### 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)

**Literature:**

- J. F. Adams,
*Stable homotopy and generalised homology* - S. O. Kochman,
*Bordism, stable homotopy, and Adams spectral sequences* - R. M. Switzer,
*Algebraic Topology – Homotopy and Homology*

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 |

05.07 | — | — |

12.07 | Benedikt Fröhlich | The Steenrod algebra |

19.07 | Athanasios Koutsopagos | The Atiyah–Hirzebruch spectral sequence |