Equivariant aspects of Yang-Mills Floer theory

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.Comment: A few cosmetic changes. To appear in Topolog

Monopole Floer homology for rational homology 3-spheres

We give a new construction of monopole Floer homology for spin-c rational homology 3-spheres. As applications we define two invariants of certain smooth compact 4-manifolds with b_1=1 and b^+=0.Comment: 60 pages, to appear in Duke J. Math. v2: Minor improvements concerning orientations on pages 18-21. v3: Two new sections 4 and 14 have been added

On the existence of representations of finitely presented groups in compact Lie groups

Given a finite, connected 2-complex $X$ such that $b_2(X)\le1$ we establish two existence results for representations of the fundamental group of $X$ into compact connected Lie groups $G$, with prescribed values on certain loops. If $b_2(X)=1$ we assume $G=SO(3)$ and that the cup product on the first rational cohomology group of $X$ is non-zero.Comment: 22 pages, to appear in `Topology and its Applications'. v2: The title was changed, reflecting the fact that Cor. 1.1 was already known. The old Theorem 1.5 was omitted, as it is easily proved using a result in the new appendix. v3: Only minor changes. v4: The proof of Prop. 2.1 was omitted, because the result was already known. Minor changes following referee's suggestion