in Geometry, Topology

Morse Homology

Morse theory studies the topology of smooth manifolds via the smooth functions that live on it. More precisely, by looking at the critical points of special height functions, so called Morse functions. Consider for instance the example of a sphere:

Example of Morse functions on S^2

Let us assign each critical point p a number according to its type, called its \textbf{index}:

    \[ \mu(p) = \left \{\begin{array}{ll} 2 & \text{, if } p \text{ is a local maximum} \\1 & \text{, if } p \text{ is a saddle point } \\0 & \text{, if } p \text{ is a local minimum} \end{array}\right.\]

Then we can consider the sum:

    \begin{equation*} \eta(S^2) = \sum_{p \text{ critical point}} (-1)^{\mu(p)} \end{equation*}

If we calculate \eta(S^2), we will find that in both cases its equal to 2. This turns out not to be a coincidence, since no matter which Morse functions we choose on S^2, this number remains 2. If we consider the 2-torus \mathbb{T}^2, i.e. a donut, and a Morse function on it, such as the one on the cover, we see that

    \begin{equation*} \eta(\mathbb{T}^2) = 0 \end{equation*}

Again, changing the Morse function will not change \eta(\mathbb{T}^2). Now one can play this game also for genus g surfaces \mathbb{T}g and sees that:

    \[ \eta(\mathbb{T}_g) = 2-2g\]

Hence, by the classification of 2-manifolds (which interestingly can also be proven using Morse theory), we see that for any Riemann surface M, we have:

(1)   \begin{equation*} \eta(M) = \chi(M) \end{equation*}

where \chi(M) is the Euler characteristic of M. This is again no coincidence! Let us rewrite the formula for \eta as follows:

(2)   \begin{equation*} \eta(M) = \sum_{k \ge 0} (-1)^kC_k, \quad \quad C_k = |\{p : p \text{ critical point of index } k\}|\end{equation*}

This compares nicely to a formula from aglebraic topology. Let M_0 \subset M_1 \subset M_2 = M be a cell complex structure on M. Then we have:

    \[ \chi(M) = \sum_{k \ge 0} (-1)^kN_k, \quad \quad N_k = \text{Number of } k \text{-cells}\]

It turns out that under certain conditions a Morse function will give rise to a cell complex structure with exactly C_k cells of dimension k. If we for assume this result for a moment, we immediately get:

    \[ \chi(M) = \sum_{k \ge 0} (-1)^kN_k = \sum_{k \ge 0} (-1)^kC_k = \eta(M)\]

for manifolds which admit these sufficiently regular Morse functions. This means that we can express \textbf{topological invariants} such as the Euler characteristic solely \textbf{in terms of critical points} of height functions. In fact, a single Morse function, by giving rise to a cell complex structure, suffices in principle to classify manifolds up to homeomorphism type. What we will look at is the so called \textbf{Morse homology} of a manifold, which is in fact just the homology of a cell complex obtained by a Morse function, from the classical point of view. But that is not how we are going to define it, as we will take a more abstract modern route. We will look at the so called \textbf{Morse chain complex } \mathrm{CM}_\bullet(M;f) associated to a manifold with a Morse function f on it. The k-th chain groups will be free \Z_2-modules generated by the critical points of index k, and the boundary operator \partial will be defined by counting solutions to a differential equation. Both approaches lead to the same result, however modern Morse homology has several advantages over classical Morse homology. One of them is that it lends itself better to our axiomatic treatment for showing its equivalence to singular homology. We will prove that Morse homology satisfies a number of axioms, called the \textbf{Eilenberg-Steenrod-axioms}. Then we will invoke a uniqueness theorem that tells us that any homology theory satisfying the ES-axioms has to be isomorphic to singular homology. Let us now take a quick look at the structure of this thesis. The first chapter will focus on defining the Morse homology groups, and in particular show that it is well defined. We will extend our notion of index to general manifolds, and make concrete the expression “counting solutions to equations” with the so called trajectory spaces. Then the second chapter will deal with extending the groups to functors and show that the ES-axioms are satisfied, so that we can conclude via the uniqueness theorem that Morse homology is in fact isomorphic to singular homology. Finally, in the last chapter we will explores a number of theorems from algebraic topology in the spirit of Morse homology. We will cover Morse cohomology and Poincaré duality, which turns out is almost trivial to prove in Morse homology. To conclude, we will give a quick proof of the existence of an Eilenberg-Zilber chain map in by exploiting an additivity property of Morse functions. The rest of the thesis can be found  here.