# Student Seminar on Equivariant Cohomology

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

#### Course listing and General Information

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

Instructors: Christian Blohmann, Annika Tarnowsky
Time: Wednesday 14:15-16:00h (not 15:45h)
Venue: Max Planck Institute for Mathematics, seminar room. Please ring the doorbell for entrance to the MPIM and register at the front desk every time (it's a very quick process and usually takes less than a minute). The institute is accessible for people with restricted mobility during the opening hours, but if you let us know beforehand in case you can't use stairs it would streamline the process.

#### 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 you 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 are expected 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. Lie groups and Lie algebras (12 Oct 2022), Chair: Christian Blohmann
a) Lie groups and Lie algebras, Speaker: Tommaso Vassura
b) Lie group actions and Lie algebra actions, Speaker: Annika Tarnowsky
2. No meeting (19 Oct 2022)
3. Properties of Lie groups (26 Oct 2022), Chair: Annika Tarnowsky
a) The Lie-group-Lie-algebra-correspondence, Speaker: Peter Moody
b) Quotients of Lie group actions, Speaker: Kai Gerd Müller
4. Invariant objects (2 Nov 2022), Chair: Christian Blohmann
a) Integration on compact connected Lie groups, Speaker: Alessandro Nanto
b) The Maurer-Cartan form, Speaker: Kai Gerd Müller
5. Principal bundles (9 Nov 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Principal bundles, Speaker: David Aretz (guest speaker)
b) Connections on principal bundles, Speaker: David Aretz (guest speaker)
6. Equivariant cohomology (16 Nov 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Universal bundles, Speaker: Hao Xiao
b) Equivariant cohomology, Speaker: Federico Astolfi
7. Computing equivariant cohomology (23 Nov 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Spectral sequences, Speaker: Maximilian Hauck
b) Computations with spectral sequences, Speaker: You Zhou
8. Universal bundles generalised (30 Nov 2022), Chair: Annika Tarnowsky
a) Model categories, Speaker: Anton Engelmann
b) The Weil algebra, Speaker: Hao Xiao
9. The equivariant de Rham theorem (14 Dec 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) The equivariant de Rham theorem and its applications, Speaker: Samuel Lee
b) The proof of the equivariant de Rham theorem, Speaker: Alessandro Nanto
10. Locally free actions and torsion (21 Dec 2022), Chair: Christian Blohmann/Annika Tarnowsky
a) Algebraic methods to study group actions, Speaker: Maximilian Hauck
b) Free and locally free actions, Speaker: Matteo Del Vecchio
11. Circle actions: Localisation (11 Jan 2023), Chair: Christian Blohmann/Annika Tarnowsky
a) Embedded manifolds and manifolds with boundary, Speaker: Kexing Chen
b) Borel localisation for circle actions, Speaker: You Zhou
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: Malcom Fack
b) The proof of the equivariant localisation formula, Speaker: Tor-Haakon Gjone
13. Applications of Equivariant Cohomology (25 Jan 2023), Chair: Christian Blohmann/Annika Tarnowsky
a) An application in topology, Speaker: Yifan Song
b) An application in physics, Speaker: Peter Moody
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 * (possibly up to ***) require more engagement with the material, either because of increased 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.

1a: Lie groups, Lie algebras, Lie algebras from Lie 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":

• Definition of a Lie group, some examples
• Definition of Lie group homomorphisms
• Recall: Definition of a Lie algebra, Lie algebra of vector fields
• Example 8.36 (b) and Theorem 8.37 (only sketch the proof)
• Some examples for Lie algebras of a Lie group
• Define the induced Lie Algebra Homomorphism of Theorem 8.44 (only sketch how the map is defined) and orally state the properties of Proposition 8.45
• Recall: Definition of integral curve
• Statement: Any maximal integral curve of a left-invariant vector field is defined on all of $\mathbb{R}$. (Sketch the proof, it is included in the proof of Theorem 20.1.)
• Present Proposition 20.8. and comment orally

1b: 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 and the notions that accompany them ([2], ch. 7 "Group Actions and Equivariant Maps", [1], ch.1.1-1.3), 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"), 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, or [2] "Infinitesimal Generators of Group Actions").

• Define (smooth) left- and right actions of groups
• Define orbits, stabilisers and fixed points
• Define transitive and free actions
• Give examples, especially the example of conjugation (also called adjunction), which in [2] is 7.22 (d) and the induced action of a group on its own Lie algebra (basically only the first paragraph following the header "The Adjoint Representation", do not include the definition of a representation in this talk)
• Define equivariant maps
• Define a Lie algebra action on a manifold
• Define a fundamental vector field
• Show Lemma 20.14
• Formulate Theorem 20.15 in terms of a Lie algebra action on a manifold and prove it
• Show [1], Proposition 11.6, state 11.7 & 11.8 (you can orally sketch the proof) and show 11.9

3a**: Universal covering groups, 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 previous chapters:

• Recall: Theorem 8.44 from talk 2a
• State Theorem 20.16, describe how the action is defined - there is a graphic interpretation, think on it! (then define the distribution $D$, explain the method stated in the first sentence of the proof and mention how actions and orbit maps correspond to each other)
• Work through Theorem 20.19, sketch the main ideas and arguments of the proof
• State Corollary 8.50 without proof
• Explain Theorem 7.7 (only present the idea of the proof, not the technicalities)
• State Theorem 19.26 and orally (or with a picture) explain the idea of the proof
• Formulate and prove Theorem 20.21

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".

