# 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 $\varphi :\mathbb{R}\times M\to M$ such that

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

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

$${W}_{p}^{s}:=\{x\in M|\underset{t\to \mathrm{\infty}}{lim}\varphi (t,x)=p\}\text{and}{W}_{p}^{u}:=\{x\in M|\underset{t\to -\mathrm{\infty}}{lim}\varphi (t,x)=p\}$$ |

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 ${\mathrm{Crit}}_{k}(f)$ 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 ${\mathrm{Crit}}_{k}(f)$. 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 $\frac{Z}{2}$.)
To make this a chain complex^{} we must define the differential map.
The map ${\delta}_{k}:{C}_{k}\to {C}_{k-1}$ 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.

Title | Morse homology |
---|---|

Canonical name | MorseHomology |

Date of creation | 2013-03-22 15:21:09 |

Last modified on | 2013-03-22 15:21:09 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 17 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 58A05 |