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Sheaves on fixed point sets and equivariant cohomology. (English) Zbl 0873.55006

For \(G\) a finite group, the author interprets the ordinary equivariant cohomology groups of a paracompact \(G\)-space \(X\) with coefficients in a contravariant coefficient system, in terms of the cohomology of a suitable Grothendieck topos, whose objects are families of sheaves on the fixed point sets \(X^K\) for all subgroups \(K\) of \(G\). As an application, in the final section of the paper, he considers the cohomology theory of topoi and obtains a spectral sequence associated to a \(G\)-map \(f:X\to Y\). In case \(Y\) is a point this gives a spectral sequence whose \(E_2\)-term depends on the non-equivariant cohomology groups \(\overline H^q(X^K;m(G/H))\), converging to the equivariant cohomology \(\overline H^n_G(X;m)\). In particular, if \(G\) acts freely on \(X\), this spectral sequence reduces to the Cartan-Leray spectral sequence of the covering space \(X\to X\smallsetminus G\).
The author also proves that if \(f: X\to Y\) is a \(G\)-fibration with \(Y\) a \(G\)-CW complex then the \(E_2\)-term of the spectral sequence of \(f\) is the cohomology of the topos associated to \(Y\) with coefficients in a family of locally constant sheaves whose stalks are isomorphic to the cohomology of the fixed point sets of the fiber of \(f\). Thus the spectral sequence can be regarded as an equivariant Serre spectral sequence.

MSC:

55N91 Equivariant homology and cohomology in algebraic topology
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