We introduce a gauge-theoretic integer valued lift of the Rohlin invariant of a smooth 4-manifold $X$ with the homology of $S^1×S^3$. The invariant has two terms: one is a count of solutions to the Seiberg–Witten equations on $X$, and the other is essentially the index of the Dirac operator on a non-compact manifold with end modeled on the infinite cyclic cover of $X$. Each term is metric (and perturbation) dependent, and we show that these dependencies cancel as the metric and perturbation vary in a generic 1-parameter family.
"Seiberg-Witten Equations, End-Periodic Dirac Operators, and a Lift of Rohlin's Invariant." J. Differential Geom. 88 (2) 333 - 377, June 2011. https://doi.org/10.4310/jdg/1320067650