On the cohomology of Coxeter groups and their finite parabolic subgroups (Q1903380)

The author relates the cohomology of a Coxeter group to the cohomology of its finite parabolic subgroups. If \((W,S)\) is a Coxeter system and if \(T\subseteq S\) let \(W_T\) denote the (parabolic) subgroup generated by \(T\). For any \(W\)-module \(A\), let \({\mathcal H}^* (W,A)=\varprojlim H^* (W_F, A)\), where \(F\) ranges over the poset \({\mathcal F}=\{F\subseteq S\mid W_F\) is finite\}, and let \(\rho_A\) be the natural homomorphism \(H^*(W,A)\to {\mathcal H}^* (W,A)\) induced by restriction maps. The main result are the following: Theorem 1: Let \(A=k\) be a commutative ring with unity, with the trivial \(W\)-action. Then \(\rho=\rho_k\) is a ring homomorphism and it satisfies the following two properties: (i) If \(u\in\text{ker }\rho\), then \(u\) is nilpotent. (ii) Suppose that \(k\) is a field of characteristic \(p > 0\). For every \(v\in {\mathcal H}^* (W,k)\), there is an integer \(n\geq 0\) such that \(v^{p^n}\in\text{im }\rho\). A Coxeter group \((W,S)\) is said to be aspherical if every three distinct elements of \(S\) generate a parabolic subgroup of infinite order. Theorem 2: If \(W\) is an aspherical Coxeter group, then \(\rho_A\) is surjective for any abelian group \(A\) (with the trivial \(W\)-action). Since the mod 2 cohomology ring of any finite Coxeter group has no nilpotent elements, the author deduces: Corollary: For any Coxeter group \(W\), \(\rho\) induces a monomorphism \(H^* (W,\mathbb{Z}/2)/\sqrt 0\to{^*(W,\mathbb{Z}/2)}\), (where \(\sqrt 0\) denotes the nilradical), which is an isomorphism if \(W\) is aspherical. The main tool of the paper is a spectral sequence derived from the action of a Coxeter group on a contractible complex due to M. W. Davis. The author points out that the results remain valid if ordinary cohomology is replaced by Farrell cohomology.