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The Codimension-One Cohomology of Sln Z

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The Codimension-One Cohomology of Sln Z The codimension-one cohomology of SLn Z Thomas Church and Andrew Putman∗ August 2, 2015 Abstract ( ) n −1 n (2) Z Q We prove that H (SLn ; ) = 0, where 2 is the cohomological dimension of SLn Z, and similarly for GLn Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GLn. These theorems are derived from a presentation of the Steinberg module for SLn Z whose generators are integral apartment classes, generalizing Manin's presentation for the Steinberg module of SL2 Z. This presentation was originally constructed by Bykovskii. We give a new topological proof of it. 1 Introduction The cohomology of SLn Z plays a fundamental role in many areas of mathematics. The Borel k Stability Theorem [Bo] determines H (SLn Z; Q) when k is sufficiently small (conjecturally, for k < n − 1). However, little is known outside this stable range. Recall that if Γ is a virtually torsion-free group, the virtual cohomological dimension of Γ is vcd(Γ) := maxfk j Hk(Γ; V ⊗ Q) =6 0 for some Γ-module V g: Borel{Serre [BoSe] proved that ( ) n vcd(SL Z) = vcd(GL Z) = : n n 2 ( ) Z n The cohomology of SLn in degrees near 2 is thus the \most unstable" cohomology. In (n) 1976, Lee{Szczarba [LSz, Theorem 1.3] proved that H 2 (SLn Z; Q) = 0. This vanishing was (n) recently extended to H 2 (SLn Z; Vλ) = 0 for rational representations Vλ of the algebraic group GLn by Church{Farb{Putman [CFP2]. n Our main theorem concerns the cohomology in codimension 1. For λ = (λ1; : : : ; λn) 2 Z ≥ · · · ≥ Q with λ1 P λn, let Vλ be the rational representation of GLn with highest weight λ. k k n − Define λ = i=1(λi λn). Theorem A (Codimension-one Vanishing Theorem). For any rational representa- tion Vλ of GLn Q, we have (n)−1 (n)−1 H 2 (SLn Z; Vλ) = H 2 (GLn Z; Vλ) = 0 for all n ≥ 3 + kλk:: In particular, (n)−1 (n)−1 H 2 (SLn Z; Q) = H 2 (GLn Z; Q) = 0 for all n ≥ 3: ∗The first author was supported by NSF grants DMS-1103807 and DMS-1350138. The second author was supported by NSF grant DMS-1255350 and the Alfred P. Sloan Foundation. 1 Remark 1.1. Theorem A proves the k = 1 case of a conjecture of Church{Farb{Putman (n)−k [CFP1] which asserts that for all k ≥ 0, we have H 2 (SLn Z; Q) = 0 for n > k + 1. (n)−k Steinberg module. To compute H 2 (SLn Z; Q) for small k, the crucial object to un- derstand is the Steinberg module, which we now discuss. The Tits building Tn for GLn Q is the simplicial complex whose p-simplices are flags of subspaces n 0 ( V0 ( ··· ( Vp (Q : By the Solomon-Tits theorem, Tn is homotopy equivalent to an infinite wedge of (n − 2)- dimensional spheres. The group GLn Q naturally acts on Tn by simplicial automorphisms, and the Steinberg module for GLn Q is the GLn Q-module e Stn := Hn−2(Tn; Z): Borel{Serre duality. While SLn Z does not satisfy Poincar´eduality, Borel{Serre [BoSe] proved that it does satisfy virtual Bieri{Eckmann duality with rational dualizing module Stn. This means that for any Q SLn Z-module V and any k ≥ 0, we have ( ) ( ) (n)−k ∼ H 2 SLn Z; V = Hk SLn Z; Stn ⊗V : (1) A similar result holds for GLn Z (with the same dualizing module Stn). Presentation of Stn. To prove Theorem A, we compute the right-hand side of (1) using a presentation of the GLn Z-module Stn. The following theorem was originally proved by Bykovskii [By] and generalizes the presentation of St2 given by Manin in 1972 [Man, Theorem 1.9]. The second purpose of this paper is to offer a new proof of it. Theorem B (Presentation of Stn). For n ≥ 2, the Steinberg module Stn is the abelian n group with generators [v1; : : : ; vn], one for each ordered basis fv1; : : : ; vng of Z , subject to the following three families of relations. R1. [v1; v2; v3; : : : ; vn] = [v1; v1 + v2; v3; : : : ; vn] + [v1 + v2; v2; v3; : : : ; vn]. R2. [v1; v2;:::; vn] = [v1; v2; : : : ; vn] for any choices of signs. σ R3. [vσ(1); vσ(2); : : : ; vσ(n)] = (−1) · [v1; v2; : : : ; vn] for any σ 2 Sn. The GLn Z-action on Stn is defined by M · [v1; : : : ; vn] = [M · v1;:::;M · vn]. The relation R2 can be replaced by the single relation [−v1; v2; : : : ; vn] = [v1; v2; : : : ; vn]. Relations R1 and R2 then involve only the first two vectors; these relations are the \stabi- lization" of Manin's relations for St2 from GL2 Z to GLn Z. In this light, what Theorem B says is that no additional