitely many primes p, which will prove the theorem. 102 C. A. Jensen Let p≥max{q,25}with p≡3 (mod 4). (Note that there are infinitely many possibilities for p, as there are infinitely many primes that are greater than a given number and congruent to 3 modulo 4.) Because H4k+i(Out(Fp+1); Z) has no p-torsion, there is also no p-torsion in H4k+i(Out(Fp+1); Z(p)). From the calculation of Glover and Mislin in [4]oftheE2-page of the equivariant spectral sequence used to calculate H∗(Out(Fp+1); Z(p)), we know that this E2-page, in the rows 0 ≤s<2(p−1), is given by Er,s 2=             Hr(ˆ Qp+1;Z(p))s=0 Z/p r = 0 and s=4k>0,k∈Z+ (np)Z/p r = 1 and s=4k>0,k∈Z+ 0 otherwise where np=(p−1)/12 −pand p∈{0,1}. Since p≥25, note that np≥1. Hence a class ˆα∈Ei,4k 2in the E2-page survives at least until the E4k+1-page. The class ˆα∈Ei,4k 2cannot survive to the E∞page, however, because there is no p-torsion in the finite (since H4k+i(Out(Fp+1); Q) = 0) additive group H4k+i(Out(Fp+1); Z(p)). It follows that there is p-torsion in E4k+i+1,0 4k+1 =H4k+i+1(ˆ Qp+1;Z(p)). Thus H4k+i+1(ˆ Qp+1;Z) has p-torsion. 4. Symmetry groups of graphs We will use spectral sequence (6) to compute a portion of the cohomology of Aut(Fn). Since our coefficient ring is Z(p), we have already remarked that for the terms in the spectral sequence above the horizontal axis, we are concerned only with simplices whose stabilizers are divisible by p. In addition, the stabilizer of a simplex consists of graph automorphisms that respect the forest collapses in the simplex. We will find which simplices arise in the case n=2p−1. In other words, we want to calculate which graphs Gwith a Z/p action on them have π1(G)∼ =Fn. Recall that a Z/p-graph Gis reduced if it contains no Z/p-invariant subforests. Integral Cohomology of Moduli Spaces of Graphs 103 We now examine the cohomology of the quotient Qnof the spine Xn of auter space. There are natural inclusions QmQm+1, and it is known [6] that the induced map Hi(Qm+1;Q)→Hi(Qm;Q)isan isomorphism for m>3i/2. Our goal is to show that, in contrast, H5(Qm;Z) never stabilizes as m→∞. This is done by showing that as mincreases, torsion from increasingly higher primes is introduced in H5(Qm;Z). To this end, we do specific calculations in the spectral sequence (5) applied to the action of Aut(Fn)onXnfor n=2p−1. The Er,0 2-term of this spectral sequence is Hr(Qn;Z(p)), and the sequence converges to Hr(Aut(Fn); Z(p)). Results from Hatcher and Vogtmann [6] on the cohomology of Aut(Fn) are then used to obtain the result. In this section, we do the ground work necessary to compute the E1-page of the spectral sequence: we find all simplices of Xnwith p-symmetry and compute the cohomology of the stabilizers of these simplices with coefficients in Z(p). In Section 5 we will compute the E2-page of the spectral sequence, and use this calculation to obtain the result. Unless otherwise stated, p≥5 will be prime and n=2p−1. The assumption that p≥5 is for convenience more than any other reason, as the main results will only consider arbitrarily large primes pand so we should not devote extra time to the (fairly easy to resolve) complications introduced by considering the prime p= 3. These complications arise from the fact that the dihedral group D6is the same as the symmetric group S3, so that we cannot distinguish between dihedral and symmetric symmetry in that case. We now define some graphs that we will need for this section. (Refer to Figures 1 and 2 for illustrations of most of these graphs.) Let Θp−1 be the graph with two vertices and pedges, each