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.
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[1] 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.
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[2] 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 [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 Seiberg-Witten invariant, which was
conjectured by Turaev.
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[3] Reidemeister torsion in generalized Morse theory,
Forum Mathematicum 14 (2002), 209-244. pdf
This paper re-proves the results of [1] and [2], 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 [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. Yi-Jen Lee
has generalized the invariance proof in [3] to some infinite
dimensional (Floer theoretic) contexts.
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