Seminar Algebraic Topology: The Immersion Conjecture and Cobordism Theory
In Spring 2022, Gijs Heuts, Lennart Meier, and Max Blans will be running a seminar for MSc students on the immersion conjecture and cobordism theory. Here is a syllabus. We meet on Mondays, 15:15-17:00. See the schedule below for the weekly location.
The aim of the seminar is to understand (part of) Ralph Cohen's proof of the immersion conjecture. Along the way, we will learn about a number of topics that are indispensable for any student of algebraic or differential topology, such as Stiefel-Whitney classes, spectra, and cobordism theory.
Resources
Ralph L. Cohen, Immersions of manifolds and homotopy theory.
Ralph L. Cohen and Ulrike Tillmann, Lectures on immersion theory. In Differential geometry and topology (Tianjin, 1986-87), volume 1369 of Lecture Notes in Math., pages 71-124. Springer, Berlin, 1989.
Allen Hatcher, Vector bundles & K-theory.
Cary Malkiewich, The stable homotopy category.
Lennart Meier, Introduction to stable homotopy theory.
Haynes Miller, Notes on cobordism.
John W. Milnor, Topology from the differentiable viewpoint.
John W. Milnor and James D. Stasheff, Characteristic classes.
Schedule
Feb 7, in Minnaert 0.14. Lennart Meier: Introduction and overview.
Feb 14, in Minnaert 0.14. Ryan Quinn: Immersions and vector bundles. Ryan's exercises and notes.
Feb 21, in Minnaert 0.14. Antonie de Potter: Stiefel-Whitney classes. Antonie's exercises.
Feb 28, in Minnaert 0.14. Bouke Jansen: Classification of vector bundles via Grassmannians. Bouke's exercises.
Mar 7, via MS Teams. Floor ter Haar: Cohomology ring of Grassmannians.
Mar 14, in Minnaert 0.14. Mick Schilder: The Thom isomorphism and existence of Stiefel-Whitney classes.
Mar 21, in Minnaert 0.14. César Iglesias: The Pontryagin-Thom construction. César's exercises.
Mar 28, in Minnaert 0.14. Dennis Hilhorst: Spectra 1. Dennis's exercises and notes.
Apr 4, in Minnaert 0.14. Bas Duijmelings: Spectra 2. Bas's exercises and notes.
Apr 11, in Minnaert 0.14. Floor & Dennis: The cohomology of MO. Exercises.
Apr 18, Easter
Apr 25, no seminar.
May 2, in Minnaert 0.14. Antonie & Bas: Generators of the homotopy groups of MO. Exercises.
May 9, in Androclus C023. Bouke & Mick: Cobordism and Immersions. Exercises.
May 16, no seminar.
May 23, in Androclus C023. Ryan & César: Relations among Stiefel-Whitney classes.
May 30, in Androclus C023. Anna Fokma: Hirsch-Smale theory.
June 20, in Androclus C023. Max Blans: Overview of the proof of the immersion conjecture.