of which goes from one vertex to the other (see Figure 1). Say the “leftmost vertex” of Θp−1is the basepoint. Hence when we write Θp−1∨Rp−1then we are stipulating that the rose Rp−1is attached to the non-basepointed vertex of Θp−1, while when we write Rp−1∨Θp−1then we are saying that the rose is attached to the basepoint of Θp−1. Let Φ2(p−1) be a graph with 3pedges a1,...,a p,b1,...,b p,c1,...,c p, and p+ 3 vertices v1,...,v p,x,y,z. The basepoint is xand each of the edges aibegin at xand end at vi. The edges biand cibegin at yand z, respectively, and end at vi. Note that there are obvious actions of Z/p on Θp−1and Φ2(p−1), given by rotation, and that these actions are unique up to conjugacy. Let Ψ2(p−1) be the graph obtained from Φ2(p−1) by collapsing all of the edges aito a point. 104 C. A. Jensen Let Ω2(p−1) be the graph obtained from Φ2(p−1) by collapsing either the edges bior the edges ci(the resulting graphs are isomorphic) to a point. Note that the only difference between Ψ2(p−1) and Ω2(p−1) is where the basepoint is located. Rp−1Θp−1Θp−1∨Rp−1 Rp−1∨Θp−1Φ2(p−1) Ψ2(p−1) Ω2(p−1) Figure 1. Some graphs with p-symmetry Given a finite subgroup Gof Aut(Fn) for some integer n, we say that a marked graph η1:Rr→Γ1 is a G-equivariant blowup in the fixed point space XG rof a marked graph η2:Rr→Γ2 if there is a 1-simplex η1>η 2in XG r. Integral Cohomology of Moduli Spaces of Graphs 105 ΞpΥ2p−1 Υ1 2p−1 Υ2 2p−1 Figure 2. Some graphs with D2p-symmetry Let Υ1 2p−1and Υ2 2p−1be the two possible graphs that can be obtained from Υ2p−1by equivariantly blowing up the pvalence 4 vertices into 2pvalence 3 vertices. That is, Υ1 2p−1can be obtained by first taking ap-gon and then attaching pfree edges to the pvertices of the p-gon. Say each of these new edges eibegins at the vertex xiand ends at the vertex yi, and suppose that the vertices xiare the ones that are attached to the p-gon. Now form the 1-skeleton of the double cone or suspension over the pvertices yi. This gives the graph Υ1 2p−1. The graph Υ2 2p−1 can be thought of as follows: First take a p-gon and cone off over the pvertices of the p-gon. Now also cone off over the pmidpoints of the pedges of the p-gon. Note that there is an obvious Zp-action on each of Θp−1,Ξ p,Υ 2p−1,Υ 1 2p−1, and Υ2 2p−1. Let Ξpbe the 1-skeleton of the cone over a p-gon, so that Ξphas p+1 vertices and 2pedges, one vertex has valence pand the other pvertices all have valence 3. Let Υ2p−1be the 1-skeleton of the suspension of a p-gon. Hence Υ2p−1has p+ 2 vertices and 3pedges; two of the vertices have valence pand the other phave valence 4. 112 C. A. Jensen the symmetric group Σpacts on the collections of edges defined above, and so the cohomology of the group of graph automorphisms of the graph is the same as that of the symmetric group. If |{y0,z 1,z 2}| = 2 then the only edges in the graph are the ei,fi, and giand the graph is either Υ2a 2p−1or Υ2b 2p−1. On the other hand, if |{y0,z 1,z 2}| = 3, then the graph has one additional edge besides the ei,fi,orgi. Accordingly, the graph looks like a Φ2p−1(see Figure 1) with one additional edge added. This additional edge can go from any of the {y0,z 1,z 2}to any other one, including possibly the same one. In any case, it is definitely true that the graph has automorphism group with the same cohomology as Σp. The lemma follows. For an example of what we were trying to avoid in the proof of the above lemma, refer to the three examples given in the Figure 4. The graphs pictured have an obvious Z/p-symmetry given by rotation about the basepoint, which is