Equivariant aspects of Yang-Mills Floer theory

Abstract

We use Floer's exact triangle to study the u-map (cup product with the 4-dimensional class) in the Floer cohomology groups of admissible SO(3) bundles over closed, oriented 3-manifolds. In the case of non-trivial bundles we show that (u^2-64)^n = 0 for some positive integer n. For homology 3-spheres Y the same holds for a certain reduced Floer group, which is obtained from the ordinary one by factoring out interaction with the trivial connection. This leads to a new proof (in the simply-connected case) of the finite type conjecture of Kronheimer and Mrowka concerning the structure of Donaldson polynomials. In the case of rational coefficients, interaction with the trivial connection is measured by a single integer h(Y), which is additive under connected sums and depends only on the rational homology cobordism class of Y.