It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology.
is Morse–Smale if the stable and unstable manifolds associated to all of the critical points of f intersect each other transversely.
After modding out by these reparametrizations, the quotient space is zero-dimensional — that is, a collection of oriented points representing unparametrized flow lines.
The set of chains is the Z-module generated by the critical points.
The differential d of the complex sends a critical point p of index i to a sum of index-
critical points, with coefficients corresponding to the (signed) number of unparametrized flow lines from p to those index-
The fact that the number of such flow lines is finite follows from the compactness of the moduli space.
) follows from an understanding of how the moduli spaces of gradient flows compactify.
Unparametrized index-2 flows come in one-dimensional families, which compactify to compact one-manifolds with boundaries.
The fact that the boundary of a compact one-manifold has signed count zero proves that
It can be shown that the homology of this complex is independent of the Morse–Smale pair (f, g) used to define it.
One can define a map to singular homology by sending a critical point to the singular chain associated to the unstable manifold associated to that point; inversely, a singular chain is sent to the limiting critical points reached by flowing the chain using the gradient vector field.
The isomorphism with singular homology can also be proved by demonstrating an isomorphism with cellular homology, by viewing an unstable manifold associated to a critical point of index i as an i-cell, and showing that the boundary maps in the Morse and cellular complexes correspond.
This approach to Morse theory was known in some form to René Thom and Stephen Smale.
It is also implicit in John Milnor's book on the h-cobordism theorem.
From the fact that the Morse homology is isomorphic to the singular homology, the Morse inequalities follow by considering the number of generators — that is, critical points — necessary to generate the homology groups of the appropriate ranks (and by considering truncations of the Morse complex, to get the stronger inequalities).
Edward Witten came up with a related construction in the early 1980s sometimes known as Morse–Witten theory.
Morse homology can be extended to finite-dimensional non-compact or infinite-dimensional manifolds where the index remains finite, the metric is complete and the function satisfies the Palais–Smale compactness condition, such as the energy functional for geodesics on a Riemannian manifold.
Sergei Novikov generalized this construction to a homology theory associated to a closed one-form on a manifold.
Morse homology is a special case for the one-form df.
To describe the resulting chain complex and its homology, introduce a generic Morse function on each critical submanifold.
This approach to Morse–Bott homology appeared in the context of unpublished work for contact homology by Bourgeois, in which the critical submanifolds are the sets of Reeb orbits, and the gradient flows between the critical submanifolds are pseudoholomorphic curves in the symplectization of a contact manifold asymptotic to Reeb orbits in the relevant critical manifolds of Reeb orbits.