Non-commutative homotopy theorySemester
WiSe 2022 / 23
Type of course (Veranstaltungsart)
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 non-commutative homotopy theory. The most important homotopical invariant is the K-theory of C^*-algebras, respectively its categorial variants called KK- or E-theory.
The goal of this course is to provide an introduction to KK and E-theory 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 non-technical way, we will construct KK and E-theory by performing a couple of localizations of the category of C*-algebras enforcing
homotopy invariance, K-stability, 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.
Blackadar: K-Theory for operator algebras
Cisinski: Higher categories and homotopical algebra
Recommended previous knowledge
Analysis: Banach spaces
Topology: Homotopy groups, Homology
Algebra: Homological algebra
Mo, Do 10-12
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Course work (Studienleistungen)
- Registration for the exercise classes:
- Registration for course work/examination/ECTS: FlexNow
- Oral examination (without grade): Duration: 25 min, Date: N.N.
- Oral exam: Duration: 25 min, Date: N.N., re-exam: Date:
BV, MV, MGAGeo