Brian Hepler, Ph.D.

Over the past few decades, Morse theory has undergone many generalizations, into many different fields. At the moment, I only know of a few, and I understand even fewer. Well, let’s begin at the beginning:

Classical Morse theory (CMT)
Stratified Morse theory (SMT)
Micro-local Morse theory (MMT)

The core of these theories is, of course, the study of Morse functions on suitable spaces and generalizations/interpretations of theorems in CMT to these spaces. For CMT, the spaces are smooth manifolds (or, compact manifolds, if your definition of Morse function doesn’t require properness). SMT looks at Morse functions on (Whitney) stratified spaces, usually real/complex varieties (either algebraic or analytic), and more generally, subanalytic subsets of smooth manifolds. MMT deals with both cases, but from a more “meta” perspective that I’m not going to tell you about right now.

The overarching theme is pretty simple: one can investigate the (co)homology of $X$ by examining the behavior of level sets of Morse functions as they “pass through” critical values. First, we’ll need some notation. Let $M$ be a smooth manifold, $a < b \in \mathbb{R}$ , and let $f: M \to \mathbb{R}$ be a smooth function. Then, set

$M_{\leq a} := f^{-1}(-\infty,a]$
$M_{< a} := f^{-1}(-\infty,a)$
$M_{[a,b]} := f^{-1}[a,b]$

In CMT, this overarching idea is described by two “fundamental” theorems:

Fundamental Theorem of Classical Morse theory, A (CMT;A):

Suppose $f$ has no critical values on the interval $[a,b] \subseteq \mathbb{R}$ . Then, $M_{\leq a}$ is diffeomorphic to $M_{\leq b}$ , and the inclusion $M_{\leq a} \hookrightarrow M_{\leq b}$ is a homotopy equivalence (that is, $M_{\leq a}$ is a deformation-retract of $M_{\leq b}$ ).

Homologically speaking, this last point can be rephrased as $H_*(M_{\leq b},M_{\leq a}) = 0$ (for singular homology with $\mathbb{Z}$ coefficients).

Fundamental Theorem of Classical Morse theory, B (CMT;B):

Suppose that $f$ has a unique critical value $v$ in the interior of the interval $[a,b] \subseteq \mathbb{R}$ , corresponding to the isolated critical point $p \in M$ of index $\lambda$ . Then, $H_k(M_{\leq b},M_{\leq a})$ is non-zero only in degree $k = \lambda$ , in which case $latex H_\lambda(M_{\leq b},M_{\leq a}) \cong \mathbb{Z}$.

So, if $c \in \mathbb{R}$ varies across a critical value $a < v < b$ of $f$ , the topological type of $M_{\leq c}$ “jumps” somehow. If we want to compare how topological type of $M_{\leq b}$ differs from that of $M_{\leq a}$ , the obvious thing to do is consider them together as a pair of spaces $(M_{\leq b}, M_{\leq a})$ and look at the relative (co)homology of this pair. CMT;A and CMT;B together tell us that we’re only going to get non-zero relative homology of this pair when there is a critical value between $a$ and $b$ , and in that case, the homology is non-zero only in degree $\lambda$ .