Information AboutEquivariant Cohomology |
| CATEGORIES ABOUT EQUIVARIANT COHOMOLOGY | |
| algebraic topology | |
| homotopy theory | |
| symplectic topology | |
| group actions | |
|
Specifically, given a group (discrete or not), a Topological Space and an action : equivariant cohomology determines a Graded Ring
the ''equivariant cohomology ring''. If is the Trivial Group , this is just the ordinary Cohomology Ring of , whereas if is Contractible , it reduces to the group cohomology of . OUTLINE CONSTRUCTION Equivariant cohomology can be constructed as the ordinary cohomology of a suitable space determined by and , called the ''homotopy Orbit Space '' : of 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
Construction of the homotopy orbit space The homotopy orbit space 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 . |
|
|