Group actions on algebraic cell complexes. (English) Zbl 1216.55007

It is well known that a \(G\)-map \(f:X\to Y\) between \(G\)-complexes which induces a homotopy equivalence \(X^H\to Y^H\) between the \(H\)-fixed point sets for every subgroup \(H\leq G\), is a \(G\)-homotopy equivalence. In this article, an algebraic version of that result is proved.
Given a \(G\)-set \(\Delta\) and a ring \(S\), \(S\Delta\) denotes the free \(S\)-module on \(\Delta\). An \(SG\)-map \(S\Delta \to S\Delta '\) is called admissible if it maps \(S[\Delta ^H]\) into \(S[\Delta '{}^H]\) for every \(H\leq G\). A special \(SG\)-complex consists of a sequence \(\Delta _0, \Delta _1, \dots\) of \(G\)-sets together with admissible maps \(S\Delta _i \to S\Delta _{i-1}\) which define a chain complex. The main result of the paper states that an admissible map between special \(SG\)-complexes is an \(SG\)-chain homotopy equivalence (between the augmented chain complexes) provided that it induces chain homotopy equivalences between the complexes of free \(S\)-modules on the \(H\)-fixed point sets of the \(G\)-sets.
As a consequence, the authors obtain a generalization of a result of S. Bouc [J. Algebra 220, 415–436 (1999; Zbl 0940.55020)] originally formulated for simplicial complexes, which claims that if \(G\) is finite, \(S\) does not have \(|G|\)-torsion and \(X\) is a finite-dimensional \(S\)-acyclic special \(SG\)-complex, then (the augmented chain complex of) \(X\) is \(SG\)-split. Some interesting alternative proofs of known results are shown as an application of this version of Bouc’s theorem.


55U15 Chain complexes in algebraic topology
57Q05 General topology of complexes
20J05 Homological methods in group theory


Zbl 0940.55020
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