indicated by a solid dot. But they have no dihedral flip, and their basepoint-preserving automorphism groups are all exactly Z/p, where p= 5 in the examples pictured and where obvious analogues exist for other odd primes. The ranks of the fundamental groups of the graphs pictured are 3p−1, 3p, and 2p, respectively. The last rank, 2p, is the lowest rank possible where one can have a graph with exactly Z/p symmetry. Corollary 9. A vertex in the p-singular locus of Xnhas at most dihedral symmetry if its cohomology is the same as that of D2por D2p×Σp, and vertices in the p-singular locus will never have exactly Z/p symmetry. In the figures below, a dotted line or a hollow dot indicates that the given edge or vertex, respectively, does not have the indicated property. A solid dot, a solid line, or a 2-simplex with an X in it, means that the given vertex, edge, or 2-simplex, respectively, does have the indicated property. By analyzing the Z/p-invariant subforests of all of the graphs explicitly listed in the proof of Lemma 7, we can see what types of stabilizers higher dimensional simplices (rather than just vertices) have. We will show that the simplices with at most dihedral symmetry will fall into two (exhaustive but not disjoint) categories. The first category consists of those that are listed in Figure 5. The second category consists of simplices whose maximal vertex (recall that Xnis the realization of a poset) has the form Ξp∨Γp−1where Γp−1is some basepointed graph with fundamental group of rank p−1, the wedge does not necessarily Integral Cohomology of Moduli Spaces of Graphs 113 take place at the basepoint, and where the forest collapses of the simplex respect the Z/p action on Ξp. Ξp∨Θp−1Υ1 2p−1Θp−1∨Ξp Rp∨Θp−1Θp−1∨Rp Υ2p−1 Θ2p−1 Υ2 2p−1 Υ2a 2p−1Υ2b 2p−1 Figure 5. Some simplices with at most dihedral symmetry Rp∨Θp−1Υ2p−1Θp−1∨Rp Υ2 2p−1 Υ2a 2p−1Υ2b 2p−1 Figure 6. Simplices with exactly Z/p symmetry 114 C. A. Jensen We will also show that the simplices listed in Figure 6 are the only ones with exactly Z/p symmetry. Corollary 10. Let p≥5be prime, n=2p−1, and consider the p-singular locus of the spine Xnof auter space. •The only simplices with at most dihedral symmetry are either: (i) listed in Figure 5; or (ii) have maximal vertex of the form Ξp∨ Γp−1. •The only simplices with exactly Z/p symmetry are those listed in Figure 6. Proof: We examine each of the graphs listed in Lemma 7 separately. By enumerating the Z/p invariant subforests of each of these graphs, one can list all of the simplices in the p-singular locus of Xn. We can ignore the graphs in Lemma 7 that do not have dihedral symmetry, as all of their symmetry comes from symmetric groups. When you collapse invariant subforests of these graphs, you still get graphs with symmetry coming from the symmetric group. So we are left with analyzing the graphs from Lemma 7 with dihedral symmetry, which were: •Θp−1∨Ξp. (There are actually two possibilities here as the enumeration in Lemma 7 did not specify basepoints. The central vertex of Ξpcould be attached to either the basepoint of Θp−1or the other vertex of Θp−1.) •Γp−1∨Ξp, where Γp−1is a basepointed graph with fundamental group of rank p−1 which has no p-symmetry (or where Γp−1is Θp−1but the central vertex of Ξpis attached to the midpoint of an edge of Γp−1). •Υ2p−1. •Υ1 2p−1. •Υ2 2p−1. For the first two types of graphs, you can obtain simplices with dihedral symmetry by collapsing all of the spokes of Ξpand/or any forest in the other graph of the wedge sum (either Θp−1or Γp−1). The resulting simplex with maximal