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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
[TODO]
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