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We introduce a gauge-theoretic integer lift of the Rohlin invariant of asmooth 4-manifold X with the homology of $S^1 \times S^3$. The invariant hastwo 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-compactmanifold with end modeled on the infinite cyclic cover of X. Each term ismetric (and perturbation) dependent, and we show that these dependencies cancelas the metric and perturbation vary in a 1-parameter family.

Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant