# Seminar on stable homotopy theory (Fall 2024)

The goal of this seminar is to introduce the notion of spectra and study its basic properties. We will discuss 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.

This seminar is intended for graduate student. The participants are encouraged to prepare and give a talk.

- When : Monday 12:00 pm ~ 1:00 pm
- Where : Mathematics Hall, Room 528
- Organizer : Sangmin Ko
- References
- [Ada74] J.F. Adams, Stable homotopy and generalized homology, Part III, The University of Chicago Press, 1974, pdf
- [Boa99] J.M. Boardman, Conditionally Convergent Spectral Sequences, 1999, pdf
- [DP84] A.Dold and D.Puppe, Duality, trace and transfer, Proc. Steklov Inst. Math. 154 (1984), pp.85-103, pdf
- [Dug22] D.Dugger, Stable categories and spectra via model categories, 2022, pdf
- [Koc96] S.O. Kochman, Bordism, stable homotopy and Adams spectral sequecnes, American Mathematical Society, 1996
- [MMSS00] M.A. Mandell, J.P. May, S.Schewde, B.Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society Volume 82, Issue 2, 2000, pdf
- [Mal23] C. Malkiewich, Spectra and stable homotopy theory (draft version, first 6 chapters), pdf
- Math 8803 Stable homotopy theory (Spring 2015) taught by Kirsten Wickelgren, course link
- Spectra and stable homotopy theory (Fall 2012) taught by Michael Hopkins and notes by Akhil Mathew, pdf

## Schedule

Date | Speaker | Title |
---|---|---|

9/9 | Sangmin Ko | Spectra: definition and examples |

9/16 | Sangmin Ko | The homotopy category of spectra |

9/23 | TBD | Smash product |

9/30 | TBD | Homology, Cohomology and products |

10/7 | Alex Scheffelin | Derived inverse limit and the Milnor exact sequence |

10/14 | TBD | Spainer-Whitehead duality and Atiyah duality |

10/21 | TBD | The Atiyah-Hirzebruch spectral sequence |

10/28 | TBD | The Adam spectral sequence |

11/4 | TBD | The Pontryagin–Thom construction |

11/11 | TBD | The classification of smooth manifolds up to cobordisms |

11/18 | TBD | TBD |

11/25 | TBD | TBD |

12/2 | TBD | TBD |

## Talk List & Abstract

### Talk 1

- Title : Spectra : definition and examples (note)
- Abstract : In this talk, we will define spectra and Omega spectra and explain how generalized cohomology theories gives rise to Omega spectra via Brown representability. We will discuss some examples of spectra and its basic properties.

### Talk 2

- Title : The homotopy category of spectra
- Abstract : In this talk, we will define the homotopy category of spectra and discuss its stability properties. We will prove that suspension induces a self-equivalence of the stable homotopy category. We will define (co)fiber sequence of spectra and construct the associated long exact sequences. We will explain how the stable homotopy category forms a triangulated category.