Sponsored by
Boston College
HMI
NSF
SFI
|
The 10th William
Rowan
Hamilton Geometry and
Topology Workshop
on
Homological
invariants
in
low-dimensional
topology
and
geometry
August 26-30, 2014
The Hamilton Mathematics
Institute,
Trinity College Dublin
Mini-Course
on
Floer homology and low-dimensional topology
with
Jacob Rasmussen (Cambridge) and Liam Watson (Glasgow)
August
26-27, 2014
Schedule of Mini-Course:
All minicourse lectures will be held in the Maxwell Lecture Hall in the
Hamilton Building (see campus map).
Time
|
Speaker
|
Talk
|
Tuesday
|
|
9:30
-
10:30
|
Jacob Rasmussen (Cambridge)
|
Floer Homology and
low-dimensional topology |
10:30 - 11:00
|
COFFEE & DISCUSSION
|
11:00
-
12:00
|
Liam Watson (Glasgow) |
Left-orderable
groups and three-manifolds |
12:00
-
2:30
|
LUNCH |
2:30
-
3:30
|
Jacob Rasmussen (Cambridge)
|
Basic structure of
Heegaard Floer homology |
3:30 - 4:00
|
DISCUSSION
|
4:00-5:00
|
Liam Watson (Glasgow) |
L-spaces |
Wednesday
|
|
| 9:30
-
10:30
|
Jacob Rasmussen (Cambridge) |
Heegaard Floer
homology and the Thurston norm, I |
10.30 - 11:00
|
COFFEE &
DISCUSSION |
| 11:00
-
12:00 |
Jacob Rasmussen (Cambridge) |
Heegaard Floer
homology and the Thurston norm, II |
| 12:00
-
2:30 |
LUNCH |
| 2:30
-
3:30 |
Liam Watson (Glasgow) |
Foliations and a
theorem of Ozsváth and Szabó.
|
| 3:30 - 4:00 |
DISCUSSION |
4:00 - 5:00
|
TBA
|
TBA
|
Overview of Minicourse
Floer Homology and low-dimensional topology
Lecture 1 (Rasmussen):
Floer homologies for three-manifolds
• HF as a TQFT
• Spin^c structures
• \hat,+,-,\infty
• gradings
Lecture 2 (Watson):
Left-orderable groups and three-manifolds
• left-orders, bi-orders and circular orders
• definition via <; definition via positive cones (exercise: these
are equivalent)
• basic properties, examples
• subgroups of Homeo(R), actions on R
• Burns-Hale criterion
• proof of the Boyer-Rolfsen-Wiest theorem for orientable 3-manifolds
Lecture 3 (Rasmussen):
Basic structure of Heegaard Floer homology
• Heegaard diagrams
• Generators, domains
• Spin^c structures
• Lipshitz's formula
Lecture 4 (Watson):
L-spaces
• definition of HFred; two definitions of L-spaces.
• examples
• Conjecture
• Prove something about one of the classes of examples (exercise:
two-fold branched covers)
• Predictions made by this conjecture: surgeries, covers.
Lecture 5 (Rasmussen):
Heegaard Floer homology and the Thurston norm, I
• Link Floer homology
• HF of surgeries on knots
• Thurston norm
Lecture 6 (Rasmussen):
Heegaard Floer homology and the Thurston norm, II
• Exact triangles
• Ozsvath-Szabo mapping cone
• property R
• Theorem of Gordon and Leucke
Lecture 7 (Watson):
Foliations and a theorem of Ozsváth and Szabó.
• Definitions
• interactions with orders
• Calegari-Dunfeild result, Boileau-Boyer result
• a non-trivial class in HFred
Problem Session or
Possible Lecture 8
Lecture
Series
on
Homological
invariants
in
low-dimensional
topology
and
geometry
August
28-30,
2014
Schedule of Talks:
All talks will be held in the MacNeil Lecture Hall in the
Hamilton
Building, TCD (see campus map).
