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.<\/p>\n\n\n\n
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 \u2014 for example, it is not homotopy-invariant. The definition of equivariant cohomology goes through Milnor\u2019s 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\u2019s model, which combines de Rham differential forms with elements of the Lie algebra.<\/p>\n\n\n\n
The seminar will take place 2 \u00d7 90 minutes per week, namely on Tuesdays<\/strong> from 14:00\u201315:30h<\/strong> and 16:00\u201317:30h<\/strong> in Seminarraum 1 (005)<\/strong>, 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 \u2014 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.<\/p>\n\n\n\n A tentative schedule for the three first weeks of the seminar is given below:<\/p>\n\n\n\n 14.04. 21.04. 28.04.
\n \u21b5 <\/span>\n14:00h\u201314:45h (Jorn) Overview<\/strong> [Tu20, \u00a71.1,1.2,1.4]<\/span>.
\n\n \u21b5 <\/span>\n14:55h\u201315:40h (Arne-Jakob) Homotopy<\/strong> [Tu20, \u00a72.1], [Hat02, \u00a71\u20134,.21\u201334,337\u2013346], [Lee11, \u00a7208\u2013209], [L\u00fcc, \u00a72.2\u20132.3].
\n\n \u21b5 <\/span>\n15:50h\u201316:35h (Christoph) CW Complexes<\/strong> [Tu20, Abschn. 2.4-2.6], [Hat02, \u00a75\u20138, 519\u2013525], [Lee11, \u00a7127\u2013143], [L\u00fcc05, \u00a73.1].
\n\n \u21b5 <\/span>\n16:45h\u201317:30h (Christoph) Whithead’s Theorem and Cellular approximation theorem<\/strong> [Tu20, \u00a72.5, 2.7], [Hat02, \u00a7346\u2013351], [L\u00fcc, Ch. 4\u20135, \u00a712.4].\n\n<\/p>\n\n\n\n
\n \u21b5 <\/span>\n14:00h\u201314:45h (Jeremias) Singular Homology<\/strong> [Hat02, \u00a7108\u2013113], [Lee11, \u00a7339\u2013351], [L\u00fcc05, \u00a72.2\u20132.3]<\/span>.
\n\n \u21b5 <\/span>\n14:55h\u201315:40h (Arne-Jakob) Singular Cohomology<\/strong> [Hat02, \u00a7190\u2013204], [Lee11, \u00a7374\u2013378], [L\u00fcc05, \u00a75.1\u20135.3].
\n\n \u21b5 <\/span>\n15:50h\u201316:35h (Tobias) Fibrations and Fibre bundles<\/strong> [Tu20, \u00a72.2\u20132.3], [Hat02,\u00a7375\u2013384], [L\u00fcc, \u00a79.1, 10.1].
\n\n \u21b5 <\/span>\n16:45h\u201317:30h (Jorn) Principal bundles<\/strong> [Tu20, Ch. 3], [Lee12, \u00a7540\u2013547].\n\n<\/p>\n\n\n\n
\n \u21b5 <\/span>\n14:00h\u201314:45h (Jeremias) Homotopy Quotients and Equivariant Cohomology<\/strong> [Tu20, Ch.. 3], [Lee12, \u00a7540\u2013547]<\/span>.
\n\n \u21b5 <\/span>\n14:55h\u201315:40h (Tobias) Equivariant Cohomology<\/strong> [Tu20, \u00a74.4\u20134.5], [Hat02, \u00a7352\u2013357].
\n\n \u21b5 <\/span>\n15:50h\u201316:35h (Karim) Universal Bundles and Classifying Spaces<\/strong> [Tu20, \u00a75.1\u20135.2, 5.4], [Hus94, \u00a74.9\u20134.10, 4.13].
\n\n \u21b5 <\/span>\n16:45h\u201317:30h (Karim) Milnor’s Construction<\/strong> [Tu20, \u00a75.3] [Hus94, \u00a74.11\u20134.12].\n\n<\/p>\n<\/div><\/div>\n\n\n\n