The equivariant cohomology of a 
-space 
is the ordinary cohomology
of the homotopy quotient space 
. It combines the topology of 
with the action of 
. For a point it gives the cohomology of the classifying
space 
. If a Lie group 
acts smoothly, freely, and properly
on a manifold 
, then 
is a principal bundle,
and the equivariant cohomology is the cohomology
of the orbit space (Meinrenken 2003, Theorems 1.21 and 2.11).
Equivariant Cohomology
See also
Cohomology, Group Action, Hikita Conjecture, Nakajima Quiver Variety, Quotient SpaceExplore with Wolfram|Alpha
References
Guillemin, V. and Sternberg, S. Supersymmetry and Equivariant de Rham Theory. Berlin, Germany: Springer-Verlag, 1999.Meinrenken, E. "Group Actions on Manifolds." Lecture Notes. Toronto, Canada: University of Toronto, 2003. https://www.math.toronto.edu/mein/teaching/LectureNotes/action.pdf.Cite this as:
Weisstein, Eric W. "Equivariant Cohomology." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EquivariantCohomology.html