Cohomology is among the classical invariants that one associates to a topological space. Cohomology is a functor from the category of topological spaces to the category of rings, which is homotopy-invariant. There are many models that can be used to define cohomology, and there is a plenitude of techniques to compute it. For smooth manifolds, the most standard model for computing cohomology was developed by de Rham and uses smooth differential forms.
Equivariant cohomology is an invariant associated with a continuous action of a topological group on a topological space. A first approximation of this invariant is the cohomology of the quotient space. However, unless the action is principal, the cohomology of the quotient fails to produce the correct result — for example, it is not homotopy-invariant. The definition of equivariant cohomology goes through Milnor’s beautiful construction of universal bundles for topological groups. In the smooth category, the equivariant cohomology of the action of a compact Lie group can be defined using Cartan’s model, which combines de Rham differential forms with elements of the Lie algebra.
The seminar will take place 2 × 90 minutes per week, namely on Tuesdays from 14:00–15:30h and 16:00–17:30h in Seminarraum 1 (005), and during the first three weeks of the semester it will be functioning as a Student seminar. In the rest of the semester, the seminar will be continued as previous Learning Seminars, where the group of PhD students in the Poisson Geometry group will be in charge of the talks — although, of course, any participant is more than welcome to contribute with talks if they so desire (if that is the case, please contact Jorn van Voorthuizen). The seminar will also be split into two portions: the first half, we will focus only in the topological category whilst the second half we shall be working in the smooth category.
A tentative schedule for the seminar is given below:
14.04.
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14:00h–14:45h (Jorn) Overview [Tu20, §1.1,1.2,1.4].
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14:55h–15:40h (Arne-Jakob) Homotopy [Tu20, §2.1], [Hat02, §1–4,.21–34,337–346], [Lee11, §208–209], [Lüc, §2.2–2.3].
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15:50h–16:35h (Christoph) CW Complexes [Tu20, Abschn. 2.4-2.6], [Hat02, §5–8, 519–525], [Lee11, §127–143], [Lüc05, §3.1].
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16:45h–17:30h (Christoph) Whithead’s Theorem and Cellular approximation theorem [Tu20, §2.5, 2.7], [Hat02, §346–351], [Lüc, Ch. 4–5, §12.4].
21.04.
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14:00h–14:45h (Jeremias) Singular Homology [Hat02, §108–113], [Lee11, §339–351], [Lüc05, §2.2–2.3].
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14:55h–15:40h (Arne-Jakob) Singular Cohomology [Hat02, §190–204], [Lee11, §374–378], [Lüc05, §5.1–5.3].
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15:50h–16:35h (Tobias) Fibrations and Fibre bundles [Tu20, §2.2–2.3], [Hat02,§375–384], [Lüc, §9.1, 10.1].
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16:45h–17:30h (Jorn) Principal bundles [Tu20, Ch. 3], [Lee12, §540–547].
28.04.
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14:00h–14:45h (Jeremias) Homotopy Quotients and Equivariant Cohomology [Tu20, Ch.. 3], [Lee12, §540–547].
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14:55h–15:40h (Tobias) Equivariant Cohomology [Tu20, §4.4–4.5], [Hat02, §352–357].
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15:50h–16:35h (Karim) Universal Bundles and Classifying Spaces [Tu20, §5.1–5.2, 5.4], [Hus94, §4.9–4.10, 4.13].
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16:45h–17:30h (Karim) Milnor’s Construction [Tu20, §5.3] [Hus94, §4.11–4.12].
05.05. (Rodrigo) Spectral sequences and an example of equivariant cohomology [Tu20, Ch. 6–7] [Hat, pp. 519–532, 542–551] [Lüc, Ch. 19, 23, 26].
12.05. (Christoph) Universal bundles and properties of equivariant cohomology [Tu20, §1.3, Ch. 8–9].
19.05. (Jorn) Differential geometry of principal bundles [Tu20, Ch. 10–12, 14–17].
26.05. No seminar due to lecture-free week of Pentecost.
02.06. (Christoph) Weil model and the Cartan model for circle actions [Tu20, Ch. 18–20].
09.06. (Rodrigo) Cartan model in general and the equivariant de Rham theorem [Tu20, Ch. 21, 22, App. A].
16.06. (Rodrigo) Integration, Localisation, and locally free group actions [Tu20, Ch. 13, 23, 24].
23.06. (Jorn) Borel localisation and representation theory [Tu20, Ch. 25–27].
30.06. (Christoph) Localisation formula for a circle action [Tu20, Ch. 28, 29, 31, Sec. 30.1, 30.2].
07.07. (Rodrigo) Characteristic classes and the localisation formula for a torus action [Tu17, Ch. 23–25] [Tu20, §30.3, 30.4] [DHJvdH19, §9.4–9.10].
14.07. (Jorn) Hamiltonian group actions and equivariant cohomology [Tu20, §32.4, 32.5] [DHJvdH19, Ch. 1, 2, 10].
References
[DHJvdH19] Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, and Theo van den Hurk. ‘Hamiltonian Group Actions and Equivariant Cohomology’. SpringerBriefs in Mathematics. Springer Cham, 2019.
[Hat] Allen Hatcher. Spectral Sequences. https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf.
[Hat02] Allen Hatcher. ‘Algebraic Topology’. Cambridge University Press, 2002.
[Hus94] Dale Husemoller. ‘Fibre Bundles’. Graduate Texts in Mathematics. Springer New York, 1994.
[Lee11] John M. Lee. ‘Introduction to Topological Manifolds’. Graduate Texts in Mathematics. Springer New York, 2011.
[Lee12] John M. Lee. ‘Introduction to Smooth Manifolds’. Graduate Texts in Mathematics. Springer New York, 2012.
[Lüc] Wolfgang Lück. Script for the courses Algebraic Topology I + II (WS 24/25 and SS 25): Basic Introduction to Homotopy Theory. https://him-lueck.uni-bonn.de/data/script_AlgTop.pdf.
[Lüc05] Wolfgang Lück. ‘Algebraische Topologie: Homologie und Mannigfaltigkeiten’. vieweg studium; Aufbaukurs Mathematik. Vieweg+Teubner Verlag Wiesbaden, 2005.
[Tu17] Loring W. Tu. ‘Differential Geometry: Connections, Curvature, and Characteristic Classes‘. Graduate Texts in Mathematics. Springer New York, 2017.
[Tu20] Loring W. Tu. ‘Introductory Lectures on Equivariant Cohomology’. Annals of Mathematics Studies. Princeton University Press, 2020.