# Student Seminar on Equivariant Cohomology

Posted in
Organiser(s):
Christian Blohmann, Annika Tarnowsky
Date:
Mon, 10/10/2022 - 14:00 - Fri, 03/02/2023 - 14:00
Location:
MPIM Seminar Room

#### Course listing

Hauptseminar Geometrie (S2D1)
University of Bonn, Winter semester 2022/23

Instructors: Christian Blohmann, Annika Tarnowsky
Time/venue: Wednesday 14:15-16:00h (not 15:45h), Max Planck Institute for Mathematics, seminar room. Please ring the doorbell for entrance and register at the front desk every time (it's a very quick process and usually takes less than a minute).

#### Seminar description

Equivariant cohomology is a cohomology theory for spaces with a group action. In modern terminology, it can be defined as the cohomology of the homotopy quotient or as the cohomology of the quotient stack. It is particularly useful, when the space is a manifold with a smooth action of a Lie group, so that it can be studied by infinitesimal methods in terms of the Lie algebra. In this case we obtain, for example, computable models for equivariant cohomology, a simple interpretation of the Chern-Weil map from invariant polynomials to characteristic classes, a cohomolgical characterization of hamiltonian actions on symplectic manifolds, a localization formula for the symplectic volume of a toric manifold, and more. The goal of the seminar is to learn the basic concepts of equivariant cohomology and some of its most important applications to differential geometry and topology.

#### Prerequisites

A background in basic differential geometry (manifolds, tangent spaces) and basic topology (fundamental groups, universal covers, basic homological algebra) is assumed. Some basic knowledge of Lie groups, Lie algebras, or (co)homology or concepts related to cohomology will be helpful, especially when would like to give one of the more difficult talks.

#### Seminar organization

The seminar consists of short talks (ca. 30-40 minutes) by the participants on well-defined parts of the seminar material, for example the presentation of a theorem with proof or the introduction of a new concept. There are usually two such talks per meeting, which will be framed and connected by a discussion and short introductions or summaries by the seminar organizers.

The main source of the seminar is a book by Tu ([1], see below). The University Library has an online version which can be found in the online catalogue. To download it, you must access via a university computer or via VPN. Instructions on how to set this up can be found on the webpages of the HRZ.

The schedule of the seminar will be posted and updated on this public webpage. If you do not want your name to be listed next to your talk for privacy reasons, please let us know.

#### Registration

If you are interested in participating you can send an email to blohmann@mpim-bonn.mpg.de and/or sign up at the

organizational meeting on 21.09.2022, 14:15h-16:00h, Seminar Room of MPIM.

In addition, you will have to register officially via Basis. The talks will be distributed during the organizational meeting. If you cannot come to this meeting, please let us know your preferences for talks by email so that we can try to accomodate your wishes.

#### Course credit

In order to obtain credit you will have to participate actively. This means that you will have to

a) come to all seminar meetings unless you have to call in sick or have previously obtained our approval for your absence and

b) give at least one of the short talks listed below.

The seminar is officially registered for both Bachelor studies (S2D1) and Master studies (S4D3).

Master's students will naturally be graded differently from Bachelor's students. However, our grading criteria will be the same for both and include the correctness of the content, the conciseness of the explanations, the structure and time-management of the talk, the engagement of the audience, the use of the blackboard, ...