Time
|
Speaker
|
Talk
|
Thursday
|
|
9.00-9.45
|
Michel Boileau (Toulouse) |
Hyperelliptic
rotations of prime odd orders on closed irreducible 3-manifolds. |
9.45-10.15
|
COFFEE
|
10:15-11:00
|
Adam
Clay
(Manitoba)
|
Graph manifolds,
left-orderability and L-spaces
|
| 11:15-12:00 |
Alessandra Iozzi
(ETH Zurich)
|
Rotation number, old
and newer
|
12:00-2:30
|
LUNCH |
2:30-3:15
|
Rachael
Roberts
(Washington)
|
Approximating
continuous foliations |
3:30-4:15
|
Igor
Mineyev
(Urbana
Champaign)
|
The deep-fall
property and L^2 (co)homology |
Friday
|
|
| 9:00-9:45
|
Brendan
Owens
(Glasgow)
|
Tait graphs, Goeritz
matrices and link invariants |
| 9.45-10.15 |
COFFEE |
| 10:15-11:00 |
Kristen
Hendricks
(UCLA)
|
Localization and the
link Floer homology of doubly-periodic knots
|
| 11:15-12:00 |
Marc
Culler (U Illinois, Chicago)
|
A-polynomials and
SU(2) character varieties
|
| 12:00-2:30 |
LUNCH |
| 2:30-3:15 |
Steve
Boyer
(Montreal)
|
Characterizing the
existence of co-oriented taut foliations on graph manifolds
|
| 3:30-4:15 |
Jeff
Brock
(Brown) |
Public
Lecture
Rolling the dice on topology: what is the geometry of a random
3-manifold?
|
Saturday
|
|
| 9:00-9:45
|
Jean
Raimbault
(Max-Plank)
|
Homology and volume
of locally symmetric spaces |
| 9.45-10.15
|
COFFEE
|
| 10:15-11:00 |
Ron
Fintushel
(Michigan
State)
|
Pinwheels and
4-manifolds |
| 11:15-12:00 |
Thang Le (Georgia
Tech)
|
Homology and torsion
growth in abelian coverings |
| 12:00-2:30
|
LUNCH
|
| 2:30-3:15 |
Matt Hedden
(Michigan State)
|
Obstructing the
existence of algebraic curves with prescribed singularities
|
| END OF
WORKSHOP |
Abstracts for Lecture Series
Michel Boileau (Toulouse)
Title: Hyperelliptic
rotations of prime odd orders on closed irreducible 3-manifolds
Abstract: A classical way to construct some closed orientable
3-manifolds is by taking finite coverings of the 3-sphere $S^3$
branched along knots. Let call hyperelliptic rotation a
periodic diffeomorphism of a closed orientable $3$-manifold
corresponding to the covering translation of a cyclic covering of
$S^3$ branched along a knot. In this talk we combine geometry and
finite group theory to show that the orientation preserving
diffeomorphism group of a closed irreducible 3-manifold which is not
homeomorphic to $S^3$ contains at most $6$ conjugacy classes of cyclic
subgroups generated by a hyperelliptic rotation of prime odd order. At
the moment we do not know examples, other than $S^3$, whith more
than $3$ distinct conjugacy classes.
This is a joint work with Clara Franchi, Mattia Mecchia, Luisa
Paoluzzi and Bruno Zimmermann.
Steve
Boyer (Montreal)
Title: Characterizing the existence of co-oriented taut foliations on
graph manifolds.
Abstract: The problem of determining when a graph manifold admits a
(topological) co-oriented taut foliation is an interesting one which
can be refined by requiring the foliation to be horizontal, or strongly
rational, or smooth. In this talk we describe how the existence of such
foliations on the manifold in the topological, resp. horizontal, resp.
strongly rational, case is equivalent to the existence of certain types
of left-orders on its fundamental group, and also equivalent to the
existence of certain types of representations of its fundamental group
with values in Homeo_+(R). We will also discuss some results and
questions concerning the smoothness of the foliations as well as
connections to the Heegaard-Floer homology of the manifold.
This is joint work with Adam Clay.
Adam
Clay (Manitoba)
Title: Graph manifolds, left-orderability and L-spaces
Abstract: It is a conjecture of Boyer, Gordon and Watson that an
irreducible rational homology 3-sphere is an L-space if and only if its
fundamental group is non-left-orderable. Towards verifying that this
conjecture holds for graph manifolds, we show that a graph manifold
which is a rational homology 3-sphere has a left-orderable fundamental
group if and only if it admits a co-orientable taut foliation. In
particular, we construct foliations on graph manifolds by ensuring that
the Seifert fibred pieces admit foliations obeying certain gluing
conditions. These gluing conditions can be
translated into a purely Heegaard-Floer theoretic language, allowing us
to say exactly which rational homology 3-sphere graph manifolds must be
L-spaces in order for the conjecture to hold true in this case. This is
joint work with Steve Boyer.
Marc
Culler (U Illinois, Chicago)
Title: A-polynomials and SU(2) character varieties
Abstract: I will discuss the related computations of
A-polynomials and SU(2) character varieties for knots. I will
define the PE character variety, which contains the SU(2)
character variety but is in some ways more natural, and describe
how its structure is reflected by several topological invariants.
Finally, I will describe some experiments related to the L-space
conjecture which were inspired by the PE character variety.
