The papers here might not be the most up-to-date versions; they are however usually more recent than the versions on www.arxiv.org. E-mail me if you have problems accessing any of the following. Research supported in part by a NSF grant DMS 0333163.
(with M. Hutchings) Circled-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of three manifolds
Topology 38, 861--888 (1999)This paper defines a Morse theoretic torsion I that counts both gradient flow lines between critical points and closed orbits. We show that I is equivalent to the combinatorially defined Reidemeister torsion of the underlying manifold, and in 3-dimensions, it is related to the Seiberg-Witten invariant of 3-manifolds.(with M. Hutchings) Circle-valued Morse theory and Reidemeister torsion
Geometry and Topology 3, 369--396 (1999)This paper refines the previous one using a different proof.Seiberg-Witten theory on three-manifolds with euclidean ends
Comm. Analysis and Geometry, 13, no. 1 (2005), 1--88 (accepted April 2002)This long and technical paper lays the foundation of a version of Seiberg-Witten theory on 3-manifolds with euclidean ends. (The analysis required here is very different from the more typical cylindrical situation). The set up was partly inspired by Taubes' program relating the 4-dimensional Seiberg-Witten invariant to a singular Gromov invariant and has related potential applications. The last section and the appendix of the Math.Arxiv version have been deleted to shorten the paper.Torsion invariants in symplectic Floer theory
Proceedings of the International Congress of Chinese Mathematicians 2001, 25 pages (to appear)This is an expanded version of the talk I gave at ICCM 2001; it surveys the background and results in the next three papers.Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori, I
Journal of Symplectic Geometry, vol.3, no.2 (2005), 221--311.A even longer and more technical paper on foundations. Modelling on the finite dimensional Morse-theoretic torsion I defined in [HL1] and [HL2], this paper defines a Floer theoretic torsion I_F which counts both (perturbed) pseudo-holomorphic cylinders ending at periodic orbits, and pseudo-holomorphic tori. I_F is invariant under hamiltonian isotopies, and under certain monotonicity conditions, also under general symplectic isotopies. When the symplectic manifold is monotone and when the symplectomorphism is isotopic to the identity, I_F reduces to the classical Reidemeister torsion of the symplectic manifold. The bulk of the paper consists of the bifurcation analysis, namely, how the moduli spaces of pseudo-holomorphic cylinders and tori vary under symplectic isotopies. This is the first time such analysis appears in its entirety in the Floer theory literature, and it is required for construction of non-homotopy invariants of the Floer complex, of which the torsion is only the simplest example. Note: The original article math.DG/0111313 is split into two parts. This is the first part.Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori, II
Journal of Symplectic Geometry, vol.3, no.3 (2005) 385--480.(with M. Sullivan) Reidemeister torsion in Floer theory of Lagrangian intersections
unfinishedThe original plan is to adapt the previous long paper [L5] to the lagrangian intersection version, substituting the lengthy gluing theorems involving degenerate critical points there with a generalization of the stabilization method of Mike's thesis. A simplified version of these results, following the original proof of [L5], is included in the next article.Noncontractible periodic orbits, Gromov invariants, and Floer-theoretic torsions
submitted, 73 pages.Finally, some applications. The existence of periodic orbits is an important problem in dynamical systems, and in symplectic topology such existence results is typically a manifestation of the `rigidity' phenomenon. However, to date most of such results are restricted to contractible periodic orbits. We obtain some existence results for noncontractible period orbits (of symplectic vector fields) using variants of the Floer-theoretic torsions I_F defined in [L5]. The heart of the proof is an intriguing relation of I_F with (open or closed) Gromov-type invariants.Noncontractible periodic orbits, Gromov invariants, and Floer-theoretic torsions
These are copies of transparencies of a talk I gave at IPAM and elsewhere.Heegaard splittings and Seiberg-Witten monopoles
Geometry and topology of manifolds, 173--202, Fields Inst. Commun., 47, Amer. Math. Soc., 2005This paper outlines a long program to relate the Seiberg-Witten and Ozsvath-Szabo Floer theories, and surveys some partial results towards this goal, to appear in the next few papers.Heegaard splittings and Seiberg-Witten monopoles
slides for some recent survey talks.A Seiberg-Witten model of Heegaard Floer homologies
in progressThis article and the next are the first two parts of a long program that aims to relate the Seiberg-Witten theory and the Ozsvath-Szabo theory. (There should be one or two more parts...) The main idea extends the circle of ideas that led to my earlier papers [HL1], [L3].Seiberg-Witten monopoles on 3-manifolds of cylindrical ends
in progress
All files on this page are in pdf format. They can be viewed/printed with Adobe Acrobat Reader, ghostview, or gv. On most UNIX platforms, the command is (after you download and save a file):
acroread <filename>orghostview <filename>orgv <filename>For best on-screen reading, use Adobe Acrobat Reader.