Localisation

We have seen that we can define a sensible cohomology theory for group actions that aren't necessarily free. It would be nice however if there were also some computational advantages. We might hope for that to be true since we have seen that the equivariant cohomology of a point is usually non-trivial and can be in fact infinite-dimensional. If for example the action of $G$$G$ on $X$$X$ has finitely many fixed points we might hope to get an inclusion $H_{G}^{*}(X)\hookrightarrow H_{G}^{*}(X^{G})$$H_{G}^{*}(X)\hookrightarrow H_{G}^{*}(X^{G})$, where $X^{G}$$X^{G}$ is the fixed point set in $X$$X$ under $G$$G$. This isn't true in general, but under fairly weak assumptions and modulo torsion (in a very general sense) this is actually an isomorphism. Thus we can in some sense localise to the fixed point locus.

For simplicity of exposition assume that $G=S^{1}$$G=S^{1}$ acts on a compact $n$$n$-dimensional manifold $X$$X$ in a way such that every orbit is either free or fixed. We don't loose generality here since we work with $\mathbb {C}$$\mathbb {C}$ coefficients from now on and so any torsion caused by stabilisers in $S^{1}$$S^{1}$ is killed anyways. Denote by $F\subseteq X$$F\subseteq X$ the subset of fixed points.

Exercise 5.6

$F\subseteq X$$F\subseteq X$ is a submanifold.

 

Proof

Let $g$$g$ be a Riemannian metric on $X$$X$. Averaging the metric $g$$g$ by $G$$G$ ($G=S^{1}$$G=S^{1}$ or any compact Lie group), we can suppose that $G$$G$ acts by isometry, i.e. $\tau _{g}:X\rightarrow X$$\tau _{g}:X\rightarrow X$, $x\to g\cdot x$$x\to g\cdot x$, is an isometry of $(X,g)$$(X,g)$.

Suppose that the point $x\in F$$x\in F$ is not isolated and define the space of invariant vectors

$$H=\{v\in T_{x}X|\,d\tau _{g}(v)=v\quad \forall g\in G\}.$$

$$H=\{v\in T_{x}X|\,d\tau _{g}(v)=v\quad \forall g\in G\}.$$

Note that $\exp _{x}(H)\subseteq F$$\exp _{x}(H)\subseteq F$. Indeed, since $\tau _{g}$$\tau _{g}$ is an isometry, we have

$$\tau _{g}\cdot \operatorname {exp} _{x}(h)=\operatorname {exp} _{\tau _{g}x}(d\tau _{g}h)=\operatorname {exp} _{x}(h).$$

$$\tau _{g}\cdot \operatorname {exp} _{x}(h)=\operatorname {exp} _{\tau _{g}x}(d\tau _{g}h)=\operatorname {exp} _{x}(h).$$

Let $y\in F\cap U$$y\in F\cap U$, $y\neq x$$y\neq x$, where $U$$U$ is a neighbourhood of $x$$x$ such that $\operatorname {exp} _{x}:\operatorname {exp} ^{-1}(U)\subseteq T_{x}X\rightarrow U$$\operatorname {exp} _{x}:\operatorname {exp} ^{-1}(U)\subseteq T_{x}X\rightarrow U$ is a diffeomorphism. Since $\operatorname {exp} _{x}$$\operatorname {exp} _{x}$ is surjective onto $U$$U$, there exists $v\in T_{x}X$$v\in T_{x}X$ such that $y=\operatorname {exp} _{x}(v)$$y=\operatorname {exp} _{x}(v)$. Being $y$$y$ a fixed point,

$$\operatorname {exp} _{x}(v)=\tau _{g}\cdot \operatorname {exp} _{x}(v)=\operatorname {exp} _{\tau _{g}x}(d\tau _{g}v)=\operatorname {exp} _{x}(d\tau _{g}v).$$

$$\operatorname {exp} _{x}(v)=\tau _{g}\cdot \operatorname {exp} _{x}(v)=\operatorname {exp} _{\tau _{g}x}(d\tau _{g}v)=\operatorname {exp} _{x}(d\tau _{g}v).$$

By the injectivity of $\operatorname {exp} _{x}|_{\operatorname {exp} ^{-1}(U)}$$\operatorname {exp} _{x}|_{\operatorname {exp} ^{-1}(U)}$, we conclude that $v\in H$$v\in H$. Hence,

$$F\cap U=\operatorname {exp} _{x}(H)\cap U$$

$$F\cap U=\operatorname {exp} _{x}(H)\cap U$$

and $\operatorname {exp} _{x}|_{H}$$\operatorname {exp} _{x}|_{H}$ is a local chart for $F$$F$.

