Circlevalued Morse theory and Reidemeister torsion
Abstract
Let X be a closed manifold with zero Euler characteristic, and let f: X > S^1 be a circlevalued 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) 679695]. We proved a similar result in our previous paper [Topology, 38 (1999) 861888], 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 Morsetheoretic interest, this work is motivated by the fact that when X is threedimensional and b_1(X)>0, the invariant I equals a counting invariant I_3(X) which was conjectured in our previous paper to equal the SeibergWitten invariant of X. Our result, together with this conjecture, implies that the SeibergWitten invariant equals the Turaev torsion. This was conjectured by Turaev [Math. Res. Lett. 4 (1997) 679695] and refines the theorem of Meng and Taubes [Math. Res. Lett. 3 (1996) 661674].
 Publication:

eprint arXiv:dgga/9706012
 Pub Date:
 June 1997
 DOI:
 10.48550/arXiv.dgga/9706012
 arXiv:
 arXiv:dgga/9706012
 Bibcode:
 1997dg.ga.....6012H
 Keywords:

 Mathematics  Differential Geometry;
 57R70;
 53C07;
 57R19;
 58F09
 EPrint:
 28 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper15.abs.html