MorseHomology 2005-06-20 18:06:38 2006-10-09 11:04:37 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\Crit}{Crit} {\em 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.