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# Morse homology

Morse homology is a tool developed by Thom, Smale, and Milnor for homology theory.

Take $M$ to be a smooth compact manifold. Throughout we assume that $f$ is a suitable Morse function, that is, all critical points of $f$ are nondegenerate. We must first make some definitions before defining the Morse homology. Choose a Riemannian metric on $M$ so that the notion of a gradient vector field makes sense. The map $\phi\colon\mathbb{R}\times M\rightarrow M$ such that

$\frac{d}{dt}\phi(t,x)=-\nabla f(\phi(t,x)),$ |

with $\phi(0,x)=\Id$, is called the negative gradient flow of $f$. Let $p$ be a critical point of $f$, and define

$W_{p}^{s}:=\{x\in M|\lim_{{t\rightarrow\infty}}\phi(t,x)=p\}\text{\ and \ }W_{% p}^{u}:=\{x\in M|\lim_{{t\rightarrow-\infty}}\phi(t,x)=p\}$ |

to be the stable and unstable manifolds respectively. Thom realized that one could decompose $M$ into its unstable manifolds and arrive at something that is homologically equivalent to its handle decomposition, but this decomposition was not a CW complex, hence it was hard to say anything about the homotopy type of $M$. But Smale realized that if we impose more conditions on the metric itself, then we can make this into a CW complex.

The pair $(f,g)$, where $f$ is a Morse function and $g$ is the Riemannian metric, is called Morse-Smale pair, if for every pair $p$, $q$ of critical points of $f$, $W_{p}^{u}$ is transverse to $W_{q}^{s}$. This is known as the Morse-Smale condition. This condition actually holds for a generic Riemannian metric on M. With this restriction, this makes Thom’s decomposition into a CW complex.

We can define a complex called the Morse complex as follows:

Let $\Crit_{k}(f)$ be the set of critical points of $f$ of index $k$. We define the chain group , $C_{k}(f)$ to be the formal linear combination with integer coefficients of elements of $\Crit_{k}(f)$. We must also keep track of the signs of the flow lines. (However, it is true if you count mod 2, the Morse complex computes homology with coefficients in $Z\over 2$.) To make this a chain complex we must define the differential map. The map $\delta_{k}:C_{k}\rightarrow C_{{k-1}}$ applied to a critical point $p$ is a formal sum of critical points with $q$ given by this number. It is possible to prove that $\delta^{2}=0$ , making this into a chain complex.

The homology of this complex is called the Morse homology. It can be shown to be isomorphic to the singular homology of $M$.

Note: There is another way of realizing the Morse homology using Hodge theory, an idea pioneered by Edward Witten. His idea is essentially to conjugate the $d$ operator by $e^{{sf}}$ and it can be shown that this conjugation again leads to another isomorphism between the set of harmonic forms and the De Rham cohomology. This parameter $s$ is like a curve of chain complexes and Witten claimed that if $s$ is large enough, then we can obtain a space whose dimension is the number of critical points of a given index and the boundary operator induced on $d$ is the number of critical paths between critical points, as before. Witten did not prove this idea rigorously, but it was done later by Helffer and Sjostrand.

## Mathematics Subject Classification

58A05*no label found*

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