Student topology seminar

I will introduce simple-homotopy theory and its application to the classification of $h$-cobordisms. If time permits, I will also talk about the geometric motivation for higher algebraic K-theory.

Following Milnor/Kervaire's Groups of Homotopy Spheres we'll refresh and remind ourselves of some basic homotopy-theoretic principles. Our goal is simple: know why $h$-cobordism classes of homotopy spheres have an abelian group structure.

I will give an introduction to spectra and talk about their relation with generalized cohomology theories. Definitions, their basic properties and examples will take most of the talk. Time permitting, I will talk about some applications of the smash product on the stable homotopy category and the complications of defining it directly on spectra.

I will present Milnor's construction of differentiable manifolds which are homeomorphic, but not diffeomorphic, to the standard 7-sphere.

We introduce basic notions from CAT(0) geometry and their relevance to geometric group theory. In particular, we explore polyhedral cell complexes of non-positive curvature and illustrate how they can arise as Dehn complexes of alternating knot projections. Finally, we will explain the surprising relevance of these constructions to the study of configuration spaces of metamorphic robots.

I will give a gentle introduction to spin geometry. Starting with construction of spin groups, I will talk about spin structures on manifolds.

I will present some parts of a paper by Dwyer. The problem is to express, up to mod p cohomology, the classifying space BG of a finite group G as a homotopy colimit of classifying spaces of smaller groups.

Reference: W. G. Dwyer, Classifying spaces and homology decompositions.

This is a continuation of the previous talk.

I will explain the basics of topology in the category of $G$-spaces and $G$-maps and introduce Bredon cohomology by doing two important computations (Smith Theory, cohomology of the torus).

We'll describe Weil's group-cohomological description of local deformations of the representation spaces $\operatorname{Hom}(\Gamma, G)$, where $\Gamma$ is a finitely-generated group and $G$ a Lie group.

Lecture notes

We will review some basic ideas of classical Morse theory and give a general overview of the intuitive key aspects of Morse homology. The talk will be relatively self-contained.

References:

• Raoul Bott, Morse theory indomitable
• Lecture notes by Alexander Ritter
• Dietmar Salamon, Lectures on Floer homology

This will be a short talk where we will prove a theorem of P.A. Smith using equivariant cohomology.

Reference: Peter May, Equivariant Homotopy and Cohomology Theory

The Nielsen–Thurston classification is a classification of dynamics of surface automorphisms, and is related to many branches of mathematics—topology, geometry, dynamics, algebra, and combinatorics. I would like to explain various ideas behind this theory.

This is a preliminary to Dale's talk. We'll briefly discuss Heegaard diagrams, surgery on 3-manifolds and show that every closed, connected, oriented 3-manifold can be obtained by surgery on S^3 along some link.

In this talk I explain how naive ideas to prove Nielsen–Thurston classification (previous talk) leads to Thurston's compatcification of Teichmüller space. I also would like to give a short exposition of train-track method, a combinatorial and algebraic approach to Nielsen–Thurston theory.