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

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.

Talk 3

• Title : The smash prodcut
• Abstract : This talk introduces the smash product in spectra, which provides a symmetric monoidal structure on both the category of spectra and its homotopy category. We will disuss its key properties and the definition of ring and module spectra. If time permits, we will also cover the point-set level construction using symmetric spectra.

Talk 4

• Title : Homology, Cohomology and products
• Abstract : I will define the homology and cohomology of spectra and show that they satisfy the Eilenberg-Steenrod axioms along with other basic properties expected of homology theories. I will also briefly describe the external, internal, and slant products.

Talk 5

• Title : Inverse limit, lim^1, and the Milnor exact sequence
• Abstract : We will define inverse limits in the context of abelian groups then explore its derived functor, namely its first derived functor lim^1. We then will define, and give a proof (sketch) of the Milnor exact sequence involving lim^1.

Talk 6

• Title : Spanier-Whitehead duality and Atiyah duality
• Abstract : I will introduce Spanier-Whitehead duality for spectra and explain how it relates their homology and cohomology. I will also introduce Atiyah duality and outline a proof of Poincare duality using it.

Talk 7

• Title : The Atiyah-Hirzebruch spectral sequence
• Abstract : We introduce the Atiyah-Hirzebruch spectral sequence, explaining where it comes from and how to apply it. Examples will include computations of K-theory groups.

Talk 8

• Title : An introduction to E_n-Algebras
• We will give the formal definitions of E_n-Algebras and relate these to factorisation homology. If possible, we will see some applications to deformation theory.

Talk 9

• Title : The Steenrod algebra and its dual
• We will describe the Steenrod operations and the structure of its algebra and dual algebra. We will see some applications that arise from this.