Some notes on Equivariant Cohomology

Me taking notes on some stuff I'm learning about equivariant cohomology and localization
  1. References
  2. Primer on principal \(G\)-bundles, \(\mathbb EG\) and \(\mathbb BG\)
    1. Principal \(G\)-bundles
    2. The functor \(\mathrm{Bun}_G\)
    3. Construction of \(\mathbb BG\)
  3. Equivariant Cohomology
    1. The Borel Construction
    2. Equivariant cohomology of a point
    3. Examples with explanations

References

Primer on principal \(G\)-bundles, \(\mathbb EG\) and \(\mathbb BG\)

Our goal is to understand equivariant cohomology and localization in the category of algebraic varieties over \(\mathbb C\), but it's informative to first work in a more flexible category. We'll therefore first discuss principal \(G\)-bundles over topological spaces.

Principal \(G\)-bundles

Let \(G\) be a topological group. The category of \(G\)-spaces has objects given by pairs \((X,\sigma)\) where \(X\) is a topological space with a left or right \(G\) action \(\sigma:G\times X\to X\) (continuous, of course). Morphisms in this category are \(G\)-equivariant continuous maps \(\varphi: X\to Y\) which commute with the \(G\)-action of \(X\) and \(Y\). We require homotopies in this category to be equivariant as well with \(G\) acting trivially on \([0,1]\).

Definition: Let \(p:E\to S\) be a fiber bundle in the category of \(G\)-spaces where \(E\) has a right \(G\)-action and \(S\) has a trivial \(G\)-action[1]. We say \((E,p)\) is a principal \(G\)-bundle over \(S\) if \(p\) is \(G\)-equivariantly locally trivial; if \(S\) has a local trivialization \(\{S_i\}\) with \(G\)-equivariant homeomorphisms \(p^{-1}(S_i)\xrightarrow{\sim}S\times G\) where \((s,h)\cdot g := (s,hg)\).

[1] This means \(p\) is automatically equivariant
Remark: Since the action of \(G\) on \(S\times G\) is free and \(E\) is locally isomorphic to \(S\times G\), the action of \(G\) on \(E\) is necessarily also free when \((E,p)\) is a principal \(G\)-bundle. Furthermore there exists a canonical homeomorphism
\[\begin{aligned} E/G\xrightarrow{\sim} S. \end{aligned}\]
This consequently implies that every fiber of \(p\) is isomorphic to \(G\).

A map between principal \(G\)-bundles \(p:E\to X\) and \(q:F\to S\) is a continuous equivariant map \(\varphi:E\to F\) which commutes with projection to the base space.

Remark: Every morphism \(\varphi:(E,p)\to (F,q)\) of principal \(G\)-bundles over a space \(S\) is an isomorphism. Thus the category of principal \(G\)-bundles over \(S\) is a groupoid.

The functor \(\mathrm{Bun}_G\)

Suppose we have a continuous map \(\alpha:S'\to S\) and a principal \(G\)-bundle \((E,p)\). Pullback via \(\alpha^*\) gives us a principal \(G\)-bundle over \(S'\),

\[\begin{aligned} \alpha^*E = \{(e,s')\in E\times S' ~\mid~ \alpha(s') = p(e)\}, \end{aligned}\]

and it comes equipped with an equivariant map \(\alpha^*E\to E\) given by projection onto the first coordiante. This means we can define a functor

\[\begin{aligned} \mathrm{Bun}_G:\mathsf{Top}^{op} \to \mathsf{Set} \end{aligned}\]

given by sending a space \(S\) to the set of isomorphism classes of principal \(G\)-bundles over \(S\). It's contravariant because we send the isomorphism class of a bundle \(E\) to \(\alpha^*E,\) reversing the direction of the arrows.

However, pullbacks of homotopic maps are isomorphic:

Proposition: If \(\alpha,\beta:S'\to S\) are homotopy equivalent, then \(\alpha^*E\cong \beta^*E\).

...which means we can upgrade \(\mathrm{Bun}_G\) by only defining it up to homotopy:

\[\begin{aligned} \mathrm{Bun}_G:\mathsf{HTop}^{op} \to \mathsf{Set}. \end{aligned}\]

This functor is represented by a topological space \(\mathbb BG\): \(\mathrm{Bun}_G(-) = \operatorname{Hom}(-,\mathbb BG)\). In other words, specifying a principal \(G\)-bundle \(p:E\to S\) up to isomorphism is equivalent to specifying a continuous map \(S\to \mathbb BG\).

Construction of \(\mathbb BG\)

Fill in later

Equivariant Cohomology

The Borel Construction

Let \(G\) be a complex linear algebraic group ("complex" here meaning that it is a complex algebraic variety) and let \(X\) be a complex algebraic group with a left \(G\)-action. Define

\[\begin{aligned} \mathbb EG \times^G X:= (\mathbb EG\times X)/(e\cdot g, x)\sim (e,g\cdot x). \end{aligned}\]

We then define the equivariant cohomology of \(X\) to be

\[\begin{aligned} H^i_G(X) := H^i(\mathbb EG\times^G X), \end{aligned}\]

where the right term is singular cohomology. The point of using \(\mathbb EG \times^G X\) instead of the orbit space \(X/G\) is that \(\mathbb EG \times X\) is homotopy equivalent to \(X\) since \(\mathbb EG\) is contractible but the \(G\) action on \(\mathbb EG\times X\) is free. Essentially this "fixes" the action of \(G\) on \(X\) by making it free without changing the homotopy type. In particular, if the \(G\)-action on \(X\) was already free, then \(H^i_G(X)\) is just the singular cohomology \(H^i(X/G)\).

Equivariant cohomology of a point

In the special case that \(X = \{pt\}\), then \(\mathbb EG \times^G X = \mathbb EG/G = \mathbb BG\), and so

\[\begin{aligned} H^i_G(\mathsf{pt}) = H^i(\mathbb BG). \end{aligned}\]

There are two things to note:

  1. Very often \(\mathbb BG\) has nontrivial topology, so \(H^*_G(\mathsf{pt}) \neq \mathbb Z\) in most cases. This is a key feature of equivariant cohomology.

  2. The ring \(H^*_G(\mathsf{pt}) = H^*(\mathbb BG)\) is typically interpreted as the ring of characteristic classes for principal \(G\)-bundles.

Examples with explanations

Example: Let \(G = \mathbb C^*\). In this case \(\mathbb EG = \mathbb C^\infty\setminus \{0\}\) and so \(\mathbb BG = \mathbb EG/G = \mathbb P^\infty\). Recall that
\[\begin{aligned} H^i(\mathbb P^\infty_{\mathbb C}) = \begin{cases} 0 & \text{$i$ is odd} \\ \mathbb Z & \text{$i$This has cohomology of $\mathbb Z$ in each even degree and trivial cohomology in every odd degree. As a ring, it is generated by the is even} \end{cases} \end{aligned}\]
and that the ring \(H^*(\mathbb P^\infty_{\mathbb C})\) is generated in degree 2. We then see that
\[\begin{aligned} \mathbb ^*_{\mathbb C^*}(\mathsf{pt}) = H^*\mathbb P^\infty_{\mathbb C} \cong \mathbb Z[t] \end{aligned}\]
where \(t = c_1(\mathcal O_{\mathbb P^\infty}(-1)\) is the first Chern class of the tautological bundle.
©Isaac Martin. Last modified: January 30, 2024.