Let X be a compact oriented Riemannian manifold and let φ : X → S 1 be a circle-valued Morse function. Under some mild assumptions on φ, we prove a formula relating: (a) the number of closed orbits of the gradient flow of φ of any given degree; (b) the torsion of a " Morse complex " , which counts gradient flow lines between critical points of φ; and (c) a… (More)
This is the first part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, I F. The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance—the Reidemeister torsion τ F of the Floer-Novikov… (More)
The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance—the Reidemeister torsion τ F of the Floer complex of (possibly non-hamiltonian) symplectomorphisms. τ F turns out not to be invariant under hamiltonian isotopies, but we… (More)
This is an expansion on my talk at the Geometry and Topol-ogy conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by Kronheimer-Mrowka. The center-piece of this program is the construction of an intermediate version of Floer theory, which… (More)
Protein arginine methyltransferase (PRMT) 1 is the most conserved and widely distributed PRMT in eukaryotes. PRMT8 is a vertebrate-restricted paralogue of PRMT1 with an extra N-terminal sequence and brain-specific expression. We use zebrafish (Danio rerio) as a vertebrate model to study PRMT8 function and putative redundancy with PRMT1. The transcripts of… (More)
Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface auto-morphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer cohomology and the corresponding… (More)
This is the second part of an article in two parts , which builds the foundation of a Floer-theoretic invariant, I F. (See [Pt1] for part I). Having constructed I F and outlined a proof of its invariance based on bi-furcation analysis in part I, in this part we prove a series of gluing theorems to confirm the bifurcation behavior predicted in part I. These… (More)
We construct the Seiberg-Witten theory on asymptotically flat three manifolds and describe the structure of the moduli space. The analysis should serve as the basis for many applications in 3-manifold topology, including a proof of the equivalence of the Seiberg-Witten invariant of 3-manifolds and the Reidemeister torsion .