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  • Homotopy theory

    Schedule:

    Lectures on Tuesdays 13:15 - 15:00, in Cramér, except the first three lectures on September 8, September 15, and September 20, 10:15-12:00 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 2022-10-04.

    Homework 2. Due 2022-10-18.

    Homework 3. Due 2022-11-01. Hint for the second problem: there is a map of fiber sequences from the sequence K(Z,3)→S2[3]→S2[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 2022-11-15.

    Homework 5. Due 2022-11-29.

    Homework 6. Due 2022-12-15.

    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 12-16).

    Prerequisites:

    Homology and cohomology of topological spaces, fundamental group, rudimentary knowledge of cell complexes (CW-complexes, Delta-complexes, or similar).

    Contents:

    Higher homotopy groups, fibrations and fiber bundles, cellular approximation, the theorems of Whitehead, Hurewicz, and Freudenthal, Eilenberg-Mac 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, 2022-09-08 Higher homotopy groups. Basepoint dependence. Relative homotopy groups. Long exact sequence in relative homotopy. Mapping spaces, loop spaces, smash product. Loop-smash adjunction. Notes
    Lecture 2, 2022-09-15 Cellular approximation theorem. Whitehead theorem. CW approximation theorem. Notes
    Lecture 3, 2022-09-20 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, 2022-09-27
    		 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, 2022-10-04 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, 2022-10-11 Spectral sequence of a filtered chain complex. Eilenberg-MacLane spaces. Puppe sequence. Eilenberg-MacLane spaces represent cohomology. Calculation of first stable homotopy group of spheres. Notes
    Lecture 7, 2022-10-18 Principal fibrations of Eilenberg-MacLane spaces. Moore-Postnikov towers. k-invariants. Convergence of Moore-Postnikov tower. Obstruction theory. Notes
    Lecture 8,
    2022-10-25
    		 Serre classes, Hurewicz and Whitehead theorems modulo a Serre class, rational homotopy equivalences, Q-localizations. Notes. (See §3 and §4 of the lecture notes for more details.)
    Lecture 9,
    2022-11-01
    		 Construction of Q-localizations. The transgression. Rational cohomology of Eilenberg-Mac Lane spaces. Rational homotopy groups of spheres. (See §4 and §5 of the lecture notes.)
    Lecture 10,
    2022-11-08
    		 Simplicial objects. Cochain algebras. Polynomial differential forms. Polynomial de Rham theorem. (See §6 and §7 of the lecture notes.)
    Lecture 11,
    2022-11-15
    		 Sullivan algebras. Existence of minimal models. Hirsch lemma.
    Lecture 12,
    2022-11-22
    		 Minimal models from Postnikov towers. Homotopy theory of cochain algebras. Uniqueness of minimal models.
    Lecture 13,
    2022-11-29
    		 Spatial realization of cochain algebras. Homotopy and cohomology of the spatial realization of a Sullivan algebra. Equivalence between the homotopy categories of simply connected Q-local spaces and minimal Sullivan algebras.
    Lecture 14,
    2022-12-06
    		 Survey 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.