# Student topology seminar

We meet once a week in the PIMS library (ESB 4133) to learn something about topology.

Friday, September 7, 2012 — 3:00 p.m.

Maxim Stykow

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.

Friday, September 14, 2012 — 4:00 p.m.

Homotopy spheres form a group: a homotopy refresher

Justin Martel

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.

Friday, September 21, 2012 — 3:00 p.m.

Spectra and Generalized Cohomology Theories

Galo Higuera Rojo

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.

Friday, September 28, 2012 — 4:00 p.m.

Exotic Spheres

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

Friday, October 12, 2012 — 4:00 p.m.

CAT(0) Geometry, Cubical Complexes and Metamorphic Robots

Maxime Bergeron

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.

Friday, October 19, 2012 — 4:00 p.m.

Spin Geometry

Atsushi Kanazawa

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

Friday, October 26, 2012 — 4:00 p.m.

Homology decompositions of classifying spaces

Cihan Okay

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.

Friday, November 9, 2012 — 4:00 p.m.

Homology decompositions of classifying spaces (part 2)

Cihan Okay

This is a continuation of the previous talk.

Friday, November 16, 2012 — 3:00 p.m.

Introduction to Equivariant Algebraic Topology

Maxim Stykow

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).

Friday, November 23, 2012 — 3:00 p.m.

Weil local rigidity of representations of f.g. groups into Lie groups

Justin Martel

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

Friday, November 30, 2012 — 3:00 p.m.

Morse Theory Indomitable

Maxime Bergeron

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:

Friday, December 7, 2012 — 3:00 p.m.

Smith Theory

Galo Higuera Rojo

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

Friday, March 1, 2013 — 4:00 p.m.

Around Nielsen-Thurston classification

Tetsuya Ito

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.

Friday, March 8, 2013 — 4:00 p.m.

Surgery on 3-manifolds

Huan Vo

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.