Organizers: Selman Akbulut (Michigan State University), Anar Akhmedov (University of Minnesota), Weimin Chen (University of Massachusetts, Amherst), Cagri Karakurt (University of Texas, Austin), Tian-Jun Li (University of Minnesota).The main focus of this workshop will be on the topology and invariants of smooth 4-manifolds. The workshop is a part of the FRG: Collaborative Research: Topology and Invariants of Smooth 4-Manifolds. It is funded by NSF Focused Research Grant DMS-1065955. There will be 5 days of mini-courses (July 31 - August 4), primarily for graduate students, followed by a conference during the second week (August 5 - 10).
Thanks to the generous support of the National Science Foundation, funds are available for partial support of participant expenses, such as lodging and meals. The applicants are kindly asked to seek for travel support from their home institutions. To apply for funding, you must register by Friday, April 12, 2013 , but early applications are strongly encouraged since there are a limited number of rooms available. Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Applicants are requested to register (see the registration form below), send a CV, and have one brief reference letter sent to Cagri Karakurt at firstname.lastname@example.org. The reference letter is optional for people with a Ph.D.
***Please note that we are no longer accepting application*** .
Abstract of mini courses :
4-manifolds via their handlebodies (by Selman Akbulut)
Handlebody descriptions of 3 and 4-manifolds will be discussed, and as time permits their various applications to 4-manifold problems will be given, such as carving, branch coverings, knot complements, complex surfaces, Stein manifolds, Lefschetz fibrations, BLF's, corks and plugs. From these techniques various exotic manifolds will be constructed, such as going from handlebody of logarithmic transforms to Dolgachev surfaces, and from handlebody of surface bundles over surfaces to Akhmedov-Park exotic manifolds.
Construction of exotic 4-manifolds (by Anar Akhmedov)
Lecture 1: Construction of Lefschetz fibrations via Luttinger surgery
Luttinger surgery is a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold. The surgery was introduced by Karl Murad Luttinger in 1995, who used it to study Lagrangian tori in R^4. Luttinger's surgery has been very effective tool recently for constructing exotic smooth structures on 4-manifolds. In this talk, using Luttinger surgery, I will present a new constructions of Lefschetz fibration over 2-sphere whose total space has arbitrary finitely presented group G as the fundamental group (a joint result with Burak Ozbagci).
Lecture 2: The geography of symplectic 4-manifolds
The symplectic geography problem, originally posed by Robert Gompf, ask which ordered pairs of nonnegative integeres are realized as (chi(X), c1^2(X)) for some symplectic 4-manifold X. In this lecture we will address the geography problem of simply-connected spin and non-spin symplectic 4-manifolds with nonnegative signature or near the Bogomolov-Miyaoka-Yau line c1^2(X) = 9chi(X) (joint results with B. Doug Park).
Lecture 3: Exotic 4-manifolds with small Euler characteristics
It is known that many simply connected, smooth topological 4-manifolds admit infinitely many exotic smooth structures. However, the smaller the Euler characteristic, the harder it is to construct exotic smooth structure. We will construct exotic smooth structures on various small 4-manifolds (mostly joint results with B. Doug Park).
Group actions on 4-manifolds (by Weimin Chen)
The Kodaira dimension of symplectic 4-manifolds (by Tian-Jun Li)
The triangulation conjecture (by Ciprian Manolescu)
First lecture: The Seiberg-Witten equations in 3 dimensions
I will review the Kronheimer-Mrowka construction of monopole Floer homology for 3-manifolds, with particular emphasis on the Froyshov invariant. I will also discuss the Pin(2) symmetry that appears in the presence of a spin structure, and some notions of Pin(2)-equivariant topology.
Second lecture: Finite dimensional approximation and the Floer spectrum
I will present an alternate construction of S^1-equivariant Seiberg-Witten Floer homology, based on finite dimensional approximation and Conley index theory. The construction yields something more, an S^1-equivariant Floer spectrum. When we have a spin structure, everything can be tweaked to take into account the Pin(2)-equivariance of the equations.
Third lecture (conference talk): The triangulation conjecture
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov'’s correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that the 3-dimensional homology cobordism group has no elements of order 2 that have Rokhlin invariant one. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.
Lefschetz fibrations and symplectic fillings of contact 3-manifolds (by Chris Wendl)
Lefschetz fibrations have played a major role in symplectic topology since the 1990's, when they appeared in the topological characterization of closed symplectic 4-manifolds due to Donaldson and Gompf. A short time
later, Giroux introduced a corresponding characterization of contact manifolds in terms of open book decompositions, which can be understood as the natural structure arising at the boundary of a Lefschetz fibration over the disk. My goal in this minicourse will be to explain the relationship between symplectic/contact structures and these types of topological decompositions, and to sketch some striking applications that make use of holomorphic curve technology, e.g. the fact that all weak fillings of planar contact manifolds are blow-ups of Stein fillings (a joint result with Niederkrueger), and a recent generalization of that result in joint work with Lisi and Van Horn-Morris. My perspective on this subject is rather more geometric than topological: instead of following the topologist's standard approach of understanding Lefschetz fibrations and Stein structures in terms of handle decompositions, I will explain a version of the "Thurston trick" that gives existence and uniqueness (up to deformation) of symplectic and Stein structures on Lefschetz fibrations over arbitrary surfaces with boundary. I will then sketch some of the holomorphic curve techniques that give results in the other direction.
Here is an approximate outline of the content of the lectures:
LECTURE 1: "Basic notions and results"
Definitions of contact structures and weak/strong/exact/Stein fillings, Lefschetz fibrations, spinal open books and contact structures, a theorem on fillings of partially planar spinal open books, examples and applications.
LECTURE 2: "Symplectic geometry from topological decompositions"
Existence/uniqueness of contact structures on spinal open books (generalizing a result of Thurston-Winkelnkemper), existence/uniqueness of symplectic and Stein structures on (allowable) Lefschetz fibrations (generalizing results of Thurston and Gompf).
LECTURE 3: "Topological decompositions from symplectic geometry"
Stable Hamiltonian structures, holomorphic curves in completed symplectic fillings, reconstructing a Lefschetz fibration from a planar open book at the boundary.