Homology and Homotopy of Configuration Spaces
Advanced Topics in Topology - University of Bonn
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 Structure 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 at random from the list below and will have 30 minutes to revise before giving a short presentation (15 minutes) on the selected topic. During the 30 minutes prior to the actual beginning of the exam the students are allowed to revise the topic with help of notes, colleagues etc. No notes are allowed during the presentation and during the exam. The presentation should include the relevant definitions, examples and theorem statements. This will be followed by a question session.
Along with the topics, we have included some keywords to help you prepare your presentation. Note that your presentation does not have to include all the keywords, they are just there to give you some guidance in case you need some direction.
The structure of the exam can be changed to accommodate exceptional circumstances.
Exam Topics
1. Homology and cohomology of ordered configuration spaces of Euclidean space.
Keywords: Leray-Hirsch Theorem, planetary systems, Arnold relation, tall forests, Jacobi identity for homology, homotopy groups of configuration spaces in terms of homotopy groups of wedges of spheres.
2. Invariance of (co)homology of configuration spaces and homotopy non-invariance of configuration spaces.
Keywords: Poincaré-Lefschetz duality, Massey products, the counterexample of Longoni-Salvatore, formality.
3. The Cohen-Taylor spectral sequence.
Keywords: sheaf cohomology, multiplicative spectral sequence, collapse, theorem of Totaro, theorem of Kriz.
4. Scanning.
Keywords: Barrat-Priddy-Quillen-Segal theorem, the scanning map, topological monoids, bar construction, group-completion.
5. Homological Stability.
Keywords: relation to the Group-Completion Theorem, key steps of proof of homological stability for unordered configuration spaces of Euclidean space (what are the key propositions and how are they assembled into a proof), computation of the rational homology of unordered configuration spaces of Euclidean spaces.
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 - Discussion session: relations between homology classes given by planetary systems, proof of Jacobi identity.
07/05 - Rational homology of unordered configuration space of Euclidean spaces, homotopy invariance of homology groups of configuration spaces, and introduction to the little discs operad.
09/05 - HOLIDAY
14/05 - Introduction to spectral sequences; examples of spectral sequences coming from a semisimplicial space and from an augmented semisimplicial space.
16/05 - Discussion session: Mayer-Vietoris spectral sequence and Bendersky-Gitler spectral sequence for configuration spaces.
21/05 - HOLIDAY
23/05 - HOLIDAY
28/05 - Homological and cohomological Serre spectral sequence. Leray-Hirsch theorem. Application to the cohomology of ordered configuration spaces of manifolds with vanishing diagonal class (with coefficients in a field), via the Fadell-Neuwirth fibrations.
30/05 - HOLIDAY
04/06 - Leray spectral sequence. Cohen-Taylor spectral sequence and the theorem of Totaro.
06/06 - Discussion session: computation of the E_2-page of the Cohen-Taylor spectral sequence.
11/06 - Introduction to rational homotopy theory via rational cdga's. Massey products and formality. Statements of the results of Kontsevich, Kriz, and Campos-Willwacher and Idrissi. The example of Longoni-Salvatore.
13/06 - Discussion session: examples of rational cdga's and of Massey products, study of the second configuration space of L_{7,1}.
18/06 - The Scanning Map, Statement of Barrat-Priddy-Quilen-Segal Theorem, Bar construction and classifying spaces.
20/06 - Discussion session: classifying spaces for discrete groups, quasifibrations and restriction maps of configuration spaces, and configuration space model for the classifying space of an abelian monoid.
25/06 - Proof of the Barrat-Priddy-Quilen-Segal Theorem.
27/06 - Discussion session: the proof of the Barrat-Priddy-Quilen-Segal Theorem and configuration space models for Eilenberg-MacLane spaces.
02/07 - The group completion theorem and its proof, modulo a few lemmas. The phenomenon of homology stability.
04/07 - Discussion session: complements to the proof of the group completion theorem.
09/07 - Proof of homology stability for unordered configuration spaces of R^n; description of the general strategy via augmented semisimplicial spaces
11/07 - Discussion session: homology stability of unordered configuration spaces of manifolds with boundary and of symmetric groups, split injectivity of the stabilisation maps.
16/07 - EXTRA LECTURE (not part of the material for the exam): some results about homology of unordered configuration spaces of manifolds; computation of the homology of C_k(R^2) and of the homology of unordered configuration spaces of orientable surfaces with boundary, with various field coefficients.
18/07 - EXTRA LECTURE (not part of the material for the exam): classifying space of diffeomorphism groups; Harer' stability theorems, decoupling results; relation between configuration spaces and the layers of the embedding tower.