vertex Θp−1∨Ξpor Γp−1∨Ξpwill clearly have at most dihedral symmetry, and will also just as clearly not give you a graph with exactly Z/p symmetry. In a similar manner, simplices in the p-singular locus of Xnwith maximal vertex Υ2p−1or Υ1 2p−1are (exhaustively) listed in Figure 5. Note that Υ2p−1can only be blown up (while still preserving the Z/p Integral Cohomology of Moduli Spaces of Graphs 115 action so that we stay in the p-singular locus of Xn) in two ways, to either Υ1 2p−1or Υ2 2p−1. The latter two graphs cannot be blown up at all. Finally, the simplices with maximal vertex Υ2 2p−1are listed in Figure 5 or Figure 6. Note that we can obtain edges and 2-simplices with exactly Z/p symmetry, even though no actual vertex of Xnhas exactly Z/p symmetry. This is because you can choose subforests of Υ2 2p−1 that do respect the dihedral “flip” of Υ2 2p−1. In other words, this flip will not take the subforest to itself again. Hence the resulting simplex will just have symmetry group Z/p. Last of all, note that you can also choose subforests of Υ2 2p−1which do respect the dihedral flip, and these give simplices with dihedral symmetry. 5. The integral cohomology of the quotient never stabilizes We will prove Theorem 2 in this section. As in Section 4, all primes p considered are assumed to be greater than or equal to 5. Lemma 11. For the rows 0≤s<2(p−1), the E2page of the spectral sequence (5) applied to calculate H∗(Aut(F2p−1); Z(p))is given by Er,s 2=       Hr(Q2p−1;Z(p))s=0 Z/p r =2and s=4k−2>0,k ∈Z+ 0otherwise. Proof: As Er,0 1is the cochain complex Cr(Q2p−1;Z(p)), it follows that Er,0 2=Hr(Q2p−1;Z(p)) as claimed above. None of the simplices in X2p−1contribute anything to the odd rows between 0 and 2(p−1) of the above spectral sequence, from Corollary 10. Also from Corollary 10, the ones that contribute to rows of the form 4k−2, k∈Z+, are all listed in Figure 6. Let Abe the subcomplex of Qn generated by all of the simplices pictured in Figure 6 and let Bbe the subcomplex generated by just the simplices corresponding to dotted lines or hollow dots in Figure 6. Then the row s=4k−2ontheE1page of the spectral sequence is Cr(A, B;Z/p). Examining Figure 6 we see that Hr(A, B;Z/p)=Z/p r =2 0 otherwise. Consequently the E2page is as claimed for the rows s=4k−2. 116 C. A. Jensen Our final task is to calculate the E2page for the rows s=4k. Simplices in the p-singular locus of the spine with “at most dihedral symmetry” contribute to these rows. From Corollary 10, we have a characterization of such simplices. Define the subcomplex Mof the p-singular locus of the spine X2p−1of auter space to be the subcomplex generated by simplices with “at most dihedral symmetry”. More precisely, from Corollary 10, we know it is generated by the simplices corresponding to those in Figure 5 (i.e., corresponding in the sense that we are taking M to be a subcomplex of the spine rather than its quotient and Figure 5 is a picture in the quotient) in addition to simplices whose maximal vertex has underlying graph of the form Ξp∨Γp−1(where the forest collapses in the simplices respect the Z/p action on Ξp). Recall that an r-simplex with at most dihedral symmetry contributes exactly one Z/p to Er,4k 1, while all other simplices (those without dihedral or exactly Z/p symmetry) contribute nothing to this row. Let Nbe the subcomplex of Mgenerated by simplices in Mwhich do not have at most dihedral symmetry. Observe that none of the simplices in Nhave at most dihedral symmetry. Also note that the row E∗,4k 1is the relative cochain complex C∗(M/Aut(F2p−1),N/Aut(F2p−1); Z/p). Let