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

1996/12/03 by Michael Hutchings, Hutchings, Michael, Yi‐Jen Lee +2
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #Geometric and Algebraic Topology

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

Abstract

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

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