## 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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