Symplectic fixed points and quantum cohomology

Matthias Schwarz, Stanford University


Let $(M,\omega)$ be a closed symplectic manifold and $\operatorname{Ham}(M,\omega)\subset \operatorname{Diff}^o(M,\omega)$ the group of exact symplectomorphisms. An open conjecture by V.I.~Arnold is that the number of fixed points of exact symplectomorphisms should be at least the minimal number of critical points of smooth functions on $M$.

In the case of nondegenerate fixed points the topological estimate that the number of fixed points is not less than the dimension of the rational homology of $M$ can now be proved by means of Floer homology for all closed symplectic manifolds.

In the general case, only for manifolds such that the cohomology class of $\omega$ vanishes on $\pi_2(M)$, the number of fixed points is known to be bounded below by the cup-length of $M$, a Liusternik-Shnirelman type result.

In this talk it is shown that an estimate for the number of degenerate fixed points is still possible if one replaces the purely topological quantity of cup-length by a similar number gained from the quantum cohomology ring associated to the symplectic manifold. This can again be explained in terms of Floer homology, which is a homology theory formally generated by the fixed point set. Floer homology carries the structure of a 2-dimensional topological field theory, and the underlying ring structure, the so-called pair-of-pants multiplication, is naturally isomorphic to the deformed cup-product in the quantum cohomology ring. This equivalence between Floer homology and quantum cohomology is used in order to prove a quantum cup-length estimate for degenerate fixed points which unifies all previously known estimates.