#### Seminar plan (preliminary)

1. Short organisatorial meeting (12 Oct 2022), Chair: Christian Blohmann
Introduction of organisers and participants, some information on our expectations from the talks, possible pair-up for rehearsal talks
2. Lie groups (19 Oct 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Lie groups and Lie algebras, Speaker: tba
b) Lie group actions, Speaker: tba
3. More details on Lie groups (26 Oct 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) The Lie-group-Lie-algebra-correspondence, Speaker: tba
b) Quotients of Lie group actions, Speaker: tba
4. Invariant objects (2 Nov 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Integration on compact connected Lie groups, Speaker: tba
b) The Maurer-Cartan form, Speaker: tba
5. Principal bundles (9 Nov 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Principal bundles, Speaker: tba
b) Connections on principal bundles, Speaker: tba
6. Equivariant cohomology (16 Nov 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Universal bundles and equivariant cohomology, Speaker: tba
b) Construction of universal bundles, Speaker: tba
7. Higher structures (23 Nov 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Model categories, Speaker: tba
b) Spectral sequences, Speaker: tba
8. Computing equivariant cohomology (30 Nov 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Computing an example of equivariant cohomology, Speaker: tba
b) The Weil algebra, Speaker: tba
9. The equivariant de Rham theorem (14 Dec 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) The equivariant de Rham theorem and its applications, Speaker: tba
b) The proof of the equivariant de Rham theorem, Speaker: tba
10. Localisation and fixed points (21 Dec 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Localisation and locally free actions, Speaker: tba
b) Topological properties of Lie group actions, Speaker: tba
11. Circle actions: Localisation and Integration (11 Jan 2023), Chair: Christian Blohmann/Annika Tarnowsky
a) Borel localisation for circle actions, Speaker: tba
b) Integration of equivariant forms for circle actions, Speaker: tba
12. The equivariant localisation formula for a circle action (18 Jan 2023), Chair: Christian Blohmann/Annika Tarnowsky
a) The equivariant localisation formula and its applications, Speaker: tba
b) The proof of the equivariant localisation formula, Speaker: tba
13. Applications of Equivariant Cohomology (25 Jan 2023), Chair: Christian Blohmann/Annika Tarnowsky
a) An application in topology, Speaker: tba
b) An application in physics, Speaker: tba
14. Miscellaneous (1 Feb 2023), Chair: Christian Blohmann
Possible buffer to reschedule other talks or give a glimpse into our research on Lie groupoid cohomology

Remarks: The dates of the talks are subject to changes since some topics may take more time than an entire meeting and some less. Due to the Dies Academicus, there will be no seminar on Dec 7, 2022.

#### Talks

Talks marked with * require more engagement with the material, either because of incresed difficulty or because of more independent work with the literature. Master's students are expected to choose one of these talks that have an increased workload.

Note that while choosing one of the marked talks will benefit your grading, an unmarked talk well-held can get a perfect score for Bachelor's students, the same holds for Master's students and simply marked talks.

Please also note that the descriptions are merely a guide of what to cover in your talk. You can freely add details from all sources to elaborate on the topics. However, please make sure that you do cover what is necessary for the next speakers to proceed. When in doubt what to talk about or you feel that your talk is getting too long, please contact Annika.

2a: Lie groups, Lie algebras, Lie algebras from Lie groups, universal covering groups, the exponential map

A review of [2], ch. 7 "Basic Definitions", "Lie group homomorphisms", ch. 8 "The Lie algebra of a Lie group", ch. 20 "One-parameter subgroups and the exponential map"

2b: Lie group actions and Lie algebra actions, equivariant maps, fundamental vector fields, adjoint actions and representations

Introduce the concept of a Lie group action and equivariant maps ([2], ch. 7), introduce the relevant notions of [1], ch.1.1-1.3 (you can also include some of the statements), use the adjoint action of a group on itself as an example and derive the adjoint action on the Lie algebra known as the adjoint representation ([2], somewhere in ch. 20 "Normal Subgroups", you don't need to actually explain what a representation is), introduce the concept of Lie algebra action/infinitesimal group action ([2], ch. 20 "Infinitesimal Generators for group actions") and use it to define fundamental vector fields ([1], ch. 11.1, you can also include the statements from 11.2).

3a*: Integrating Lie algebra homomorphisms, integrating Lie algebras, the Lie-group-Lie-algebra correspondence

Present the statements and proofs from [2], ch. 20 "The Lie correspondence", sketch what is necessary from [2], ch. 19.