• Define the orbit space
• Give examples of "good" and "bad" orbit spaces
• Define proper actions
• State Proposition 21.4 and sketch the idea of the proof
• State Proposition 21.5 without proof (you can comment on it orally)
• State Propositions 21.7 and Corollary 21.8 and sketch the idea of the proof
• Work through Theorem 21.10, explain the outline of the proof and the main arguments and ideas

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

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

• Define left-invariant and right-invariant
• State Proposition 13.1 and Corollary 13.2 and sketch the proof of each
• Define the Haar measure
• Define the average of a function over $G$
• Orally comment on the notion of a smooth family of $k$-forms
• Define the average of a family of $k$-forms over $G$
• State Proposition 13.6 and sketch the idea of the proof, deduce Corollary 13.7
• Prove Proposition 1 from the Blogpost of Terence Tao (ignore the Peter-Weyl-Theorem part at the appropriate point)

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

Review [1], ch. 14+15

• Define vector-valued covectors and vector-valued $k$-forms
• Define the Lie-bracket of Lie-algebra-valued forms (only very shortly comment on shuffles)
• State Lemma 14.4 without proof
• Explain the correspondence of invariant forms on a group $G$ to exteriour powers of the dual to the Lie algebra
• Define the exteriour differentiation and compute its expressions in coordinates
• State and prove Theorem 15.2 and 15.3
• There are two equations presented as "The Maurer-Cartan-equation". Comment orally.

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

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

• Recall the definition of fibre bundles and define principal bundles
• State Theorem 3.3 and sketch the proof
• Recall the definition of the Lie derivative and interiour multiplication of differential forms and the rules of the Cartan calculus
• Define a basic form on a principal bundle
• Define invariant forms on a principal bundle
• Show Theorem 12.2
• Define the vertical tangent space (Definition 11.11)
• Show Proposition 12.4
• Prove Theorem 12.5 and Corollary 12.6

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

Review [1], ch. 16+17

• Define a horizontal distribution on a principal bundle
• Define an invariant horizontal distribution
• State Proposition 16.4 and sketch the proof
• State Proposition 16.2 and comment on the proof orally
• Explain the transformation of a horizontal distribution into a vector-valued 1-form
• State Theorem 16.5 and sketch the proof of (ii), comment on the bijectivity of the construction and (i) orally
• Define a connection
• State Theorem 16.7 and sketch the proof
• State Proposition 16.8 without proof
• Define the Maurer-Cartan connection on the product bundle and describe orally to which horizontal distribution it corresponds
• Define the curvature
• State Theorem 17.2 and sketch the proof
• State Theorem 17.4 without proof

6a*: Universal bundles

Review [1], ch. 5+8

• Define a universal bundle
• State Theorem 5.2 without proof
• Comment orally on the name "classifying space"
• Define the join of two topological spaces
• State Theorem 5.8 without proof
• Define $EG$ and the action of $G$ on $EG$ and argue why $EG \rightarrow BG$ has a principal bundle structure
• Define the Stiefel variety
• State Proposition 8.1 and prove it
• State Proposition 8.2 and prove it
• Define Grassmannians
• Define infinite Stiefel varieties and infinite Grassmannians
• Sketch the proof of Theorem 8.3 and Theorem 8.4

6b: Equivariant cohomology and homotopy quotients

Review [1], ch. 4+9

• Motivate equivariant cohomology as cohomology of the quotient
• Recall talk 3b and its message about "good" and "bad" quotients
• State Lemma 4.3 and sketch the proof
• Define the homotopy quotient
• Define equivariant cohomology
• Compute that the equivariant cohomology of a point is BG
• Define Cartan's mixing space
• State Proposition 4.5 and sketch its proof
• Define Cartan's mixing diagram
• State Theorem 9.5 and Corollary 9.6 and sketch the proof
• Explain how equivariant morphisms induce morphisms on the homotopy quotient and hence on equivariant cohomology

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

Review [1], ch. 6.1+6.2

• Give the definitions of differential group, spectral sequence, first-quadrant spectral sequence
• Draw pictures to make clear how everything works
• Explain why the values get stationary in first-quadrant spectral sequences
• Define filtrations etc.
• Explain Leerays Theorem
• Read through the given example, but do not present it in class
• Instead work through the practice problem 6.3 and present it in class
• Comment on the subject of practice problem 6.4 orally

7b**: The equivariant cohomology of $S^1$ acting on $S^2$

Review [1], ch. 6.3+7

• Work through the example presented in chapter 6.3
• Comment on how this example would change for $\mathbb{C}P^\infty$
• Explain the calculations presented in chapter 7, comment why we can apply Leray's theorem

8a***: 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 $X \times EG/G$.

