April 11: Dan Christensen on "Sphere bundles and their invariants"

[Emily Riehl]

The HoTTEST seminar continues this week, on Thursday, on its summer schedule: 11:30am EDT = 15:30 UTC.

This week we're delighted to finally hear from our very own Dan Christensen, whose talk is titled "Sphere bundles and their invariants". The talk will be 60 minutes long, followed by up to 30 minutes for questions. The abstract is below.

The Zoom link is https://zoom.us/j/994874377

Further information, including the list of upcoming speakers, and videos and slides from past talks, is at:

https://www.math.uwo.ca/faculty/kapulkin/seminars/hottest.html

Emily

(On behalf of the HoTTEST organizers: Carlo Angiuli, Dan Christensen, Chris Kapulkin, and Emily Riehl.)

Title: Sphere bundles and their invariants

Abstract:

Sphere bundles arise naturally in many contexts, often as the unit sphere bundles associated to vector bundles. For example, a manifold has a sphere bundle associated to its tangent bundle. In order to distinguish these bundles and to learn more about the manifolds, it's important to understand invariants of sphere bundles, such as the Euler class and the Thom class which live in cohomology groups. This talk will begin with a review of bundles and oriented bundles, and will then take a detour to discuss central types. We'll use central types to produce new models of Eilenberg-Mac Lane spaces. One of the themes of the talk will be that by using an appropriate model of an Eilenberg-Mac Lane space, we can give concrete descriptions of its H-space structure (which represents addition in cohomology) and of the cup product operation. Moreover, we'll show that by using these models, we can give very simple descriptions of the Euler class and Thom class of an oriented sphere bundle, and can prove theorems about them, such as the relationship between the two and the Whitney sum formula. I will also mention our work constructing the tangent sphere bundles of spheres and on the hairy ball theorem, which shows that the tangent sphere bundle of the n-sphere has a section if and only if n is odd.

This is joint work with Ulrik Buchholtz, David Jaz Myers and Egbert Rijke, and most of our results have been formalized using the Coq-HoTT library.


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Professor of Mathematics (she/her)
Johns Hopkins University
emilyriehl.github.io