# 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.

•  Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds (with Y-J. Lee),
Topology 38 (1999), 861-888. Postscript PDF
We show that by suitably counting closed orbits and flow lines between critical points of the gradient of a circle-valued Morse function on a manifold, one recovers a form of topological Reidemeister torsion. On a three-manifold with positive first Betti number, we conjecture that a finer version of the Morse theory invariant is equal to the Seiberg-Witten invariant, by analogy with Taubes' ``Seiberg-Witten = Gromov'' theorem in four dimensions. Combining our theorem with this conjecture, we recover the Meng-Taubes formula relating part of the Seiberg-Witten invariant to Milnor torsion.

•  Circle-valued Morse theory and Reidemeister torsion (with Y-J. Lee),
Geometry and Topology 3 (1999), 369-396. Postscript PDF.
This paper proves a refinement of the main theorem of , using a different method. In the three dimensional case, combining this result with the conjecture in , we obtain a formula for the full Seiberg-Witten invariant, which was conjectured by Turaev.

•  Reidemeister torsion in generalized Morse theory,
Forum Mathematicum 14 (2002), 209-244. pdf
This paper re-proves the results of  and , and extends them from circle-valued functions to closed 1-forms. The strategy is to first give an a priori proof, by bifurcation analysis, that our Morse-theoretic analogue of Reidemeister torsion is a topological invariant. We then use invariance to reduce to the easier case of real-valued functions. It is hoped that the a priori proof of invariance will provide a model for the possible construction of torsion invariants in Floer theory.
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  indirectly prove the conjecture of . 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  to some infinite dimensional (Floer theoretic) contexts.

You are vistor number to this page since my web counters got reset.