Morse Homology


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Morse Homology Theorem

Theorem. For $M$ compact and $h : M \to \mathbb{R}$ Morse,
$$MH_\bullet(h;\; \mathbb{F}_2) \cong H_\bullet(M;\; \mathbb{F}_2)$$
regardless of the choice of $h$.

The conceptually simplest proof involves an isomorphism to the cellular homology of $M$, where the cell structure is that given by the stable manifold $W^u(p)$ for $p \in \mathrm{Cr}(h)$. The Stable Manifold Theorem from dynamical systems asserts that these unstable manifolds are all cells homeomorphic to $\mathbb{R}^{\mu(p)}$, and transversality argument provides acceptable attaching maps.

The proofs are clearest for Morse-Smale functions for which all stable and unstable manifolds of critical points are transverse. Morse-Smale functions, like Morse functions, are generic, and for such there is an isomorphism at the level of chain complexes.

For example, consider a particularly simple height function $h$ on $\mathbb{S}^2$ with two maxima, one minimum, and a saddle. The Morse complex in $\mathbb{F}_2$ coefficients is of the form,

$$0 \to \mathbb{F}_2^2 \to \mathbb{F}_2 \to \mathbb{F}_2 \to 0$$

Thus, $MH_2(h) \cong \mathbb{F}_2$, $MH_1(h) \cong 0$, $MH_0(h) \cong \mathbb{F}_2$, one notes that in this example, $MH_\bullet(h) \cong H_\bullet(\mathbb{S}^2; \mathbb{F}_2)$.

Morse Polynomial

One efficient means of encoding all the critical point data of a Morse function is by means of the Morse polynomial of $h$, defined as:

$$M_h(t) := \sum_{p \in \mathrm{Cr}(h)} t^{\mu(p)}$$

This polynomial in the abstract variable $t$ has coefficients $c_i \in \mathbb{N}$ the number of critical points of $h$ with Morse index $i$. There is a relationship between Morse polynomial and the Poincare polynomial (which encodes the singular homology of the manifold):

$$P(t) = \sum_i \dim H_i(M)t^i$$

Such that $M_h(t)$ is greater or equal to $P(t)$ coefficient-wise. Furthermore, for $h : M \to \mathbb{R}$ Morse,

$$M_h(t) = P(t) + (1+t)Q(t),\; Q \in \mathbb{N}[t]$$

By corollary, Morse polynomial gives the Euler characteristic:

$$\chi(M) = \sum_{p \in \mathrm{Cr}(h)} (-1)^{\mu(p)}$$

Meaning Of Poincare Duality

Poincaré duality

Theorem. The $\mathbb{F}_2$-homology of a compact $n$-dimensional manifold $M$ is symmetric in its grading. That is for all $k,n$, $0 \leq k \leq n$: $$H_k(M; \mathbb{F}_2) \cong H_{n-k}(M; \mathbb{F}_2)$$



Topics in Topological Data Analysis

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