363 research outputs found
Defining and classifying TQFTs via surgery
We give a presentation of the -dimensional oriented cobordism category
with generators corresponding to diffeomorphisms and surgeries
along framed spheres, and a complete set of relations. Hence, given a functor
from the category of smooth oriented manifolds and diffeomorphisms to an
arbitrary category , and morphisms induced by surgeries along framed
spheres, we obtain a necessary and sufficient set of relations these have to
satisfy to extend to a functor from to . If is symmetric
and monoidal, then we also characterize when the extension is a TQFT.
This framework is well-suited to defining natural cobordism maps in Heegaard
Floer homology. It also allows us to give a short proof of the classical
correspondence between (1+1)-dimensional TQFTs and commutative Frobenius
algebras. Finally, we use it to classify (2+1)-dimensional TQFTs in terms of
J-algebras, a new algebraic structure that consists of a split graded
involutive nearly Frobenius algebra endowed with a certain mapping class group
representation. This solves a long-standing open problem. As a corollary, we
obtain a structure theorem for (2+1)-dimensional TQFTs that assign a vector
space of the same dimension to every connected surface. We also note that there
are nonequivalent lax monoidal TQFTs over that do
not extend to (1+1+1)-dimensional ones.Comment: 68 pages, 4 figures, to appear in Quantum Topolog
Spectral order for contact manifolds with convex boundary
We extend the Heegaard Floer homological definition of spectral order for
closed contact 3-manifolds due to Kutluhan, Mati\'c, Van Horn-Morris, and Wand
to contact 3-manifolds with convex boundary. We show that the order of a
codimension zero contact submanifold bounds the order of the ambient manifold
from above. As the neighborhood of an overtwisted disk has order zero, we
obtain that overtwisted contact structures have order zero. We also prove that
the order of a small perturbation of a Giroux torsion domain has order
at most two, hence any contact structure with positive Giroux torsion has order
at most two (and, in particular, a vanishing contact invariant).Comment: 18 pages, 5 figures, to appear in Algebraic and Geometric Topolog
Contact handles, duality, and sutured Floer homology
We give an explicit construction of the Honda--Kazez--Mati\'c gluing maps in
terms of contact handles. We use this to prove a duality result for turning a
sutured manifold cobordism around, and to compute the trace in the sutured
Floer TQFT. We also show that the decorated link cobordism maps on the hat
version of link Floer homology defined by the first author via sutured manifold
cobordisms and by the second author via elementary cobordisms agree.Comment: 86 pages, 54 figures, to appear in Geometry and Topolog
Knot cobordisms, bridge index, and torsion in Floer homology
Given a connected cobordism between two knots in the 3-sphere, our main
result is an inequality involving torsion orders of the knot Floer homology of
the knots, and the number of local maxima and the genus of the cobordism. This
has several topological applications: The torsion order gives lower bounds on
the bridge index and the band-unlinking number of a knot, the fusion number of
a ribbon knot, and the number of minima appearing in a slice disk of a knot. It
also gives a lower bound on the number of bands appearing in a ribbon
concordance between two knots. Our bounds on the bridge index and fusion number
are sharp for and , respectively. We
also show that the bridge index of is minimal within its concordance
class.
The torsion order bounds a refinement of the cobordism distance on knots,
which is a metric. As a special case, we can bound the number of band moves
required to get from one knot to the other. We show knot Floer homology also
gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon
knots with arbitrarily large ribbon distance from the unknot.Comment: 21 pages, 7 figures, to appear in the Journal of Topolog
Sutured Floer homology, fibrations, and taut depth one foliations
For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we
describe how sutured Floer homology () can be used to determine all
fibered classes in . Furthermore, we show that the of a balanced
sutured manifold detects which classes in admit a taut
depth one foliation such that the only compact leaves are the components of
. The latter had been proved earlier by the first author under the
extra assumption that . The main technical result is that we can
obtain an extremal -structure (i.e., one that is
in a `corner' of the support of ) via a nice and taut sutured manifold
decomposition even when , assuming the corresponding group
has non-trivial Euler characteristic.Comment: 30 pages, improved expositio
Holomorphic discs and sutured manifolds
In this paper we construct a Floer-homology invariant for a natural and wide
class of sutured manifolds that we call balanced. This generalizes the Heegaard
Floer hat theory of closed three-manifolds and links. Our invariant is
unchanged under product decompositions and is zero for nontaut sutured
manifolds. As an application, an invariant of Seifert surfaces is given and is
computed in a few interesting cases.Comment: This is the version published by Algebraic & Geometric Topology on 4
October 200
Functoriality of the EH class and the LOSS invariant under Lagrangian concordances
We show that the EH class and the LOSS invariant of Legendrian knots in
contact 3-manifolds are functorial under regular Lagrangian concordances in
Weinstein cobordisms. This gives computable obstructions to the existence of
regular Lagrangian concordances.Comment: 13 pages, 1 figure, to appear in Algebraic and Geometric Topolog
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