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The Farrell-Tate and Bredon homology for \(\operatorname{PSL}_4(\mathbb{Z})\) via cell subdivisions. (English) Zbl 1428.11097
There are two competing notions of cellular group action on a CW-complex \(X\). The weaker notion imposes that the group action respects the cell structure of \(X\). The stronger and more common notion includes the additional assumption that if a group element stabilizes a cell, then it stabilizes the cell pointwise. In the article under review the latter actions are called rigid. The authors present subdivision algorithms which allow to refine a cell complex with a (weakly) cellular action into a rigid one, with the aim of avoiding the creation of many new cells.
The research is motivated by computations of the cohomology of arithmetic groups. To compute the homology of an arithmetic group \(\Gamma\) it is usually sufficient to have a cell structure on the associated symmetric space such that the action of \(\Gamma\) is (weakly) cellular. The authors argue that the computation of Bredon homology and Farrell-Tate homology requires rigid cell structures. To this end they describe three algorithms which produces suitable subdivisons and briefly discuss their advantages and disadvantages.
The methods are applied to compute some examples. The authors discuss the Farrell-Tate cohomology modulo \(2\) of \(\mathrm{SL}_3(\mathbb{Z})\), and the Farrell-Tate cohomology modulo \(3\) and \(5\) of \(\mathrm{Sp}_4(\mathbb{Z})\). In particular, they describe the Bredon homology and the Farrel-Tate cohomology modulo \(3\) and \(5\) of \(\mathrm{PSL}_4(\mathbb{Z})\).
11F75 Cohomology of arithmetic groups
Full Text: DOI
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