15 research outputs found
Instanton Floer homology and the Alexander polynomial
The instanton Floer homology of a knot in the three-sphere is a vector space
with a canonical mod 2 grading. It carries a distinguished endomorphism of even
degree,arising from the 2-dimensional homology class represented by a Seifert
surface. The Floer homology decomposes as a direct sum of the generalized
eigenspaces of this endomorphism. We show that the Euler characteristics of
these generalized eigenspaces are the coefficients of the Alexander polynomial
of the knot. Among other applications, we deduce that instanton homology
detects fibered knots.Comment: 25 pages, 6 figures. Revised version, correcting errors concerning
mod 2 gradings in the skein sequenc
Exact Triangles for SO(3) Instanton Homology of Webs
The SO(3) instanton homology recently introduced by the authors associates a finite-dimensional vector space over the field of two elements to every embedded trivalent graph (or "web"). The present paper establishes a skein exact triangle for this instanton homology, as well as a realization of the octahedral axiom. From the octahedral diagram, one can derive equivalent reformulations of the authors' conjecture that, for planar webs, the rank of the instanton homology is equal to the number of Tait colorings.National Science Foundation (U.S.) (Grant DMS-0805841)National Science Foundation (U.S.) (Grant DMS-1406348
Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant
We introduce a gauge-theoretic integer lift of the Rohlin invariant of a
smooth 4-manifold X with the homology of . The invariant has
two terms; one is a count of solutions to the Seiberg-Witten equations on X,
and the other is essentially the index of the Dirac operator on a non-compact
manifold with end modeled on the infinite cyclic cover of X. Each term is
metric (and perturbation) dependent, and we show that these dependencies cancel
as the metric and perturbation vary in a 1-parameter family.Comment: Revision after referee report; section on negative definite manifolds
revised. To appear in JD
PU(2) monopoles and a conjecture of Marino, Moore, and Peradze
In this article we show that some of the recent results of Marino, Moore, and
Peradze (math.DG/9812042, hep-th/9812055) -- in particular their conjecture
that all closed, smooth four-manifolds with b_2^+ > 1 (and Seiberg-Witten
simple type) are of `superconformal simple type' -- can be understood using a
simple mathematical argument via the PU(2)-monopole cobordism of Pidstrigach
and Tyurin (dg-ga/9507004) and results of the first and third authors
(dg-ga/9712005, dg-ga/9709022).Comment: 13 pages, 1 figure. Improved exposition, typographical slips
corrected, figure and references added. Minor correction on page 2. To appear
in Mathematical Research Letter
Witten's conjecture and Property P
Let K be a non-trivial knot in the 3-sphere and let Y be the 3-manifold
obtained by surgery on K with surgery-coefficient 1. Using tools from gauge
theory and symplectic topology, it is shown that the fundamental group of Y
admits a non-trivial homomorphism to the group SO(3). In particular, Y cannot
be a homotopy-sphere.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper7.abs.html Version 5: links
correcte
L-2-topology and Lagrangians in the space of connections over a Riemann surface
We examine the L²-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl lemma of harmonic analysis, and deduce local pathwise connectedness of the gauge orbits. Based on a quantitative version of the connectivity, we generalize compactness results for anti-self-dual instantons with Lagrangian boundary conditions to general gauge-invariant Lagrangian submanifolds. This provides the foundation for the construction of instanton Floer homology for pairs of a 3-manifold with boundary and a Lagrangian in the configuration space over the boundary