 

Let $c$$c$ be the codimension of $F$$F$ in $X$$X$. Now consider the long exact sequence for cohomology of a pair:

$$\cdots \longrightarrow H^{*}(X,X\setminus F)\longrightarrow H^{*}(X)\longrightarrow H^{*}(X\setminus F)\longrightarrow H^{*+1}(X,X\setminus F)\longrightarrow \cdots .$$

$$\cdots \longrightarrow H^{*}(X,X\setminus F)\longrightarrow H^{*}(X)\longrightarrow H^{*}(X\setminus F)\longrightarrow H^{*+1}(X,X\setminus F)\longrightarrow \cdots .$$

By excision and the Thom isomorphism we have

$${\begin{aligned}H^{*}(X,X\setminus F)&\cong H^{*}(\nu _{F},\partial \nu _{F})\\&\cong H_{\text{compact}}^{*}(\nu _{F})\\&\cong H^{*-c}(F),\end{aligned}}$$

$${\begin{aligned}H^{*}(X,X\setminus F)&\cong H^{*}(\nu _{F},\partial \nu _{F})\\&\cong H_{\text{compact}}^{*}(\nu _{F})\\&\cong H^{*-c}(F),\end{aligned}}$$

where $\nu _{F}$$\nu _{F}$ is the normal bundle of $F$$F$ in $X$$X$. By assumption $X\setminus F$$X\setminus F$ has a free $G$$G$-action. Thus if we pass to equivariant cohomology we obtain

$$\cdots \longrightarrow H_{G}^{*-c}(F)\longrightarrow H_{G}^{*}(X)\longrightarrow H^{*}((X\setminus F)/G)\longrightarrow H_{G}^{*-c+1}(F)\longrightarrow \cdots .$$

$$\cdots \longrightarrow H_{G}^{*-c}(F)\longrightarrow H_{G}^{*}(X)\longrightarrow H^{*}((X\setminus F)/G)\longrightarrow H_{G}^{*-c+1}(F)\longrightarrow \cdots .$$

This should be understood as a long exact sequence of $\mathbb {C} [t]$$\mathbb {C} [t]$-modules. Note that $H^{*}((X\setminus F)/G)$$H^{*}((X\setminus F)/G)$ is finite dimensional as a $\mathbb {C}$$\mathbb {C}$-vector space. Since it is also a $\mathbb {C} [t]$$\mathbb {C} [t]$-module it must be of the form $\mathbb {C} [t]/(t^{k})$$\mathbb {C} [t]/(t^{k})$ for some $k\in \mathbb {N}$$k\in \mathbb {N}$. We introduce the notation

$${\hat {H}}_{G}^{*}(X):=qH_{G}^{*}(X)\otimes _{\mathbb {C} [t]}\mathbb {C} (t),$$

$${\hat {H}}_{G}^{*}(X):=qH_{G}^{*}(X)\otimes _{\mathbb {C} [t]}\mathbb {C} (t),$$

and call this localisation. Note that $\mathbb {C} (t)$$\mathbb {C} (t)$ is a flat $\mathbb {C} [t]$$\mathbb {C} [t]$-module and so if we localise the above exact sequence we obtain another exact sequence where ${\hat {H}}_{G}^{*}(X\setminus F)=0$${\hat {H}}_{G}^{*}(X\setminus F)=0$ and thus we obtain an isomorphism

$${\hat {H}}_{G}^{*-c}(F){\overset {\cong }{\longrightarrow }}{\hat {H}}_{G}^{*}(X).$$

$${\hat {H}}_{G}^{*-c}(F){\overset {\cong }{\longrightarrow }}{\hat {H}}_{G}^{*}(X).$$

Similarly by considering the cohomology long exact sequence for the pair $(X,F)$$(X,F)$ we obtain another isomorphism

$${\hat {H}}_{G}^{*}(X){\overset {\cong }{\longrightarrow }}{\hat {H}}_{G}^{*}(F).$$

$${\hat {H}}_{G}^{*}(X){\overset {\cong }{\longrightarrow }}{\hat {H}}_{G}^{*}(F).$$

