

Globus Seminar
April 12, 2012 15:40, Moscow, IUM (Bolshoi Vlas'evskii per., 11)






What is a Morse complex? (A survey from R. Thom to K. Fukaya, via S. Smale and E. Witten)
F. Laudenbach^{} ^{} CNRS — Laboratoire de Mathématiques Jean Leray,
Département de Mathématiques,
Universite de Nantes

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Abstract:
Given a compact manifold $M$ equipped with a realvalued function $f$, whose critical points are quadratic nondegenerate and whose critical values are distinct, Marston Morse was able to deduce the socalled Morse inequalities from a local study of the homology near the critical points only. In his 1949 note, his first publication indeed, Rene Thom used the gradient of a Morse function for decomposing $M$ into cells. But, the transversality condition of the stable and unstable manifolds was missing.
This condition on the gradient vector field, now called the MorseSmale condition, was pointed out by Steve Smale in the early sixties. In high dimension and when $M$ is simply connected, Smale was able to cancel out pairs of the critical points which do not contribute to the homology. On this occasion, Smale discovered the socalled Morse complex, an algebraic $Z$complex based on the critical points of $f$ whose differentials count the gradient orbits which connects critical points of successive indices.
In 1982, Edward Witten rediscovered this complex by deforming the Laplacian by means of the given Morse function. The de RhamWitten complex, that is, the space $\Omega^*(M)$ of differential forms on $M$ equipped with a coboundary operator deformed «a la Witten», has the Morse complex as its semiclassical limit. This approach allowed analysts to take the case when $M$ has a nonempty boundary into account. More recently, I found a direct topological approach to this, that is, a right notion of pseudogradient adapted to the boundary allowing me to build a complex which calculates the homology of $M$, or the relative homology $H_*(M,\partial M)$, according to the type of adapted pseudogradient used.
All these complexes are stable under small perturbations of the function $f$ and its pseudogradient $X$. This is the starting point of Kenji Fukaya when $M$ is closed. By multiintersecting the stable/unstable manifolds of $X$ with stable/unstable manifolds of perturbations of $X$, Fukaya equipped the Morse complex with a rich mutiplicative structure, an $A_\infty$structure indeed.
This also works in the case of nonempty boundary. For instance, consider the Borromean rings $L$ in $S^3$ endowed with the standard height function and define $M$ as the complementary in $S^3$ to a small open tubular neighborhood of $L$. In this case, the third product in the infinite series yields a Morse approach to the Massey product.

