### 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)

**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 |
---|---|---|

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 |