relations are needed to present Stn, once the permutations Sn are taken into account. Apartment classes and Bykovskii's proof of Theorem B. The general theory of spherical buildings automatically provides a generating set for Stn. Namely, every rational n n−2 basis fw1; : : : ; wng for Q determines a spherical apartment in Tn homeomorphic to S e whose fundamental class determines an apartment class [w1; : : : ; wn] 2 Hn−2(Tn; Z) = Stn, and the general theory implies that Stn is generated by these rational apartment classes. However, the generating set in Theorem B is much smaller: it consists only of the integral n apartment classes [v1; : : : ; vn], i.e. those for which fv1; : : : ; vng is an integral basis for Z . 2 That Stn is generated by these integral apartment classes was proved by Ash{Rudolph [AR] in 1979. To do this, they gave an algorithm for expressing an arbitrary rational apartment class as a sum of integral apartment classes. Bykovskii proved Theorem B by carefully examining Ash{Rudolph's algorithm, which requires making many arbitrary choices, and showing that the only ambiguity in its output comes from the relations in Theorem B. We remark that from this perspective, Theorem B appears as the integral analogue of Lee{Szczarba's presentation of Stn as a GLn Q-module [LSz]. Our proof of Theorem B. Our proof of Theorem B is quite different. It is inspired by our alternate proof of Ash{Rudolph's theorem in Church{Farb{Putman [CFP2] and by Manin's original proof of Theorem B for St2. We use topology to show directly that the homology of Tn is generated by integral apartment classes; non-integral apartment classes never show up in our proof. The key is the complex of partial augmented frames for Zn defined below, which provides an \integral model" for the Tits building Tn. We begin first with the more familiar complex of partial frames. Definition 1.2. Let V be a finite-rank free abelian group. • A line in V is a 2-element set fv; −vg of primitive vectors in V ; we denote it by v. • f g f g A frame for V is a set v1 ; : : : ; vn of lines such that v1; : : : ; vn is a basis for V . • A partial frame for V is a frame for a direct summand of V , or equivalently a set of lines in V that can be completed to a frame for V . n The complex of partial frames for Z , denoted Bn, is the simplicial complex whose p- simplices are partial frames for Zn of cardinality (p + 1). The complex Bn is (n − 1)-dimensional, and Maazen [Maa] proved that Bn is (n − 2)- connected. This connectivity is what we used in [CFP2] to prove Ash{Rudolph's theorem on generators for Stn. However, to obtain a presentation for Stn this is not enough; we need to attach higher-dimensional cells to Bn to improve its connectivity. Improving connectivity: the complex of partial augmented frames. To motivate the cells we add, we recall how Manin found his presentation for St2. The simplicial complex B2 is the following graph. • The vertices are lines (a; b), where (a; b) 2 Z2 is a primitive vector. • Vertices (a; b) and (c; d) are joined by an edge exactly when f(a; b); (c; d)g is a basis for Z2, or equivalently when ad − bc = 1. This is exactly the classical Farey graph; see Figure 1. This graph is connected, but not e simply-connected. The reduced 0-chains C0(B2; Z) can be identified (essentially by defini- e tion) with St2 = H0(T2; Z), and it is not hard to see that the 1-chains C1(B2; Z) have a presentation just like that in Theorem B, but where we impose only the relations R2 and R3. What this means is that the first homology H1(B2; Z) measures exactly the additional relations beyond R2 and R3 needed to present St2. As is clear from Figure 1, the group H1(B2; Z) is spanned by the boundaries of the evident triangles in the Farey graph. Under our identification of B2 with the Farey graph, f g f g these triangles are formed by triples of vertices v1 ; v2 ; v3 such that v1; v2 is a basis for 2 Z and v1 v2 v3 = 0 for some choice of signs (in which case fv2; v3g and fv1; v3g are also 2 bases for Z ). Reordering and changing the signs of the vi, we can assume that v3 = v1 +v2. f g The relation in St2 corresponding to the boundary of the triangle v1 ; v2 ; (v1 + v2) is [v1; v1 + v2] − [v1; v2] + [v1 + v2; v2] = 0; 3 1=0 −2=1 2=1 −1=1 1=1 −1=2 1=2 0=1 Figure 1: The complex B2 is isomorphic to the Farey graph under the identification that takes the 2 Z2 a 2 Q [ f1g line (a; b) to b .
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