Homology and Homotopy of Configuration Spaces

Advanced Topics in Topology - Homology and Homotopy of Configuration Spaces

Summer 2024, Uni Bonn and MPIM 

Place and time: 

Tuesdays 10:15 - 12:00 in N0.007 and 

Thursdays 10:15 - 12:00 in 0.006

Instructors

Course description

This course is about configuration spaces of manifolds. We will begin from the basics of the topic: main definitions, examples, and classic results such as relation to braid groups. We will then explore the results on the homology and homotopy of configuration spaces, leading to recent developments in the subject such as homological stability and their rational homotopy.

Audience and Prerequisites

The course is aimed at advanced bachelor’s students, master’s students, PhD students, and researchers who are not experts. We expect students to have familiarity with the topics of the courses Topologie I and Topologie II. This includes fundamental group, covering spaces, (singular) homology and cohomology, cup product, Künneth isomorphism, and universal coefficient theorem. While some knowledge of higher homotopy group and manifolds (notions of homeomorphisms and diffeomorphisms, embeddings, a bit of transversality) is useful, we will recall these notions during the course.

Problem Sheets

Problem sheets will be assigned throughout the semester, and students are invited to collaborate in groups of two or three people and submit a solution to at most one problem per week; we will give feedback on the solutions, but they will not be graded and they will contribute neither to the admission to the exam nor the final grade. Students are encouraged to prepare for the oral exam by consistently working on the problem sheets throughout the semester and by attending discussion sessions.

Exam and Evaluation

An oral exam will be held in the week 22-26 July, covering definitions, theorem statements, and the problem sheets. The exam will be conducted at the university or the MPIM by both instructors, and each will last approximately 30 minutes. Students will draw a topic from a pre-announced list and have 30 minutes to revise before making a short presentation (15 minutes) on the given topic, at the end of which there will be a question session.

The structure of the exam can be changed to accommodate exceptional circumstances.

Course Outline (to be updated during the course)

09/04 - Basic definitions, examples, motivations, and outline of the course.

11/04 - Discussion session: main examples, topology of configuration spaces, and braid groups of high dimensional manifolds. 

16/04 - Fadell-Newirth fibrations, configuration spaces of manifolds with boundary, and showing configuration spaces of acyclic surfaces are aspherical.

18/04 - Discussion session: sections of Fadell-Newirth fibrations, functoriality of configuration spaces, manifold structure and orientability of configuration spaces.

23/04 - Cohomology ring of ordered configuration space of Euclidean spaces.

25/04 - Discussion session: relations in the cohomology ring of ordered configurations of Euclidean space, surface braid groups, and an example of two homotopy equivalent open surfaces with non-homotopy equivalent configuration spaces.

30/04 - Homology of ordered configuration space of Euclidean spaces through planetary systems.

02/05 - 

07/05 - 

09/05 - HOLIDAY

14/05 - 

16/05 - 

21/05 - HOLIDAY

23/05 - HOLIDAY

28/05 - 

30/05 - HOLIDAY

04/06 - 

06/06 - 

11/06 - 

13/06 - 

18/06 - 

20/06 - 

25/06 - 

27/06 - 

02/07 - 

04/07 - 

09/07 - 

11/07 - 

16/07 - 

18/07 -