Equivariant cohomology

From Conservapedia

This article or section needs to be written in plain English, using plain English that most of our readers can understand. Articles that depend excessively on technical terms accessible only to specialists are useless for our purposes, so writers are admonished to avoid jargon
This article has been nominated for deletion. See Conservapedia:AFD Equivariant_cohomology

Borel Model

Suppose that a Lie group $G$ acts on a manifold $M$ . We would like to form a good notion of a quotient $M / G$ . The problem is that if $G$ has fixed points, the quotient space $M / G$ will likely no longer be a manifold. However, if we could find a contractible space $E G$ on which $G$ acted freely, then $M\times EG$ would have the same homotopy type as $M$ , and the induced $G$ action on the product would be free.

It is an interesting fact that such an $E G$ always exists. For example, $G = \mathbb{Z}$ acts freely via translations on the contractible space $\mathbb{R}$ , so $E\mathbb{Z} = \mathbb{R}$ . In general, however, the $E G$ s are infinite-dimensional spaces. For example, for $G = \mathbb{Z}_2$ , $EG = S^\infty$ , and similarly for $G = S 1$ .

Thus $M_G = (M\times EG) /G$ is usually an infinite-dimensional manifold. There is a natural projection map from $M G$ onto $B G = E G / G$ which realizes $M G$ as a fiber bundle over $B G$ with fiber $M$ .

We define the equivariant cohomology $H^*_G(M) = H^*(M_G)$ . Note first that if the action is free, then $M_G = M/G \times EG$ , so the equivariant cohomology coincides with $H * (M / G)$ . On the other side of the spectrum, if the $G$ -action is trivial, then $M_G = M\times BG$ , and therefore $H^*_G(M) = H^*(M)\otimes H^*(BG)$ . For example, if $G = S 1$ , $BG = S^\infty/S^1$ , which is just $\mathbb{C}P^\infty$ . Hence, the equivariant cohomology (with complex coefficients) is $H^*(M)\otimes\mathbb{C}[u]$ . Thus, in some sense, the presence of large stabilizers of the group action get measured by the size of the equivariant cohomology group.

Localization Theorem

Let us now restrict ourselves to the case where $G = S 1$ . Now the natural projection map onto a point: $M\rightarrow p$ gives rise to a morphism $H^*_G(p)\rightarrow H^*_G(M)$ , and in this way makes $H^*_G(M)$ a $\mathbb{C}[u]$ -module. Clearly if $F$ is a fixed component of $M$ , the $\mathbb{C}[u]$ -module $H^*_G(F)$ is simply $H^*(F)\otimes\mathbb{C}[u]$ .

The interesting fact is that if we make $H^*_G(M)$ into a $\mathbb{C}(u)$ -module via localization, then this localized cohomology group is isomorphic to $\bigoplus_F H^*(F)\otimes\mathbb{C}(u)$ , where the sum runs over the fixed components of $M$ . This means that, modulo torsion, we can understand the equivariant cohomology completely from the fixed point data.

Equivariant cohomology

From Conservapedia

Borel Model

Localization Theorem

Views

Personal tools

Search

Popular Links

Help

World History

edit console