WiSe 21/22 Seminar: Introduction to stable homotopy theory

Time and place: Thursday 14-16 in M009


Originally a subfield of algebraic topology that emerged in the second half of the 20th century, stable homotopy theory nowadays plays a much larger role in mathematics and has applications to various fields such as geometric topology, algebraic geometry, and even number theory.

The goal of this seminar is to introduce the central notion of spectrum and study its basic properties. Among other things, we will discuss the equivalence between spectra and generalized cohomology theories, the smash product of spectra, Spanier–Whitehead duality, Atiyah duality, the Steenrod algebra, the Atiyah–Hirzebruch and Adams spectral sequences, and the relationship between stable homotopy and bordism of smooth manifolds.

Recommended prerequisites: Algebraic Topology I and II (in particular: homology groups, homotopy groups, CW complexes, Eilenberg–Mac Lane spaces)


Date Speaker Topic
21.10 Marc Hoyois Introduction
28.10 Simon Lang Spectra: definitions and examples
04.11 Giovanni Sartori The homotopy category of spectra I: Construction
11.11 Georg Thurner The homotopy category of spectra II: Properties
18.11 Axel Sixt The smash product
25.11 Areeb Shah-Mohammed Homology, cohomology, and products
02.12 Tobias Wagenpfeil Derived inverse limits and the Milnor exact sequence
09.12 Jonas Linssen Spanier–Whitehead duality
16.12 Alessandro de Innocentiis Atiyah duality
23.12 Ludovico Morellato Spectral sequences
13.01 Jonas Wittman The Atiyah–Hirzebruch spectral sequence
20.01 Chiara Sabadin The Steenrod algebra and its dual
27.01 Ritheesh Krishna Thiruppathi The Adams spectral sequence
03.02 Pier Federico Pacchiarotti The Pontryagin–Thom construction
10.02 Niklas Kipp The classification of smooth manifolds up to cobordism