Local Cohomology and Morse Data

This post is next in my series of posts on Morse theory and its various generalizations. I last talked about “Classical Morse Theory,” (CMT) which studies the how the topology of a (real) smooth manifold is related to critical points of Morse functions defined on that manifold. Say the manifold is called $M$ , and we have some Morse function, $f: M \to \mathbb{R}$ ; (CMT;A) says that the topological type of the set $M_{\leq c} := f^{-1}(-\infty,c]$ remains constant unless $c \in \mathbb{R}$ passes over a critical value of $f$ . In the case where $v$ is a critical value, we study the change in topology of $M_{\leq c}$ with a pair of spaces, $(A,B)$ , which we call local Morse data for for f at p (which I’ll write as $|LMD(f,p)|$ ), defined as follows: let $\epsilon > 0$ be “sufficiently small,” $B_\epsilon(p)$ a small ball centered at $p$ of radius $\epsilon$ (say, with respect to some Riemannian metric on $M$ ). Then, for $\delta > 0$ “sufficiently small,”

$|LMD(f,p)| := (B_\epsilon(p) \cap f^{-1}[v-\delta,v+\delta], B_\epsilon(p) \cap f^{-1}(v-\delta) )$

(thankfully, the topological type of $|LMD(f,p)|$ is independent of the choice of metric, and independent of $\epsilon,\delta$ , provided that they’re chosen to be sufficiently small).

Say $p \in M$ is a non-degenerate critical point of $f$ of index $\lambda$ (recall that this means the Hessian of $f$ at $p$ in non-singular, and has $\lambda$ negative eigenvalues), $f(p) =v$ the corresponding critical value. Since critical values are locally isolated in $\mathbb{R}$ , there exists some $\delta > 0$ small so that $v$ is the only critical value of $f$ in the interval $[v-\delta,v+\delta]$ . Then, (CMT;B) says that $M_{\leq v + \delta}$ is obtained as a topological space from $M_{\leq v-\delta}$ by attaching the space $latex B_\epsilon(p) \cap f^{-1}[v-\delta,v+\delta]$ along the space $latex B_\epsilon(p) \cap f^{-1}(v-\delta)$. More specifically,

$|LMD(f,p)| \cong (D^\lambda \times D^{n-\lambda}, \partial D^\lambda \times D^{n-\lambda})$

where $n = \dim M$ , and $D^k$ is the $k$ -dimensional disk.

Now, me being me, I need to see how this fits into more general machinery. Thankfully, the way has already been paved for us: the Morse theory for constructible sheaves explored in Topology of Singular Spaces and Constructible Sheaves by J. Schurmann. There, local Morse data is framed, functorially, in terms of local cohomology groups:

(*) $LMD(f,p) := R\Gamma_{\{f \geq v\}}(\mathbb{Z}_M^\bullet)_p$

where $\mathbb{Z}_M^\bullet$ is the constant sheaf with stalk $\mathbb{Z}$ on $M$ , considered as a complex of sheaves concentrated in degree zero. $R\Gamma_{f \geq v}(-)$ is the derived functor “sections with support in $\{f \geq v\} := f^{-1}[v, \infty)$ , and we take the stalk at the point $p$ . This all seems a bit complicated, and it is at first for everybody. Worth investigating though, since understanding the LMD(f,p) construction is instrumental in generalizing the ideas of CMT and SMT to the “microlocal” setting; in particular, to the derived category (of bounded, constructible complexes of sheaves), and the construction of the “microsupport”of a complex of sheaves.

Let’s investigate (*). By constructibility of $R\Gamma_{\{\geq f \geq v\}}(\mathbb{Z}_M^\bullet)$ , there is an $\epsilon > 0$ such that

$R\Gamma_{\{ f \geq v\} }(\mathbb{Z}_M^\bullet)_p \cong R\Gamma (B_\epsilon(p);R\Gamma_{\{ f \geq v\}}(\mathbb{Z}_M^\bullet))$

which is isomorphic to

$R\Gamma(B_\epsilon(p),B_\epsilon(p) \cap f^{-1}[v-\delta,v); \mathbb{Z}_M^\bullet)$

where $\delta > 0$ is such that $f(\Sigma f) \cap [v-\delta,v+\delta] = \{v\}$ . Then, by (CMT;A), there is a homeomorphism of pairs $(B_\epsilon(p),B_\epsilon(p) \cap f^{-1}[v-\delta,v)) \overset{\thicksim}{\to} (B_\epsilon(p) \cap f^{-1}[v-\delta,v+\delta],B_\epsilon(p) \cap f^{-1}(v-\delta))$ , inducing the isomorphism

$R\Gamma_{\{f \geq v\} }(\mathbb{Z}_M^\bullet)_p \cong R\Gamma(B_\epsilon(p) \cap f^{-1}[v-\delta,v+\delta],B_\epsilon(p) \cap f^{-1}(v-\delta);\mathbb{Z}_M^\bullet)$

In short,

$LMD(f,p) \cong R\Gamma(|LMD(f,p)|;\mathbb{Z}_M^\bullet)$

Local Cohomology and Morse Data

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