Mbe the subcomplex of Mgenerated by Nand by simplices whose maximal vertex is Υ2 2p−1. Hence Mis the subcomplex consisting of Nand the bottom two thirds of Figure 5. There is an Aut(F2p−1)-equivariant deformation retraction of Monto M, given on the vertices of the poset by: •Contracting the spokes of the graph Ξpin Θp−1∨Ξp. •Contracting the spokes of the graph Ξpin Γp−1∨Ξp, where Γp−1 has no p-symmetry. •Contracting the poutward radiating edges attached to the p-gon in the center of the graph Υ1 2p−1. In the terminology used at the beginning of Section 4 while defining Υ1 2p−1, we are contracting the edges ei. That it is a deformation retraction follows from the Poset Lemma in [10] attributed to Quillen. As the homotopy retracting Mto Mis Aut(F2p−1)-invariant, it descends to a deformation retraction of M/Aut(F2p−1)toM/Aut(F2p−1). Hence the relative cohomology groups H∗(M/Aut(F2p−1),N/Aut(F2p−1); Z/p) Integral Cohomology of Moduli Spaces of Graphs 117 and H∗(M/Aut(F2p−1),N/Aut(F2p−1); Z/p) are isomorphic. Now referring to Figure 5, we see that Ht(M/Aut(F2p−1),N/Aut(F2p−1); Z/p)=0 for all tbecause we can contract all of the simplices in M/Aut(F2p−1) uniformly into N/Aut(F2p−1). An immediate consequence is Proof of Theorem 2: From [6], if m≥8k+3,then the standard map H4k(Aut(Fm+1); Z)→H4k(Aut(Fm); Z) is an isomorphism. Observe that H4k(Aut(F8k+3);Z)=H4k(Aut(F∞);Z) is a finitely generated abelian group. If it contains a torsion free summand isomorphic to Z, then we are done and H4k(Aut(F∞); Q)=0. Otherwise, choose a prime qsuch that 2q−1≥8k+ 3 and so that for all primes p≥qthere is no p-torsion in H4k(Aut(F8k+3); Z). We will show that H4k+1(Q2p−1;Z) has p-torsion for all primes p≥q, which will prove the theorem. Let p≥q. From the lemma above, if we use the standard equivariant spectral sequence to calculate H∗(Aut(F2p−1); Z(p)), then a class α∈ E2,4k−2 1in the E1-page survives at least until the E4k−1-page. Because H4k(Aut(F2p−1); Z) has no p-torsion and H4k(Aut(F2p−1); Q) = 0, we have H4k(Aut(F2p−1); Z(p)) = 0. Hence the class α∈E2,4k−2 1 cannot survive to the E∞page. It follows that there is p-torsion in E4k+1,0 4k−1. Recall that Er,0 1corresponds to the cellular chain complex with Z(p)coefficients for Q2p−1. The p-torsion in E4k+1,0 4k−1, therefore, would have to have been created when going from the E1to E2pages, because any of the torsion above the horizontal axis of the spectral sequence could not map onto a torsion free element on the horizontal axis. So H4k+1(Q2p−1;Z(p)) has p-torsion, and thus H4k+1(Q2p−1;Z) has p-torsion. References [1] G. Baumslag and T. Taylor, The centre of groups with one defining relator, Math. Ann. 175 (1968), 315–319. [2] K. S. Brown,“Cohomology of groups”, Graduate Texts in Mathematics 87, Springer-Verlag, New York-Berlin, 1982. [3] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84(1) (1986), 91–119. 118 C. 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Vogtmann, A generating function for the Euler characteristic of Out(Fn), in: “Proceedings of the Northwestern conference on cohomology of groups” (Evanston, Ill., 1985), J. Pure Appl. Algebra 44(1–3) (1987), 329–348. [13] J. Smillie and K. Vogtmann, Automorphisms of graphs, p-subgroups of Out(Fn) and the Euler characteristic of Out(Fn), J. Pure Appl. Algebra 49(1–2) (1987), 187–200. [14] R. G. Swan, The p-period of a finite group, Illinois J. Math. 4 (1960), 341–346. Department of Mathematics University of New Orleans New Orleans, LA 70148 U.S.A. E-mail address:[email protected] Rebut el 7 de mar¸c de 2001.