Noncommutative homotopy theory Semester WiSe 2022 / 23
Lecturer Ulrich Bunke
Type of course (Veranstaltungsart) Vorlesung
German title Nichtkommutative Homotopietheorie
Contents In view of Gelfand duality between locally compact topological spaces and commutative C^*algebras the homotopy theory of topological spaces can be interpreted as the homotopy theory of commutative C^*algebras. Giving up commutativity we enter the field of noncommutative homotopy theory. The most important homotopical invariant is the Ktheory of C^*algebras, respectively its categorial variants called KK or Etheory.
The goal of this course is to provide an introduction to KK and Etheory from the homotopical point of view. We will start with an introduction of the basic notions of C^*algebra theory. Based on a consequent use of the language of infinity categories, which will be introduced on fly in a nontechnical way, we will construct KK and Etheory by performing a couple of localizations of the category of C*algebras enforcing
homotopy invariance, Kstability, exactness and stability. As this approach is new and not yet documented in the literature we will provide a detailed script.
We will then reconnect with the classical constructions by interpreting Kasparov modules and extensions in the new picture.
This course will be continued in SS2023 with the emphasis on applications to index theory, assembly maps and more.
Literature Blackadar: KTheory for operator algebras
Cisinski: Higher categories and homotopical algebra
Recommended previous knowledge Analysis: Banach spaces
Topology: Homotopy groups, Homology
Algebra: Homological algebra
Time/Date Mo, Do 1012
Location M 102
Course homepage https://bunke.app.uniregensburg.de (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)
Registration Registration for the exercise classes:
 Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen) Oral examination (without grade): Duration: 25 min, Date: N.N.
Examination (Prüfungsleistungen) Oral exam: Duration: 25 min, Date: N.N., reexam: Date:
Modules BV, MV, MGAGeo
ECTS 9
