Circle-valued Morse theory and Reidemeister torsion

Abstract

Let $X$ be a closed manifold with $\chi \left(X\right)=0$, and let $f:X\to S^1$ be a circle-valued Morse function. We define an invariant $I$ which counts closed orbits of the gradient of $f$, together with flow lines between the critical points. We show that our invariant equals a form of topological Reidemeister torsion defined by Turaev [Math. Res. Lett. 4 (1997) 679–695].

We proved a similar result in our previous paper [Topology 38 (1999) 861–888], but the present paper refines this by separating closed orbits and flow lines according to their homology classes. (Previously we only considered their intersection numbers with a fixed level set.) The proof here is independent of the previous proof, and also simpler.

Aside from its Morse-theoretic interest, this work is motivated by the fact that when $X$ is three-dimensional and $b_1\left(X\right)>0$, the invariant $I$ equals a counting invariant $I_3\left(X\right)$ which was conjectured in our previous paper to equal the Seiberg–Witten invariant of $X$. Our result, together with this conjecture, implies that the Seiberg–Witten invariant equals the Turaev torsion. This was conjectured by Turaev and refines the theorem of Meng and Taubes [Math. Res. Lett 3 (1996) 661–674].