2-4 juillet 2025

Conférenciers invités:

    • Anthony Conway (UT Austin): Embedding surfaces in 4-manifolds
    • James Davis (University of Indiana): Introduction to surgery theory
    • Mark Powell (University of Glasgow): Classification of topological 4-manifold

Horaire des mini-cours

Lectures:
Anthony Conway:

These lectures are concerned with spheres in closed, simply-connected 4-manifolds. Emphasis will be placed on the case where the complement of the sphere has abelian fundamental group, i.e. so-called “simple spheres”.

Lecture 1: Definitions, examples and invariants. The goal is to discuss examples of knotted spheres as well as invariants based on the algebraic topology of the sphere exterior.

Lecture 2: Simple spheres up to isotopy. This lecture describes sufficient conditions for two simple spheres to be isotopic. This will be based on work of Lee-Wilczynski and on joint work with Powell.

Lecture 3: Realisation and enumeration. We conclude by discussing necessary and sufficient conditions for a given second homology class to be realised by a simple sphere. This will be based on work of Lee-Wilczynski and on joint work with Piccirillo and Powell.

James Davis:

Lecture 1:  Manifolds
This will be a whirlwind tour of standard topics in manifold theory including the following topics:   topological and smooth manifolds (with boundary), the surgery equivalence relation, bordism, handles, Morse theory and handlebody structure on manifolds, Dehn surgery, the Lickorish-Wallace Theorem, and Kirby diagrams for three-manifolds (surgery) and four-manifolds (handlebodies).

Lecture 2: Uniqueness and existence
This will be “surgery theory.”   The statement of the uniqueness and existence questions for homotopy structures on manifolds.    The motivating examples of S^4 (the 4d Poincare conjecture), the cone on the Poincare homology sphere, and the E_8 manifolds.  Rochlin’s and Donaldson’s theorem.  The (stable) h-cobordism theorem.    Degree one normal maps, surgery up to the middle dimension, and the surgery exact sequence.   L-groups.  Good groups.

Lecture 3:  Applications and the global structure of surgery
The first half of the talk will be applying the above theory to see why homology 3-spheres bound contractible manifolds, the 4d topological Poincare conjecture, why Alexander 1 knots (and links?) are topological slice, and the existence of *CP^2.  The second half of the lecture will indicate some of my applications of the algebraic surgery exact sequence to 4-manifolds – including how characteristic class formulae for surgery obstructions interact with stable diffeomorphisms and how the total surgery obstruction constructs aspherical 4-manifolds with boundary.

Mark Powell:

I will explain how to classify 4-manifolds up to homotopy equivalence, and how to upgrade this to a homeomorphism classification using surgery theory. I will focus on the cases of trivial and infinite cyclic fundamental group for simplicity, while describing a framework that can be applied for more general fundamental groups. Here is the plan for the lectures.

Lecture 1: Homotopy classifications.

Lecture 2: Homotopy automorphisms.

Lecture 3: Homeomorphism classifications