A relation on Floer homology groups of homology handles (Q2372710)

The paper under review is to show a relationship on the rank of the Floer homology of integral homology handles. The Floer homology for homological \(S^2\times S^1\) is defined only for manifolds admitting a unique nontrivial \(SO(3)\), and its Floer relative index is no longer mod 8, but mod 4 period. The paper starts with observations from \textit{N. Saveliev's} work on Seifert manifolds in Theorem 3. The Floer exact triangle only applies to those admissible 3-manifolds with a condition on the second Stiefel-Whitney class. Let \(N\) be the 3-manifold resulting from 0-surgery along a singular fiber in a Seifert fibered 3-manifold. Construct \(N^*\) to be diffeomorphic to \(N\) with reversed orientation via plumbing. The main result in this paper is that \((b_2- b_0)(N) = (b_2 - b_0)(N^*)\); the proof follows from various observations on Taubes' result on relating the Euler characteristic to the Casson invariant up to sign and on Saveliev's result on relating the ranks to Neumann's \(\overline{\mu}\) invariant.