# Topology Student Seminar

## Spring 2020

The seminar meets on Mondays from 11:30am to 12:45pm in the PIMS lounge (ESB 4133).

It aims to be a nexus for those students at UBC who have an interest in topology to share it. All graduate students are welcome.

## Schedule

• January 20: Characteristic Classes 1, by Santanil Jana
Characteristics classes are cohomology classes on a space that help us distinguish vector bundles over the space. The goal of this talk will be to introduce the notion of characteristic class and look at some examples, namely Stiefel-Whitney, Chern, Pontryagin and Euler classes. We'll also see some properties of the Euler class and see how it can be used to differentiate oriented real vector bundles.
• January 27: Characteristic Classes 2, by Sebastian Gant and Mihai Marian
We all like concrete things. In this talk, we will construct some vector bundles using clutching functions, give a low-tech proof that homotopy classes of clutching functions are in bijection with vector bundles and, finally, talk about characteristic classes from the point of view of obstruction theory.
• February 3: Introduction to Topological K-Theory, by Niny Arcila Maya
In this talk we give the definition of the reduced and unreduced K-group of a topological space X, namely K0(X) and $$\widetilde{K}$$0(X). We do this by using the Grothendieck group construction. We see that these definitions are related to homotopy theory.
• February 10: K-Theory 2, by Santanil Jana
Computation of the K-theory of the sphere by proving a special case of the fundamental product theorem in Bott periodicity.
• February 24: No Talk
• March 2: Heegaard-Floer Theory 0: Background on 3-Manifold Topology, by Mihai Marian
In the early 2000s, Ozsváth and Szabó developed Heegaard-Floer theory to study 3-manifold geometry and topology, which is a most wonderful topic of mathematics, by using the work of Andreas Floer, who was a most wonderful mathematician. In this first talk, I will survey the "Heegaard" part, by which I mean the classical 3-manifold theory, up until the work of William Thurston in the '80s.
• March 9: Sheaf (co)Homology 1, by Sebastian Gant
We will talk about (co)homology with local coefficients. I'll give an algebraic definition, state a generalization of Poincare duality for non-orientable manifolds, and hopefully talk about local coefficients via bundles of groups (or locally constant sheaves of groups) if time allows.
• March 16: Sheaf (co)Homology 2, by Sebastian Gant
In this talk I will define bundles of groups in a few different ways, and hash through some examples. The goal is to define homology with local coefficients in a bundle of groups and hopefully relate this new definition to the previous one from last week.