Cap-product structures on the Fintushel-Stern spectral sequence (Q2760481)

This papers describes a way of making more precise the idea that Floer homology, formally interpreted as ``middle dimensional'' homology of an infinite dimensional configuration space of gauge fields, can be paired, via a cap product construction, to ordinary (finite dimensional) cohomology of the same configuration space. The Floer theory considered here is the one arising from the dimensional reduction of \(SO(3)\) Yang-Mills theory. Floer first introduced the \({\mathbb Z}_8\) graded instanton homology, as a topological invariant of integral homology 3-spheres \(Y\). In the original formulation, instanton Floer homology is, very roughly speaking, the Morse homology of the Chern-Simons functional, that is, the homology of the complex that has the critical points of this functional as generators, while the flow lines between critical points of relative index one define the boundary map. Later, Fintushel and Stern refined this construction to an integer graded instanton Floer homology, by imposing a constraint on the values of the Chern-Simons functional. This Floer homology is related to the original one via a spectral sequence.NEWLINENEWLINENEWLINESolutions of the flow lines equation are instantons on the cylinder manifold \(Y\times {\mathbb R}\). The author considers the configuration space \(B_{Y\times {\mathbb R}}\) of connections on this cylinder manifold modulo gauge action, in order to define the cap product action of the rational cohomology of this configuration space on the integer graded instanton Floer homology, and induced actions on the higher terms in the spectral sequence, hence on the original periodically graded Floer homology. The map is first constructed at the level of cycles, by intersecting submanifolds Poincaré dual to the rational cohomology classes with the moduli spaces of flow lines. Then the author shows compatibility with the differentials of the integer graded Floer complex and with the higher differentials in the spectral sequence. The technical problem is making the necessary intersection numbers well defined, and to this purpose the author uses the integer graded Floer homology, since the bubbling phenomena in the compactification of the moduli spaces of flow lines are controlled by the additional condition on the values of the Chern-Simons functional.