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In mathematics, equivariant cohomology is a theory from algebraic topology which applies to spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory.
Specifically, given a group (discrete or not), a topological space and an action
equivariant cohomology determines a graded ring
Equivariant cohomology can be constructed as the ordinary cohomology of a suitable space determined by and , called the homotopy orbit space
on . (The 'h' distinguishes it from the ordinary orbit space .)
If is the trivial group this space will turn out to be just itself, whereas if is contractible the space will be a classifying space for .
Properties of the homotopy orbit space
- If is a free action then
- If is a trivial action then
- In particular (as a special case of either of the above) if is trivial then
Borel construction of the homotopy orbit space
The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically correct” version of the orbit space (the quotient of by its -action) in which is first replaced by a larger but homotopy equivalent space so that the action is guaranteed to be free.
To this end, construct the universal bundle for and recall that has a free -action. Then the product —which is homotopy equivalent to since is contractible—has a “diagonal” -action defined by taking the -action on each factor: moreover, this action is free since it is free on . So we define the homotopy orbit space to be the orbit space of this -action.
This construction is denoted by
- Hazewinkel, Michiel, ed. (2001), "Equivariant cohomology", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Tu, Loring W. (March 2011). "What Is . . . Equivariant Cohomology?". AMS Notices 58 (03): 423–426.