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19223811 Forschungsmodul: Topologie "Stable Homotopy Theory"

Summer Term 2023

Dozenten: Dr. Georg Lehner, Prof. Dr. Elmar Vogt

  • Time and place:  Thursday,  4pm -- 6pm, SR 115, Arnimallee 3.

  • Leistungsnachweis/criteria for proof of performance:
    Grade and credit points will be awarded based on a presentation and written summary.

Prerequisites: We assume basic knowledge of topology as taught in Topology I and II. This means concretely some familiarity with CW-complexes, generalized (co)-homology theories and basics of category theory.

Content: This seminar will cover advanced topics from topology and homotopy theory.
The category of spectra is central to many problems in modern algebraic topology and was originally introduced in 1964 by Michael Boardman. A spectrum can be thought of in many equivalent ways:

  1. As a higher-categorical analogue of an abelian group,
  2. as a generalization of a space, where continuous maps only need to exist after suitable suspensions,
  3. as an infinite loop space,
  4. as a generalized cohomology theory.

Spectra have, just like spaces, homotopy, homology and cohomology groups. There exist many spectra that encode useful information in their homotopy groups. For example, the homotopy groups of the sphere spectrum  S  are linked to the existence of exotic spheres, that is, different smooth structures on  Sn. The spectrum  KU  represents topological  K-theory, which has proven to be a very useful cohomology theory and is important for index theory. The algebraic  K-theory spectrum  K(R)  for a ring   plays a central role in both number theory as well as our understanding of manifolds in dimensions greater than four.
In this seminar, we will try a modern approach, using the newly available language of  ∞-categories, which builds on the foundational work by Lurie, 2009. This has the advantage that many proofs of the properties of the  ∞-category of spectra are "formal nonsense". The downside is that we will have to rely mostly on a handful of recently written notes on the subject. In particular we will try to follow the lecture notes of Denis Nardin.
The seminar will fall into three parts: The first will focus on elementary  ∞-category theory. The second will be about spectra and their properties. The last will deal with  E and  E-spaces, which are higher algebraic versions of monoids and commmutive monoids. There exist "group completions", which are analogous to the process of turning a monoid into a group. Just as the group completion of a commutative monoid is an abelian group, it turns out that the group completion of an  E-space is a connective spectrum. This is the content of the celebrated recognition principle.
After this we will discuss topological  K-theory and depending of the interests of the participants additional related material.


20.04. Organization and overview - Short talk on motivation. Georg Lehner/
Elmar Vogt
27.04. Talk 1: Simplicial Methods. Simplicial Sets, Kan Complexes, Homotopy Groups, Geometric Realization/Singularization, the Homotopy Category [Nardin, Ch.1, Page 5-11] Georg Lehner
04.05. Talk 2: Homotopy Limits/Colimits of spaces. [Lambrecht] N.N.
11.05. Talk 3:  ∞-categories Part 1. Definition via lifting properties, Mapping Spaces, Categories enriched in Kan complexes, Simplicial Nerve construction. The  ∞-category of spaces and the  ∞-category of  ∞-categories. [Nardin, Ch.1, Page 12-17] N.N.
18.05. Talk 4:  ∞-categories Part 2. Homotopy Category of an  ∞-category. Limits and Colimits in  ∞-categories. Definition, Commuting with Mapping spaces, Adjunctions, Yoneda Lemma. [Nardin, Ch.1, 18-21] (Further resource needed!) N.N.
25.05. Talk 5: Cohomology and Brown Representability. Eilenberg Maclane Spaces, Proof of Brown Representability Theorem. [Nardin, Ch.2, Page 25-29] N.N.
01.06. Talk 6: Spectra Part 1. Definition, Examples, Cohomology, Homotopy Groups, Limits, filtered Colimits,  Ω  and  Σ  are equivalences,  Σ∞  and  Ω∞  form an adjunction. [Nardin, Ch.2, Page 29-31] N.N.
08.06. Talk 7: Spectra Part 2. Colimits, Stability, Fiber and Cofiber Sequences, Exact Sequences, Mapping Spectra, Tensor Product [Nardin, Ch.2, Page 32-34] N.N.
15.06. Talk 8:  E1-Monoids. Definition, Classifying Space, Adjunction between   and  Ω. Statement of the Recognition Principle. [Nardin, Ch.3, Page 37-39] N.N.
22.06. Talk 9: Recognition Principle: Proof [Nardin, Ch.3, Page 39-41] N.N.
29.06. Talk 10: Commutative Monoids and Spectra. Recognition principle for Connective Spectra. [Nardin, Ch.3, Page 42-45] N.N.
06.07. Talk 11: Group Completion and  K-Theory [Nardin, Ch.3, Page 46-49] N.N.
13.07. Talk 12: Vector Bundles and Topological  K-theory [Nardin, Ch.4, Page 53-58] N.N.
20.07. Talk 13: TBA N.N.