19223811 Forschungsmodul: Topologie "Stable Homotopy Theory"
 FUStudents should register via Campus Management.
 NonFUstudents should register via MyCampus/Whiteboard.
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 CWcomplexes, 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:
 As a highercategorical analogue of an abelian group,
 as a generalization of a space, where continuous maps only need to exist after suitable suspensions,
 as an infinite loop space,
 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 S^{n}. The spectrum KU represents topological Ktheory, which has proven to be a very useful cohomology theory and is important for index theory. The algebraic Ktheory spectrum K(R) for a ring R 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_{1 } 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 Ktheory and depending of the interests of the participants additional related material.
Talks
Date  Title  Speaker 

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 511]  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 1217]  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, 1821] (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 2529]  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 2931]  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 3234]  N.N. 
15.06.  Talk 8: E_{1}Monoids. Definition, Classifying Space, Adjunction between B and Ω. Statement of the Recognition Principle. [Nardin, Ch.3, Page 3739]  N.N. 
22.06.  Talk 9: Recognition Principle: Proof [Nardin, Ch.3, Page 3941]  N.N. 
29.06.  Talk 10: Commutative Monoids and Spectra. Recognition principle for Connective Spectra. [Nardin, Ch.3, Page 4245]  N.N. 
06.07.  Talk 11: Group Completion and KTheory [Nardin, Ch.3, Page 4649]  N.N. 
13.07.  Talk 12: Vector Bundles and Topological Ktheory [Nardin, Ch.4, Page 5358]  N.N. 
20.07.  Talk 13: TBA  N.N. 
Literature:
 Nardin: Introduction to stable homotopy theory.
 Krause, Nikolaus: Higher Algebra, Session 1,2,3 and 8.
 Land: Introduction to ∞categories.
 Groth: A short course on ∞categories, arXiv:1007.2925.
 Lambrecht: A gentle introduction on Homotopy Limits and Colimits
 Lurie: Higher Topos Theory, Chapters 1 and 4.
 Lurie: Higher Algebra, Chapter 1.