89 research outputs found
A Casson-Lin type invariant for links
We define an integer valued invariant for two-component links in S^3 by
counting projective SU(2) representations of the link group having non-trivial
second Stiefel-Whitney class. We show that our invariant is, up to sign, the
linking number of the link. Our construction generalizes that of X.-S. Lin who
defined a similar invariant for knots in S^3; his invariant equals half the
knot signature.Comment: New set of generators for the braid group in Section
Instanton Floer homology for two-component links
For any link of two components in an integral homology sphere, we define an
instanton Floer homology whose Euler characteristic is the linking number
between the components of the link. We relate this Floer homology to the
Kronheimer-Mrowka instanton Floer homology of knots. We also show that, for
two-component links in the 3-sphere, the Floer homology does not vanish unless
the link is split.Comment: 10 pages, 4 figures; updated to clarify relation to the
Kronheimer-Mrowka KHI theor
Rohlin's invariant and gauge theory, I. Homology 3-tori
This is the first in a series of papers exploring the relationship between
the Rohlin invariant and gauge theory. We discuss the Casson-type invariant of
a 3-manifold with the integral homology of a torus, given by counting
projectively flat connections. We show that its mod 2 evaluation is given by
the triple cup product in cohomology, and so it coincides with a sum of Rohlin
invariants. Our counting argument makes use of a natural action of the first
cohomology on the moduli space of projectively flat connections; along the way
we construct perturbations that are equivariant with respect to this action.
Combined with the Floer exact triangle, this gives a purely gauge-theoretic
proof that Casson's homology sphere invariant reduces mod 2 to the Rohlin
invariant.Comment: Changed title to fit with succeeding papers in series. Added
reference to Turaev's wor
Casson-type invariants from the Seiberg-Witten equations
This is a survey of our recent work with Tom Mrowka on Seiberg-Witten gauge
theory and index theory for manifolds with periodic ends. We explain how this
work leads to a new invariant, which is related to the classical Rohlin
invariant of homology 3-spheres and to the Furuta-Ohta invariant originating in
Yang-Mills gauge theory. We give some new calculations of our invariant for
4-dimensional mapping tori.Comment: Slightly expanded exposition; to appear in volume "New Ideas in
Low-Dimensional Topology", edited by L. Kauffman and V. Manturo
Link homology and equivariant gauge theory
The singular instanton Floer homology was defined by Kronheimer and Mrowka in
connection with their proof that the Khovanov homology is an unknot detector.
We study this theory for knots and two-component links using equivariant gauge
theory on their double branched covers. We show that the special generator in
the singular instanton Floer homology of a knot is graded by the knot signature
mod 4, thereby providing a purely topological way of fixing the absolute
grading in the theory. Our approach also results in explicit computations of
the generators and gradings of the singular instanton Floer chain complex for
several classes of knots with simple double branched covers, such as two-bridge
knots, torus knots, and Montesinos knots, as well as for several families of
two-components links.Comment: 59 pages. Corrected a grading error in Lemma 2.5, which affected
calculations for some of the knot
Rohlin's invariant and gauge theory II. Mapping tori
This is the second in a series of papers studying the relationship between
Rohlin's theorem and gauge theory. We discuss an invariant of a homology S^1
cross S^3 defined by Furuta and Ohta as an analogue of Casson's invariant for
homology 3-spheres. Our main result is a calculation of the Furuta-Ohta
invariant for the mapping torus of a finite-order diffeomorphism of a homology
sphere. The answer is the equivariant Casson invariant (Collin-Saveliev 2001)
if the action has fixed points, and a version of the Boyer-Nicas (1990)
invariant if the action is free. We deduce, for finite-order mapping tori, the
conjecture of Furuta and Ohta that their invariant reduces mod 2 to the Rohlin
invariant of a manifold carrying a generator of the third homology group. Under
some transversality assumptions, we show that the Furuta-Ohta invariant
coincides with the Lefschetz number of the action on Floer homology. Comparing
our two answers yields an example of a diffeomorphism acting trivially on the
representation variety but non-trivially on Floer homology.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper2.abs.htm
On the deleted squares of lens spaces
The configuration space of ordered pairs of distinct points in a
manifold , also known as the deleted square of , is not a homotopy
invariant of : Longoni and Salvatore produced examples of homotopy
equivalent lens spaces and of dimension three for which and
are not homotopy equivalent. In this paper, we study the natural
question whether two arbitrary -dimensional lens spaces and must be
homeomorphic in order for and to be homotopy equivalent.
Among our tools are the Cheeger--Simons differential characters of deleted
squares and the Massey products of their universal covers.Comment: 27 pages, 10 figure
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