Reidemeister torsion in circle-valued Morse theory

In Morse theory, one starts with a generic real-valued smooth function on a closed smooth manifold. By counting gradient flow lines between critical points, one can recover the homology of the manifold. If we generalize from real-valued to circle-valued functions, a new dynamical feature appears, namely closed orbits of the gradient flow. By counting these together with gradient flow lines, one can recover the Reidemeister torsion of the manifold. This is what my PhD thesis was about. This project started when Cliff Taubes pointed out that his "Seiberg-Witten=Gromov" correspondence suggests that the Seiberg-Witten invariant of a three-manifold with positive first Betti number should be computable by counting gradient flow lines and closed orbits of a harmonic circle-valued function. The three papers below develop the Morse theoretic part of this story, which works in any number of dimensions and does not require the function to be harmonic. The third paper is my last and most definitive on this subject, but the connection with Seiberg-Witten theory is discussed only in the first two papers.

Some further developments: V. Turaev has proved his formula for the Seiberg-Witten invariant of a 3-manifold with positive first Betti number. Turaev's result and [2] indirectly prove the conjecture of [1]. T. Mark has given another proof of most of this conjecture. A. Pajitnov and D. Schuetz have introduced "noncommutative" refined versions of the Reidemeister torsion invariant which take into account the homotopy classes of the gradient flow lines and closed orbits, and not just their homology classes as in the above papers. R. Forman proved an analogue of our theorem in combinatorial Morse theory. Yi-Jen Lee has generalized the invariance proof in [3] to some infinite dimensional (Floer theoretic) contexts.

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