Reidemeister torsion in circlevalued Morse theory
In Morse theory, one starts with a generic realvalued 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 realvalued to circlevalued 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 "SeibergWitten=Gromov" correspondence suggests that the
SeibergWitten invariant of a threemanifold with positive first Betti
number should be computable by counting gradient flow lines and closed
orbits of a harmonic circlevalued 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 SeibergWitten theory is discussed only in the
first two papers.

[1] Circlevalued Morse theory, Reidemeister torsion, and
SeibergWitten invariants of 3manifolds (with YJ. Lee),
Topology 38 (1999), 861888. Postscript PDF
We show that by suitably counting closed orbits and flow lines between
critical points of the gradient of a circlevalued Morse function on a
manifold, one recovers a form of topological Reidemeister torsion. On
a threemanifold with positive first Betti number, we conjecture that
a finer version of the Morse theory invariant is equal to the
SeibergWitten invariant, by analogy with Taubes' ``SeibergWitten =
Gromov'' theorem in four dimensions. Combining our theorem with this
conjecture, we recover the MengTaubes formula relating part of the
SeibergWitten invariant to Milnor torsion.

[2] Circlevalued Morse theory and Reidemeister torsion (with
YJ. Lee),
Geometry and Topology 3 (1999), 369396. Postscript PDF.
This paper proves a refinement of the main theorem of [1],
using a different method. In the three dimensional case, combining
this result with the conjecture in [1], we obtain a
formula for the full SeibergWitten invariant, which was
conjectured by Turaev.

[3] Reidemeister torsion in generalized Morse theory,
Forum Mathematicum 14 (2002), 209244. pdf
This paper reproves the results of [1] and [2], and extends them from
circlevalued functions to closed 1forms. The strategy is to first
give an a priori proof, by bifurcation analysis, that our
Morsetheoretic analogue of Reidemeister torsion is a topological
invariant. We then use invariance to reduce to the easier case of
realvalued 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 SeibergWitten invariant of a 3manifold 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. YiJen Lee
has generalized the invariance proof in [3] to some infinite
dimensional (Floer theoretic) contexts.
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