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 , and we have some Morse function, ; (CMT;A) says that the topological type of the set remains constant unless passes over a critical value of . In the case where is a critical value, we study the change in topology of with a pair of spaces, , which we call **local Morse data for for f at p **(which I’ll write as ), defined as follows: let be “sufficiently small,” a small ball centered at of radius (say, with respect to some Riemannian metric on ). Then, for “sufficiently small,”

(thankfully, **the topological type of ** **is independent of the choice of metric, and independent of **, provided that they’re chosen to be sufficiently small).

Say is a non-degenerate critical point of of index (recall that this means the Hessian of at in non-singular, and has negative eigenvalues), the corresponding critical value. Since critical values are locally isolated in , there exists some small so that is the only critical value of in the interval . Then, (CMT;B) says that is obtained as a topological space from 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,

where , and is the -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:

(*)

where is the constant sheaf with stalk on , considered as a complex of sheaves concentrated in degree zero. is the derived functor “sections with support in , and we take the stalk at the point . 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 , there is an such that

which is isomorphic to

where is such that . Then, by (CMT;A), there is a homeomorphism of pairs , inducing the isomorphism

In short,