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Boston College

HMI

NSF

SFI

Jacob Rasmussen (Cambridge) and Liam Watson (Glasgow)

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

Floer Homology and low-dimensional topology

Lecture 1 (Rasmussen): Floer homologies for three-manifolds
• HF as a TQFT
• Spin^c structures
• \hat,+,-,\infty

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
• 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

Homological invariants in low-dimensional topology and geometry

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.

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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