We can now ask what the composition of the above two isomorphisms is. In the case of usual cohomology we have the following commutative diagram explaining the maps:

$$MISSING$$

$$MISSING$$

The vertical isomorphisms are given by Poincare' duality. If we start at the top left with $1\in H^{*-c}(F)$$1\in H^{*-c}(F)$ and go around anticlockwise we end up getting mapped to the Euler class $e(\nu _{F})\in H^{*}(F)$$e(\nu _{F})\in H^{*}(F)$. Passing to equivariant cohomology again we have that the composition

$$H_{G}^{*-c}(F){\overset {i_{*}}{\longrightarrow }}H_{G}^{*}(X){\overset {i^{*}}{\longrightarrow }}H_{G}^{*}(F)$$

$$H_{G}^{*-c}(F){\overset {i_{*}}{\longrightarrow }}H_{G}^{*}(X){\overset {i^{*}}{\longrightarrow }}H_{G}^{*}(F)$$

is given by cup product with the equivariant Euler class $e_{G}(\nu _{F})\in H_{G}^{*}(F)$$e_{G}(\nu _{F})\in H_{G}^{*}(F)$ (since $\nu _{F}$$\nu _{F}$ carries an obvious $G$$G$-action it induces a bundle on $F_{G}$$F_{G}$ and we understand its Euler class to be the equivariant Euler class of $\nu _{F}$$\nu _{F}$). By the above we now know that after localisation $e_{G}(\nu _{F})$$e_{G}(\nu _{F})$ is invertible in ${\hat {H}}_{G}^{*}(F)$${\hat {H}}_{G}^{*}(F)$. Combining everything so far we obtain the formula

$$\sigma =i_{*}\left({\frac {i^{*}\sigma }{e_{G}(\nu _{F})}}\right)$$

$$\sigma =i_{*}\left({\frac {i^{*}\sigma }{e_{G}(\nu _{F})}}\right)$$

for $\sigma \in {\hat {H}}_{G}^{*}(X)$$\sigma \in {\hat {H}}_{G}^{*}(X)$. Finally note that

$${\begin{aligned}{\hat {H}}_{G}^{*}(F)&=H_{G}^{*}(F)\otimes _{\mathbb {C} [t]}\mathbb {C} (t)\\&\cong (H^{*}(F)\otimes _{\mathbb {C} }H^{*}(BG))\otimes _{\mathbb {C} [t]}\mathbb {C} (t)\\&\cong H^{*}(F)\otimes _{\mathbb {C} }(\mathbb {C} [t]\otimes _{\mathbb {C} [t]}\mathbb {C} (t))\\&\cong H^{*}(F)\otimes _{\mathbb {C} }\mathbb {C} (t).\end{aligned}}$$

$${\begin{aligned}{\hat {H}}_{G}^{*}(F)&=H_{G}^{*}(F)\otimes _{\mathbb {C} [t]}\mathbb {C} (t)\\&\cong (H^{*}(F)\otimes _{\mathbb {C} }H^{*}(BG))\otimes _{\mathbb {C} [t]}\mathbb {C} (t)\\&\cong H^{*}(F)\otimes _{\mathbb {C} }(\mathbb {C} [t]\otimes _{\mathbb {C} [t]}\mathbb {C} (t))\\&\cong H^{*}(F)\otimes _{\mathbb {C} }\mathbb {C} (t).\end{aligned}}$$

We have thus sketched the proof of

Theorem 5.3 (Atiyah--Bott--Berline--Vergne)

Let $\sigma \in H^{*}(X)$$\sigma \in H^{*}(X)$ and assume that it comes from a ${\tilde {\sigma }}\in H_{G}^{*}(X)$${\tilde {\sigma }}\in H_{G}^{*}(X)$ under the map $H_{G}^{*}(X)\to H^{*}(X)$$H_{G}^{*}(X)\to H^{*}(X)$. Then

$$\int _{X}\sigma =\int _{F}\left[{\frac {i^{*}{\tilde {\sigma }}}{e_{G}(\nu _{F})}}\right],$$

$$\int _{X}\sigma =\int _{F}\left[{\frac {i^{*}{\tilde {\sigma }}}{e_{G}(\nu _{F})}}\right],$$

where the square brackets mean taking the constant term.

 

Exercise 5.7

Take $S^{1}$$S^{1}$ to act on $S^{2}$$S^{2}$ by rotation around the $z$$z$-axis. Lift the volume form $\mathrm {vol} \in H^{*}(S^{2})$$\mathrm {vol} \in H^{*}(S^{2})$ to $H_{S^{1}}^{*}(S^{2})$$H_{S^{1}}^{*}(S^{2})$, localise and calculate $\int _{S^{2}}\mathrm {vol}$$\int _{S^{2}}\mathrm {vol}$ using Theorem of Localisation formula.

 

Exercise 5.8

Show using the localisation theorem that a generic cubic surface in $\mathbb {P} ^{3}$$\mathbb {P} ^{3}$ contains 27 lines.