Equivariant cohomology and analytic descriptions of ring isomorphisms

Details Equivariant cohomology and analytic descriptions of ring isomorphisms Equivariant cohomology and analytic descriptions of ring isomorphisms

2007-03-05

arxiv.org/abs/math/0703083v3

Math. Z. 261 (2009), 891-908.

Mathematics

Algebraic Topology

20 pages, updated version with two references added

Scientific paper

10.1007/s00209-008-0357-y

In this paper we consider a class of connected closed $G$-manifolds with a non-empty finite fixed point set, each $M$ of which is totally non-homologous to zero in $M_G$ (or $G$-equivariantly formal), where $G={\Bbb Z}_2$. With the help of the equivariant index, we give an explicit description of the equivariant cohomology of such a $G$-manifold in terms of algebra, so that we can obtain analytic descriptions of ring isomorphisms among equivariant cohomology rings of such $G$-manifolds, and a necessary and sufficient condition that the equivariant cohomology rings of such two $G$-manifolds are isomorphic. This also leads us to analyze how many there are equivariant cohomology rings up to isomorphism for such $G$-manifolds in 2- and 3-dimensional cases.

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