Ron Fintushel (Michigan
State)
Title: Pinwheels and 4-manifolds
Abstract: I will explain the notion of a "pinwheel structure" on a
4-manifold and exhibit their utility in constucting 4-manifolds with
b^+ = 1, both simply connected and not.
Matt Hedden
(Michigan State)
Title: Obstructing the existence of algebraic curves with prescribed
singularities
Abstract: A non-singular algebraic curve in the complex projective
plane of degree d has topological genus (d-1)(d-2)/2. If the curve has
singularities, yet topologically is still an embedded surface, then the
genus will be lower. Heuristically, some of the topology gets pushed
into the singularities. This talk will examine the question of which
configurations of singularities can arise in algebraic curves of degree
d that have some fixed topological genus. I will discuss new
obstructions that imply the non-existence of algebraic curves with
certain configurations of singularities. The obstructions come from
Heegaard Floer homology. Combining our obstructions with those from
algebraic geometry leads to a classification of genus one curves with a
single simple singularity. Perhaps surprisingly, the degrees and
singularity types which arise are given by even terms in the Fibonnacci
sequence. This is joint work with Maciej Borodzik and Charles
Livingston.
Kristen
Hendricks (UCLA)
Title: Localization and the link Floer homology of doubly-periodic
knots
Abstract: A knot K in S^3 is said to be q-periodic if there is an
orientation-preserving action of Z_q on S^3 which preseves K and has
fixed set an unknot disjoint from K. There are many classical
obstructions to the possible periods of a knot, including Edmonds'
condition on the genus and Murasugi's conditions on the Alexander
polynomial. We construct localization spectral sequences on the link
Floer homology of 2-periodic knots, and show that they give a
simultaneous generalization of Edmonds' condition and one of Murasugi's
conditions. We conclude with an example in which these spectral
sequences give a stronger obstruction than these two (although not all)
classical conditions.
Alessandra Iozzi
(ETH Zurich)
Title: Rotation number, old and newer
Abstract: Rotation numbers classify orientation preserving
homeomorphisms of the circle (hence actions of the integers) up to
semiconjugacy. After recalling the classical theory, we will show how
one can generalize the notion of rotation number both to actions of
groups larger than the integers and to actions on manifolds more
complicated than the circle.
As an application, we will illustrate some rigidity results.
Thang Le (Georgia Tech)
Title: Homology and torsion growth in abelian coverings
Abstract: We discuss the growth of the homology and
the regulators in finite abelian coverings of a finite complex.
Igor Mineyev (University
of Illinois, Urbana Champaigne)
Title: The deep-fall property and L^2 (co)homology.
Abstract:
The talk will be an overview and an advertisement for certain notions.
The
deep-fall property was used in the proof of the Strengthened Hanna
Neumann
Conjecture (SHNC). It is a property of (orderable) group actions on
graphs, and more generally, on cell complexes. In the case of graphs,
the
deep-fall property can be stated combinatorially. In the general
setting
of complexes, the language of L^2 homology is used, which relates it to
some open problems about L^2 invariants (Atiyah problem,
submultiplicativity). In short, the talk will mix homological
invariants,
low-dimensional topology, and just a bit of geometry, which is,
incidentally, a good match for the title of the workshop.
Brendan Owens (Glasgow)
Title: Tait graphs, Goeritz matrices and link invariants
Abstract: I will discuss link diagrams for which the Goeritz forms are
semi-definite, and what this tells us about their homological link
invariants. This is joint work in progress with Paolo Lisca and Liam
Watson.
Jean Raimbault (Max-Plank)
Title: Homology and volume of locally symmetric spaces
Abstract: A result due to Gromov implies that the Betti numbers of
locally symmetric spaces are at most proportional to their volume (with
constants depending only on the curvature and dimension). It is not
hard to see that this bound is optimal in all generality; we will
present some settings where one can obtain stronger constraints on the
Betti numbers (especially in the case of arithmetic manifolds) and time
permitting we will say a few words about the torsion subgroup of the
integral homology.
Rachael Roberts
(Washington)
Title: Approximating continuous foliations
Abstract: Taut foliations and tight contact structures are important
topological structures on 3-manifolds that are at opposite ends of the
spectrum in the study of 2-plane distributions. One is integrable
everywhere, the other is integrable nowhere. Eliashberg and Thurston
have shown that they are closely related; in fact, every sufficiently
smooth taut foliation can be perturbed to a pair of tight contact
structures, one positive and one negative. In joint work with Will
Kazez, we show that the smoothness assumptions on the foliation can be
modified. This allows the approximation theorem to be applied to a wide
range of recent constructions of (non-smooth) continuous foliations.
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