3b*: Lie group actions and their quotients

Review [2], ch. 21 "Quotients of manifolds by group actions" and sketch the proof of the "Quotient Manifold Theorem".

4a*: Compact connected Lie groups: Haar measure, integration, application: exp is surjective

Review [1], ch. 13 and [3] up until Prop. 1.

4b: Vector-valued forms, the Maurer-Cartan form (definition, equation)

Review [1], ch. 14+15

5a: Fibre bundles(?), principal bundles and basic forms (including a review of the Cartan calculus)

Review [1], ch. 3+10+12

5b: Connections on principal bundles, example: The Maurer Cartan form, curvature

Review [1], ch. 16+17

6a*: Universal bundles, equivariant cohomology and homotopy quotients

Review [1], ch. 4+5

6b*: The universal bundle of the orthogonal group, properties of equivariant cohomology

Review [1], ch. 8+9

7a***: Model categories and homotopy categories, equivariant cohomology in terms of model categories

Watch the first two lectures of Scott Balchins series ([4]) on "Model categories by example" (you can also watch all five, they are amazing!) and review model categories and cofibrant replacement. Introduce the Serre model structure on topological spaces and state what fibrant and cofibrant objects are. Now consider the category of $G$-spaces for a topological group $G$. Can you find a functor that is right adjoint to taking the quotient by the $G$-action? Give some thought to how the weak equivalences, fibrations and cofibrations could look like in the category of $G$-topological spaces (there is a sublety there in generalising the test objects of the Serre fibrations, you are not expected to actually choose one of the options). Use that a space with a non-free action cannot map into a space with a free action, to show that under the assumption of $EG \times X \rightarrow X$ being an acyclic fibration, a $G$-space with a non-free action cannot be cofibrant, but can cofibrantly be replaced by $X \times EG$. Review homotopy categories and derived functors and show how the left derived functor of the adjunction we previously established computes the "homotopy quotient" as $X \times EG/G$.

7b**: Spectral sequences, Leray's theorem

Review [1], ch. 6

8a**: The equivariant cohomology of $S^1$ acting on $S^2$

Recall spectral sequences, review [1], ch. 7, also consider the practice problems if there is time

8b*: Differential graded algebras, the Weil algebra and the Weil map

Review [1], ch. 18+19

9a*: The Cartan model, the equivariant de Rham theorem: statement and examples

Review [1], ch. 20+21

9b***: Ideas behind the equivariant de Rham theorem

Review [1], ch. 22+Appendix A

10a*: Localisation with respect to a variable, torsion, equivariant cohomology with respect to a locally free action

Review [1], ch. 23+24

10b*: Regularity of the fixed point set, equivariant vector bundles, normal bundles and equivariant tubular neighbourhoods

Review [1], ch. 25

11a**: Borel localisation for a circle action

Review [1], ch. 26

11b*: Representation theory, Stoke's theorem for equivariant integration, integration of equivariant forms for a circle action

Review [1], ch. 27+28

12a**: The localisation formula for a circle action (need: 24.3) and the area of a sphere

Review [1], ch. 29.1+30

12b***: Spherical blow-ups and an idea of the proof of the equivariant localisation formula

Review [1], ch. 29.2+31

13a***: Computing Topological Invariants (Chern Classes) of Homogeneous Spaces

Review [1], ch. 32.3 and have a look at the original paper on this topic ([5]). Establish the necessary notions and review how equivariant cohomology is applied in this context.

13b***: Applications of Equivariant Cohomology in Physics

Review [1], ch. 32.4+32.5 and have a look at some additional sources ([6]). Review how equivariant cohomology is applied in this context.

#### Literature

Main reference (access on university library computers or via VPN):

[1] Tu, Loring W.. Introductory Lectures on Equivariant Cohomology: (AMS-204), Princeton: Princeton University Press, 2020. https://doi.org/10.1515/9780691197487