Topic outline

Schedule:
Lectures on Tuesdays 13:15  15:00, in Cramér, except the first three lectures on September 8, September 15, and September 20, 10:1512:00 in Cramér. Last lecture on December 6. Note the changed timeslot.
Teachers:
Dan Petersen (first half)
Alexander Berglund (second half)
Examination:
Six homework sets.
Homework 1. Due 20221004.
Homework 2. Due 20221018.
Homework 3. Due 20221101. Hint for the second problem: there is a map of fiber sequences from the sequence K(Z,3)→S^{2}[3]→S^{2}[2]=K(Z,2) to the sequence K(Z,3)→pt→K(Z,4). Here X[n] denotes Postnikov truncation of the space X. Consider the induced map between the respective Serre spectral sequences. Suggested solution for the second problem.
Homework 4. Due 20221115.
Homework 5. Due 20221129.
Homework 6. Due 20221215.
PhD students also need to carry out an independent project, the topic of which is decided in discussion with the teachers, to be presented in class at the end of the course, tentatively at some point during v50 (December 1216).
Prerequisites:
Homology and cohomology of topological spaces, fundamental group, rudimentary knowledge of cell complexes (CWcomplexes, Deltacomplexes, or similar).
Contents:
Higher homotopy groups, fibrations and fiber bundles, cellular approximation, the theorems of Whitehead, Hurewicz, and Freudenthal, EilenbergMac Lane spaces, Postnikov towers, cohomology operations, the Serre spectral sequence, Serre classes and localizations, rational homotopy type, homotopy theory of differential graded algebras, and Sullivan models.
Lecture plan Lecture 1, 20220908 Higher homotopy groups. Basepoint dependence. Relative homotopy groups. Long exact sequence in relative homotopy. Mapping spaces, loop spaces, smash product. Loopsmash adjunction. Notes Lecture 2, 20220915 Cellular approximation theorem. Whitehead theorem. CW approximation theorem. Notes Lecture 3, 20220920 Fiber bundles. Examples. Homotopy lifting and homotopy extension property. Definition of fibration. Fiber bundles are fibrations. Long exact sequence in homotopy of a fibration. Monodromy. Notes Zoom lecture Lecture 4, 20220927 The path space fibration. The homotopy fiber. Relative homotopy groups as homotopy groups of homotopy fiber. Introduction to spectral sequences. Statement of the Serre spectral sequence. Example calculations: Hopf bundle and homology of complex projective space. Notes Lecture 5, 20221004 Multiplicativity in spectral sequences. Calculation of cohomology ring of complex projective space and loop spaces of spheres. Proof of Hurewicz and relative Hurewicz theorem. Homology Whitehead theorem. Freudenthal suspension theorem for spheres. Construction of Serre spectral sequences, taking the spectral sequence of a filtered complex for granted. Notes (with more details in construction of Serre spectral sequence than was given in class). Lecture 6, 20221011 Spectral sequence of a filtered chain complex. EilenbergMacLane spaces. Puppe sequence. EilenbergMacLane spaces represent cohomology. Calculation of first stable homotopy group of spheres. Notes Lecture 7, 20221018 Principal fibrations of EilenbergMacLane spaces. MoorePostnikov towers. kinvariants. Convergence of MoorePostnikov tower. Obstruction theory. Notes Lecture 8,
20221025Serre classes, Hurewicz and Whitehead theorems modulo a Serre class, rational homotopy equivalences, Qlocalizations. Notes. (See §3 and §4 of the lecture notes for more details.) Lecture 9,
20221101Construction of Qlocalizations. The transgression. Rational cohomology of EilenbergMac Lane spaces. Rational homotopy groups of spheres. (See §4 and §5 of the lecture notes.) Lecture 10,
20221108Simplicial objects. Cochain algebras. Polynomial differential forms. Polynomial de Rham theorem. (See §6 and §7 of the lecture notes.) Lecture 11,
20221115Sullivan algebras. Existence of minimal models. Hirsch lemma. Lecture 12,
20221122Minimal models from Postnikov towers. Homotopy theory of cochain algebras. Uniqueness of minimal models. Lecture 13,
20221129Spatial realization of cochain algebras. Homotopy and cohomology of the spatial realization of a Sullivan algebra. Equivalence between the homotopy categories of simply connected Qlocal spaces and minimal Sullivan algebras. Lecture 14,
20221206Survey of applications: arithmeticity of the group of homotopy equivalences, characterization of rational loop spaces, rational dichotomy, homotopy groups of elliptic spaces, Halperin conjecture. Literature:
A. Hatcher, Algebraic Topology.
A. Hatcher, Spectral Sequences.
P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds. Invent. Math. 29 (1975), no. 3, 245–274.
A. Berglund, Rational homotopy theory.
J.P. Serre, Homologie singulière des espaces fibrés, Ann. of Math. 54 (1951), 425–505.