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

Review [1], ch. 18+19

• Define differential graded algebras and $\mathfrak{g}$-differential graded algebras
• Define morphisms of $\mathfrak{g}$-differential graded algebras and state Proposition 18.5 without proof
• Define the diagonal actions of interiour multiplication and Lie derivative on a tensor product and state Proposition 18.7 without proof
• Define horizontal, invariant and basic elements
• State Proposition 18.10 and comment on the proof orally
• Define the Weil algebra and the Weil map
• Explain the computation of the Weil map relative to a basis
• Argue why the Weil differential takes the form of equation (19.4)
• State Theorem 19.1 without proof
• Present Theorem 19.2 and sketch its proof
• Explain the interpretation of the Weil algebra as an algebraic model for the universal bundle (section 19.5)

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

Review [1], ch. 20+21

• Compute the Weil algebra for a circle action and its Lie derivative
• State Proposition 20.1 and 20.2 and sketch its proof
• State Theorem 20.3 without proof
• State Theorem 21.1 (mention the name!) and sketch the proof of (21.3)
• Define the Cartan model and equivariant forms
• Compute the induced differential on the Cartan model
• State Theorem 21.6 (mention the name!)
• Explain how to compute the equivariant cohomology of a circle action on a point (Section 20.5)
• Verify that the equivariantly closed extension of the volume form  given at the end of Chapter 21 is indeed closed and argue orally why it cannot be exact
• Relate this to the findings of talk 7b

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

Review [1], ch. 22+Appendix A

10a*: Localisation with respect to a variable, torsion, representation theory

Review [1], ch. 23+27

• Define a representation, an invariant subspace and irreducibility
• Define the direct sum of two representations and complete reducibility
• State Theorem 27.3 and Theorem 27.4 without proof
• Define the isotropy representation
• State Theorem 27.5 and comment shortly on the proof
• State and prove Corollary 27.6
• Define the localisation of a module
• Define induced module homomorphisms
• Define torsion
• State Proposition 23.5 and comment on the proof orally
• Define the grading on the localisation
• Comment on Proposition 23.7 orally
• State Proposition 23.8 and sketch its proof
• State Proposition 23.9 and Proposition 23.10 and comment shortly/orally on the proof

10b**: Equivariant cohomology with respect to a locally free action

Review [1], ch. 24

• Recall the definition of free, define locally free and give an example
• State and prove Theorem 24.2
• State and prove Proposition 24.5
• State Theorem 24.7 and sketch its proof
• State Theorem 24.10 and present its proof (not too formal but detailed)
• Recall the definition of a cochain homotopy
• Explain how the proof of Theorem 24.10 can be formulated in terms of a cochain homotopy
• Think on which maps this homotopy can be interpreted to relate

11a*: Regularity of the fixed point set, equivariant vector bundles, normal bundles and equivariant tubular neighbourhoods, a short digression on integration of equivariant forms for a circle action

Review [1], ch. 25.1-25.4+28

• Define regular(=embedded) submanifold (maby orally describe an example that highlights the difference to an immersed submanifold)
• State Theorem 25.1 and sketch its proof
• State Proposition 25.2 and sketch its proof
• Orally comment on Corollary 25.3
• Define a $G$-equivariant vector bundle
• State the tangent bundle as an example
• Define the normal bundle of an immersed submanifold
• Define tubular neighbourhoods and $G$-invariant submanifolds/tubular neighbourhoods
• State Theorem 25.11 and shortly explain the idea of the proof
• Recall tangent and cotangent space of a manifold with boundary
• Define the integration of equivariant forms
• Use the example from Section 28.4 to highlight that a lot of terms vanish through integration
• State Theorem 28.2 and sketch its proof

11b**: Borel localisation for a circle action

Review [1], ch. 25.5+26, throwback to ch. 7

• Recall the calculations from talk 7b and the additive structure of the equivariant cohomology groups
• State Lemma 26.2 and sketch its proof
• State Theorem 25.15 and orally give a motivation for why it holds
• If you want to use the triangle notation later, explain it
• State Theorem 26.1 and work through its proof
• State and prove Corollary 26.3
• Work through the calculations in Section 26.2

12a**: The localisation formula for a circle action and the area of a sphere

Review [1], ch. 29.1+30

• Work through the argumentation of Section 29.1 to explain why we should expect the integral to take the form of a sum
• Recall the results from talk 10a (end of Chapter 27) and the definition of the exponents of a circle action, note the remark on signs
• Recall the definition of equivariantly closed forms and explain the phrase "an equivariantly closed form with respect to $X$"
• Define the moment map of a form which admits an equivariant extension
• Comment on different moment maps for the same invariant closed form
• State Theorem 30.1
• Work through the computation of Section 30.2 and also highlight how different choices of the moment map would yield the same result
• State Theorem 30.2 and orally comment on the notions that appear

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

Review [1], ch. 29.2+31

• Explain the definition of a spherical blow-up and how to construct it (not too many formal details)
• Work through the calculations of Section 31.1 to prove the equivariant localisation formula

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 ([6], Chapter 5) 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 ([7]). 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

[2] Lee, John M. Introduction to Smooth Manifolds. New York: Springer, 2003. Print.