SoSe 23 Seminar: Introduction to stable homotopy theory

Time and place: Wednesday 16-17:30 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
19.04 Marc Hoyois Introduction
26.04 Marc Hoyois Spectra: definitions and examples
03.05 Marc Hoyois The homotopy category of spectra
10.05 Vikram Nadig The smash product
17.05 Vikram Nadig Homology, cohomology, and products
24.05 Marco Giustetto Spanier–Whitehead duality
31.05 Marco Giustetto Atiyah duality
07.06 Alexandra Pröls The Pontryagin–Thom construction
14.06 Laura Grepmair Acyclic maps and epimorphisms
21.06 Katharina Schneider Adams representability
28.06 Marc Hoyois The group completion theorem
12.07 Benedikt Fröhlich The Steenrod algebra
19.07 Athanasios Koutsopagos The Atiyah–Hirzebruch spectral sequence