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On the cohomology of SL$_n(\mathbb{Z})$

Avner Ash

Abstract

Let St denote the Steinberg module of $SL_n(Q)$ tensored with Q. Let Sh denote the sharbly resolution of St. By Borel-Serre duality, $H^{n(n-1)/2-i}(SL_n(Z),Q)$ is isomorphic to $H_i(SL_n(Z),St)$. The latter is isomorphic to the homology of the $SL_n(Z)$-coinvariants of Sh. We produce nonzero classes in $H_i(SL_n(Z),St)$ for certain small $i$ in terms of sharbly cycles and cosharbly cocycles.

On the cohomology of SL$_n(\mathbb{Z})$

Abstract

Let St denote the Steinberg module of tensored with Q. Let Sh denote the sharbly resolution of St. By Borel-Serre duality, is isomorphic to . The latter is isomorphic to the homology of the -coinvariants of Sh. We produce nonzero classes in for certain small in terms of sharbly cycles and cosharbly cocycles.
Paper Structure (10 sections, 19 theorems, 64 equations)

This paper contains 10 sections, 19 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. The Steinberg module and the sharbly resolution
  3. Composition
  4. Product
  5. Starter cases: $n=1,3$
  6. Parity
  7. Next step: $n=4$
  8. Some cases that don't compose well: $n=2,5$
  9. Next cases: $n=6,7$
  10. Larger $n$

Key Result

Theorem 1.1

Let $k\ge0$. Then (1) If $n=3k+3$, $H^{\nu_n-n}(\mathrm{SL}_{n}(\mathbb{Z}),\mathbb{Q})\ne0$; (2) If $n=3k+4$, $H^{\nu_n-(n-1)}(\mathrm{SL}_{n}(\mathbb{Z}),\mathbb{Q})\ne0$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 3.1
  • ...and 37 more