10-27 Notes

\section{Morse-Bott Functions}

\begin{defn} Let $f:M\rightarrow \mathbb{R}$ be a smooth function. We say that $f$ is Morse-Bott if the critical set $C=\{x\in M|d_x f=0\}$ can be writtin as a union of components $C=\bigcup_{\alpha}C_\alpha$ such that:

1)Each $C_\alpha$ is a submanifold of $M$

2)$\text{Ker(Hess}_x f)=T_x C_\alpha$, for all $x\in C_\alpha$ \end{defn}

Condition 2) implies that if $\tau$ is a transverse submanifold to $C$ at $x$, then $f|_\tau$ has a non-degenerate critical point at $x$, and $\tau\cap C=\{x\}$.

Let $\nu_\alpha$ be the normal bundle of $C_\alpha$. Write $\nu_\alpha=\nu^{+}_\alpha\oplus\nu^{-}_\alpha$, where $\text{Hess}_x(f)\nu^{+}_\alpha$ (resp. $\text{Hess}_x(f)\nu^{-}_\alpha$) is positive (resp. negative) definite. For each $C_\alpha$, define $W^{\pm}_ {C_\alpha}=\{x\in M|\lim_{t\rightarrow\pm\infty} \phi_{t}(x)\in C_\alpha\}$, where $\phi_{t}(x)$ is the gradient flow of $f$.

\begin{thm} $W^{\pm}_{C_\alpha}$ are submanifolds of $M$, diffemorphic to $\nu^{\pm}_\alpha$ \end{thm}

\begin{proof} We will show that for all $x\in C_\alpha$, $W^{\pm}_{C_\alpha}$ is a submanifold diffeomorphic to $\mathbb{R}^{n_\pm}$, where $n_\pm$ is the dimension of $\nu^\pm_\alpha$. Work locally and write $\mathbb{R}^n=\mathbb{R}^{n_+}\times\mathbb{R}^{n_-}\times\mathbb{R}^c$ with coordinates $(x^+,x^-,y)$, where $n$=dim $M$, $c$=dim $C_\alpha$. We can assume that $f$ is of the form $f(x^+,x^-,y)=g_y(x^+,x^-)$ where $y\mapsto g_y$ is a smooth map from $\mathbb{R}^c$ to the space of quadratic forms on $\mathbb{R}^{n_+}\times\mathbb{\R}^{n_-}$ of signature $(n_+,n_-)$, i.e. we can find Morse coordinates in a family.

Let $H_1=\{\gamma:[0,\infty)\to\mathbb{R}^n|\int(|\dot{\gamma}|^2+|\gamma|^2)<\infty\}$,

   $$H_0=\{\gamma:[0,\infty)\to\mathbb{R}^n|\int|\gamma|^2<\infty\}$$

   $$F(\gamma)=(\dot{\gamma}+\nabla_{\gamma}f):H_1\to H_0$$ 

   $$\tilde{F}:H_1\to H_+ \times \R^{n_+}$$

In the $c=0$ case, $\tilde{F}$, was a diffeomorphism in a neighborhood of zero, but here $\gamma\mapsto\frac{d}{dt}\gamma:H_1\to H_0$ doesn't even have closed range. Instead, consider $H_{1,\delta}=e^{-t\delta}H_1$, $H_{0,\delta}=e^{-t\delta}H_0$. Let $F_\delta:H_{1,\delta}=\to H_{0,\delta}=$ be the induced map. One easily chekcs that its differential at the origin is $-\delta+\text{Hess}_0(f)$, which is invertivle if $\delta$ is between zero and the first positive eigenvalue of $\text{Hess}_0(f)$. Since multiplication by $e^{t\delta}$ induces an isomorphism between $H_{i,\delta}$ and $H_i$, this implies that $\tilde{F}$ has invertible differential at $\gamma=0$, so the inverse function theorem holdsm abd $\tilde{F}^{-1}(0,x^+)$ parametrizes a collection of paths solving $\dot{\gamma}+\nabla_\gamma f=0$, which converges to $0$ as $t\to\infty$. \end{proof}

\section{The Morse Stratification of M}

A stratification (by manifolds) of a space $X$ is a decomposition $X=\coprod_{i=1}^{\infty}S_i$ such that 1) The $S_i$ are manifolds, locally closed in $X$, 2) $\bar{S_i}\smallsetminus S_i\subset\bigcup_{j<i}S_j$. Under moderate assumptions, a stratification gives a spectral sequence that computes the homology of $X$. For example, a CW-structure on $X$ is a stratification, with $S_i$ the $i$-skeleton of $X$

Given a MOrse-Bott functon $f$, let $S_0$ be the union of the minima of $f$. We construct $S_i$ inductively, letting it be the union of all unstable manifolds $W^u_{C_\alpha}$ so that $S_i\subset\bigcup_{j<i}S_j$. Let $X_i=\bigcup_{j\leq i}S_j$. We would like to understand how $H^*(X_i)$ changes as $i$ grows. Consider the exact sequence of the pair $(X_i,X_{i-1})$:

$$H^*(X_i)\to H^*(X_i,X_{i-1})\overset{\delta}{\to} H^{*+1}(X_{i-1})$$

The last two terms of this sequence are isomorphic to

$$\oplus \Z\simeq H^*(\bar{S_i},\bar{S_i}\smallsetminus S_i)\overset{\delta}{\to} H^{*+1}(\bar{S_i}\smallsetminus S_i)$$

In the Morse situation, we need to understand if a sphere in $W_x^u$ bounds a cycle in $X_{i-1}$ (see the picture in Ben's notes). In the Morse-Bott situation, $H^*(X_i,X_{i-1})\simeq H^*(\nu_-,\nu_-\smallsetminus0)$, so we can fit the last two terms of the exact sequence of the pair into the Gysin sequence:

$$H^*(\nu_-)\to H^*(\nu_-,\nu_-\smallsetminus0)\overset{\delta}{\to}H^{*+1}(\nu_-\smallsetminus0)$$

This sequence is isomorphic to

$$H^*(C_\alpha)\overset{\cup e}{\to}H^*(C_\alpha)\overset{\delta}{\to}H^{*+1}(S(\nu_-))$$

where $e$ is the Euler class of $v_-$, and $S(\nu_-)$ is the sphere bundle associated to $\nu_-\smallsetminus0$. We can guarantee that a class $\beta\in H^*(\nu_-,\nu_-\smallsetminus0)$ extends to a class $\tilde{\beta}\in H^*(X_i)$ if $\beta=\Phi^{-1}(e\cup\beta)$, where $\Phi$ is the Thom isomorphism. In the fintie dimensional case, not all classes have this form, but in the infinite dimensional equivariant case, this can happen for all classes.

In the equivariant case we would like to check: $e\in H^*_G(C_\alpha)$ if $e$ is not a zero divisor $\implies$ $H^*_G(C_\alpha)\hookrightarrow H^*_G(X)$. This is the case in YM.