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Cohomology at infinity and the well-rounded retract for general linear groups. (English) Zbl 0903.11016

Let \(GL_n(k)\) be the general linear group over an algebraic number field \(k\), and let \(\Gamma\) be a subgroup of finite index in \(GL_n({\mathcal O})\), where \({\mathcal O}\) denotes the ring of integers of \(k\). In dealing with the cohomology of \(\Gamma\) the Borel-Serre compactification \(\overline X/ \Gamma\) of the underlying locally symmetric space plays a major role. Its cohomology coincides (mod torsion) with the one of \(\Gamma\). Its boundary \(\partial (\overline X/ \Gamma)\) has a stratification given as the disjoint union of finitely many faces \(e'(P)\), one for each equivalence class mod \(\Gamma\) of parabolic \(\mathbb{Q}\)-subgroups of the \(\mathbb{Q}\)-group \(\text{Res}_{k/ \mathbb{Q}} GL_n(k)\) obtained by restriction of scalars. By use of the theory of Eisenstein series, one can give a description of the cohomology of \(\overline X/ \Gamma\) at infinity, i.e., those classes which restrict non-trivially to the cohomology of \(\partial (\overline X/ \Gamma)\). This is a description in terms of automorphic forms.
For the purpose of explicit computations another approach has turned out to be more useful. One looks for a deformation retract \(W\) of \(X\) by a \(\Gamma\)-invariant deformation retraction, so that \(W/ \Gamma\) is compact, and that \(\dim W\) equals the virtual cohomological dimension of \(\Gamma \). Then \(W\) can be given the structure of a cell complex on which \(\Gamma\) acts cellularly, and the cohomology of \(W/ \Gamma\) can be found combinatorially.
This paper extends the construction of \(W\) (called the well-rounded retract) in the case \(GL_n\) given by A. Ash [Duke Math. J. 51, 459-468 (1984; Zbl 0542.22011)] in the following sense: To each face in \(\overline X\) a closed subcomplex \(W'\) of \(W\) is associated. If \(\Gamma'\) denotes the stabilizer of \(W'\) in \(\Gamma\) then the inclusion \(W'/\Gamma\to W/\Gamma\) includes the same map on cohomology as the inclusion of the corresponding face \(e'(P)\) into \(\overline X/ \Gamma\). The main result says that there is a spectral sequence converging to \(H^* (\partial (\overline X/ \Gamma))\) whose \(E_1\) term is the direct sum of the cohomology of the various \(W'/ \Gamma'\). There is a map of spectral sequences, starting with a natural map from \(H^*(W/ \Gamma)\) to the \(E_1\) term, that induces the canonical restriction map \(H^* (\overline X/ \Gamma) \to H^* (\partial (\overline X/ \Gamma))\) up to canonical isomorphism. One can also find the maps on cohomology induced by the inclusion of a single face into \(\overline X/ \Gamma\) without any spectral sequences.
Each \(W'/ \Gamma'\) is a finite cell complex, so its cohomology can be worked out combinatorially. Observe that each \(W'\) is a subcomplex of \(W\), so that one can work with it in terms already available once one has computed \(W\) in a given case. The main technical work is done by exploring the connections between the well-rounded retraction and the geodesic action as it occurs in the work of Borel-Serre.

MSC:

11F75 Cohomology of arithmetic groups
22E41 Continuous cohomology of Lie groups
57T15 Homology and cohomology of homogeneous spaces of Lie groups

Citations:

Zbl 0542.22011
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References:

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