Localization is a powerful tool in computations involving cohomology and characteristic classes, in particular in calculating Gromov-Witten invariants. The actual localization theorem has always seemed like a bit of magic to me. This series of posts is an attempt to understand some of the surrounding ideas so I can isolate the part that seems like “magic”. In the first post, I’ll talk about equivariant cohomology, and in the next, about localization.

In what follows, let $X$ be a manifold, and $G$ a Lie group acting on $X$ (by diffeomorphisms). If the $G$-action is such that $X/G$ is a manifold, then it has a cohomology ring $H^\ast(X/G)$. When the quotient does not exist, there is still a way to associate a cohomology ring to it. One of the ways to do this, as we shall see, is equivariant cohomology.

Given a Lie group $G$, there is a principal $G$-bundle $\pi: EG \to BG$, where $EG$ is a weakly contractible space (i.e. all its homotopy groups are trivial) on which $G$ acts freely, and $BG$ is called the classifying space of $G$. This bundle is called the universal bundle. I won’t go into the construction of any of these here, but it suffices to say that they exist, and have the following universal property: whenever there is a principal $G$-bundle $P \to M$, there is a map $\phi: M \to BG$, unique up to homotopy, such that $P$ is the pullback of $E$ under $\phi$. Thus $BG$ classifies all principal $G$-bundles.

Now suppose $G$ acts on $X$ as above. Since the action need not be free, we will construct a new space, denoted by $X_G$ and called the Borel space, which in some way is supposed to capture the topology of $X/G$. It is defined as

\[X_G := EG \times_G X = (EG \times X)/G,\]

where the action on $EG \times X$ is the diagonal action: $g\cdot(e, x) = (e\cdot g, g^{-1}\cdot x)$. This action is free since the action on $EG$ is free, and so the definition makes sense as a topological space.

The projection onto the first factor induces a map $p: X_G \to BG$ which is in fact a fiber bundle (where our definition of a fiber bundle is liberal enough to allow the base to not be a manifold, since $BG$ need not be a manifold) with fiber $X$: the fiber over a point $b$ is $\pi^{-1}(b) \times_G X$ with the diagonal action, and hence diffeomorphic to $X$. Moreover, when the $G$-action on $X$ is free, the projection onto the second factor induces a map $X_G \to X/G$ with fiber $EG$ and therefore a homotopy equivalence. In that case $H^\ast(X/G)\simeq H^\ast(X_G)$, and that motivates our definition of equivariant cohomology: the equivariant cohomology of $X$ is defined to be \(H^\ast_G(X) := H^\ast(X_G).\)

I will now work out an example in some detail. Let $X = \mathbb{C}\mathbb{P}^n$ be the $n$-dimensional complex projective space, and let $G = \mathbb{T} = (\mathbb{C}^\ast)^{n+1}$ be the $(n+1)$-dimensional torus whose action on $X$ is induced from the action on $\mathbb{C}^{n+1}: (t_0, \ldots, t_n)\cdot(x_0, \ldots, x_n) = (t_0x_0, \ldots, t_nx_n)$.

Now the classifying space of $\mathbb{C}^\ast$ is $B\mathbb{C}^\ast \simeq \mathbb{C}\mathbb{P}^\infty$, while the total space of the universal bundle $E\mathbb{C}^\ast$ can be thought of as the space of all nonzero vectors in some separable Hilbert space. We know that $H^\ast(\mathbb{C}\mathbb{P}^\infty) \simeq \mathbb{Q}[\alpha]$, where $\alpha$ is the first Chern class of the hyperplane bundle $O_{\mathbb{C}\mathbb{P}^\infty}(1)$ (analogous to the situation for a finite dimensional projective space). Thus the equivariant cohomology of a point $H^\ast_{\mathbb{C}^\ast}(pt.)$, which by definition is $H^\ast(B\mathbb{C}^\ast)$, is $\mathbb{Q}[\alpha]$. Observe how the cohomology ring is torsion-free when the entire space (a point) is fixed under the group action.

