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Homotopy Quotients and Equivariant Cohomology
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This chapter investigates two candidates for equivariant cohomology and explains why it settles on the Borel construction, also called Cartan's mixing construction. Let G be a topological group and M a left G-space. The Borel construction mixes the weakly contractible total space of a principal bundle with the G-space M to produce a homotopy quotient of M. Equivariant cohomology is the cohomology of the homotopy quotient. More generally, given a G-space M, Cartan's mixing construction turns a principal bundle with fiber G into a fiber bundle with fiber M. Cartan's mixing construction fits into the Cartan's mixing diagram, a powerful tool for dealing with equivariant cohomology.
Title: Homotopy Quotients and Equivariant Cohomology
Description:
This chapter investigates two candidates for equivariant cohomology and explains why it settles on the Borel construction, also called Cartan's mixing construction.
Let G be a topological group and M a left G-space.
The Borel construction mixes the weakly contractible total space of a principal bundle with the G-space M to produce a homotopy quotient of M.
Equivariant cohomology is the cohomology of the homotopy quotient.
More generally, given a G-space M, Cartan's mixing construction turns a principal bundle with fiber G into a fiber bundle with fiber M.
Cartan's mixing construction fits into the Cartan's mixing diagram, a powerful tool for dealing with equivariant cohomology.
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