Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds

1996/12/03 by Michael Hutchings, Hutchings, Michael, Yi‐Jen Lee +3
Computer Science · Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #dg-ga #math.DG

paper · pdf · doi:10.48550/arxiv.dg-ga/9612004

35 pages, AMS-LaTeX (one hand-drawn figure available on request)

arxiv created 1996/12/03 · arxiv updated 2016/08/31

Abstract

Let X be a compact oriented Riemannian manifold and let ϕ:X→ S¹ be a circle-valued Morse function. Under some mild assumptions on ϕ, we prove a formula relating: (a) the number of closed orbits of the gradient flow of ϕ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of ϕ; and (c) a kind of Reidemeister torsion of X determined by the homotopy class of ϕ. When dim(X)=3 and b<sub>1</sub>(X)>0, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed manifolds, modulo signs) the Meng-Taubes relation between the Seiberg- Witten invariants and the ``Milnor torsion'' of X.

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