Similarly, $B\mathbb{T} \simeq (\mathbb{C}\mathbb{P}^\infty)^{n+1}$ and we have $H^\ast_\mathbb{T}(pt.) \simeq \mathbb{Q}[\alpha_0, \ldots, \alpha_n]$ where $\alpha_i = c_1(L_i)$, the first Chern class of the line bundle $O(1)$ pulled back from the $i^{th}$ factor. For the above action of $\mathbb{T}$ on $X$, the Borel space $X_\mathbb{T}$ is a $\mathbb{C}\mathbb{P}^n$ bundle over $B\mathbb{T}$, which is actually just the projective bundle $\mathbb{P}(\oplus_i L_i^\ast)$, since a fiber above $(l_0, \ldots, l_n)$ in the latter consists of equivalence classes $[v_0, \ldots, v_n]$ where $v_i \in l_i$. Since $X_\mathbb{T}$ is a projective bundle over $B\mathbb{T}$, there is a tautological line bundle $O_{X_\mathbb{T}}(-1)$ over $X_\mathbb{T}$, and we have the tautological exact sequence on $X_\mathbb{T}$:

\[0 \to O_{X_\mathbb{T}}(-1) \to p^\ast(\oplus_i {L}^\ast_i) \to Q \to 0,\]

where $Q$, the tautological quotient bundle, is by definition the cokernel of the second map. Letting $H = c_1(O_{X_\mathbb{T}}(1))$ and using $\alpha_i$ to denote $p^\ast\alpha_i$, the above sequence tells us that $(H-\alpha_0)\cdots(H-\alpha_n) = c_{n+1}(p^\ast(\oplus_i {L}^\ast_i\otimes O_{X_\mathbb{T}}(1))) = \sum_j c_j(O_{X_\mathbb{T}})c_{n+1-j}(Q\otimes O_{X_\mathbb{T}}(1)) = 0$. Therefore we have

\[H^\ast_{\mathbb{T}}(\mathbb{P}^n) \simeq \raise{.2em}{\mathbb{Q}[H, \alpha_0, \ldots, \alpha_n]}\left/\raise{-.2em}{\langle(H-\alpha_0)\cdots(H-\alpha_n)\rangle}\right.\]

The map induced by $p:X_\mathbb{T} \to B\mathbb{T}$ on cohomology is

\[\begin{aligned} p^\ast: \mathbb{Q}[\alpha_0, \ldots, \alpha_n] &\to \raise{.2em}{\mathbb{Q}[H, \alpha_0, \ldots, \alpha_n]}\left/\raise{-.2em}{\langle(H-\alpha_0)\cdots(H-\alpha_n)\rangle}\right.\\ \alpha_i &\mapsto \alpha_i \end{aligned}\]

The inclusion of a fiber $\iota: \mathbb{P}^n \hookrightarrow \mathbb{P}^n_\mathbb{T}$ induces a map $\iota^\ast:H^\ast_\mathbb{T}(\mathbb{P}^n) \to H^\ast(\mathbb{P}^n) \simeq \mathbb{Q}[H]/\langle H^{n+1}\rangle$ which maps $H$ to $H$ and each $\alpha_i$ to 0. A choice of preimage of a cohomology class of $\mathbb{P}^n$ under this map is called a lift. The action on $\mathbb{P}^n$ has $n+1$ fixed points, say $p_0, \ldots, p_n$. The inclusion of each of these induces an inclusion of Borel spaces $\iota_j:B\mathbb{T} \to X_\mathbb{T}$ which is actually a section of the map $p$. The induced maps on equivariant cohomology are

\[\begin{aligned} \iota_j^\ast: \raise{.2em}{\mathbb{Q}[H, \alpha_0, \ldots, \alpha_n]}\left/\raise{-.2em}{\langle(H-\alpha_0)\cdots(H-\alpha_n)\rangle}\right. &\to \mathbb{Q}[\alpha_0, \ldots, \alpha_n]\\ \alpha_i &\mapsto \alpha_i\\ H &\mapsto \alpha_j \end{aligned}\]

So we see that each fixed point $p_j$ corresponds in some sense to $\alpha_j$. Since each fixed point gives a section, and the class $H$ (which is the only torsion class) lives on the fiber, the fixed points don’t see the torsion in cohomology. The $\alpha_j$’s are torsion-free and we have the informal correspondence

\[\{\text{fixed points of torus action}\} \leftrightarrow \{\text{torsion-free part of equivariant cohomology}\}\]

This includes within it our earlier observation that when the action is free, $H^\ast_G(X) \simeq H^\ast(X/G)$, since the latter, being the cohomology ring of a manifold, is torsion. This suggests that if we tensor the cohomology ring with its fraction field (i.e, localize), we will be left with only the torsion-free part, which can be recovered from the fixed points. That is the idea I will explore in a later post.