Invariants of 4-manifolds with the homology of $S^1\times S^3$

Gauge theoretic invariants of a closed 4-manifold $X$ usually are defined only in the setting where the intersection form has $b_2^+ > 0$. This condition rules out reducible solutions. Many interesting questions remain, however, in the setting where there is no second homology at all. We relate several invariants that have been previously defined in this context. We relate the invariant $\lambda_{SW}(X)$ defined by Mrowka-Ruberman-Saveliev to two invariants introduced by Fr\o yshov for a rational homology sphere $Y$. One is the \emph{correction term} $h(Y)$; if $Y$ is embedded in $X$, then $h(Y)$ is actually an invariant of $X$ alone. The other is the Lefschetz number of the map on reduced monopole homology induced by the cobordism $W$ obtained from cutting X open along Y. We show that $\lambda_{SW}(X)+ h(Y) = Lef (W_*: HM_{red}(Y) \to HM_{red}(Y)).$ This is joint work with Jianfeng Lin and Nikolai Saveliev.