Lecture material
| Date | Topics
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|---|
| Jan. 21 | The homotopy category, homotopy groups
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| Jan. 23 | Action of the fundamental group, relative homotopy groups
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| Jan. 26 | Long exact sequence of relative groups, examples
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| Jan. 28 | Covering spaces, unique disc lifting (HW1 due)
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| Jan. 30 | Fibrations, the homotopy lifting property
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| Feb. 2 | Converting maps to inclusions and fibrations
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| Feb. 4 | Cofibrations (HW2 due)
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| Feb. 6 | Function spaces, cofibrations and fibrations are dual
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| Feb. 9 | Loop spaces, iterated loop spaces
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| Feb. 11 | The Puppe sequence, CW approximation (HW3 due)
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| Feb. 13 | The Whitehead theorem
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| Feb. 16 | Connective covers and Postnikov stages
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| Feb. 18 | The Postnikov tower, Eilenberg-Maclane spaces (HW4 due)
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| Feb. 20 | No class
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| Feb. 23 | The Blakers-Massey excision theorem, Freudenthal suspension
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| Feb. 25 | Stability
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| Feb. 27 | The Hurewicz theorem
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| Mar. 2 | The singular complex, equivalences in homotopy theory
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| Mar. 4 | Spectral sequences (HW5 due)
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| Mar. 6 | Spectral sequences, extension problems
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| Mar. 9 | Cohomology of matrix groups and K(Z/n,1)s
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| Mar. 11 | K(Z,2), K(Z/2,2), pi_4 and pi_5 of S^3 (HW6 due)
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| Mar. 13 | Finiteness and finite generation of homology, cohomology, and homotopy
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Mar. 16
Mar. 18
Mar. 20 | Spring break (no classes)
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| Mar. 23 | How spectral sequences arise
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| Mar. 25 | Spectral sequences associated to double complexes (HW7 due)
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| Mar. 27 | Rational homotopy groups
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| Mar. 30 | Eilenberg-Maclane spaces represent cohomology
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| Apr. 1 | Extended powers, homotopy orbits, Steenrod squares (HW8 due)
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| Apr. 3 | Adem relations, admissible monomials, excess, Hopf invariant
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| Apr. 6 | Transgressions, cohomology of Eilenberg-Maclane spaces at 2
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| Apr. 8 | Deducing homotopy groups from cohomology (HW9 due)
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| Apr. 10 | Bordism theory
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| Apr. 13 | Bordism is a homology theory, basic calculations
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| Apr. 15 | Oriented bordism and other variants (HW10 due)
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| Apr. 17 | Normal bundles, stable normal bundles
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| Apr. 20 | Thom spaces and Pontriagin-Thom collapse
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| Apr. 22 | The map from bordism to homotopy of Thom spaces
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| Apr. 24 | The map from homotopy of Thom spaces to bordism
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| Apr. 27 | Homology of Thom spaces
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| Apr. 29 | The Thom isomorphism theorem
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| May 1 | The Gysin sequence and relations between Grassmannians
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| May 4 |
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| May 6 |
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| May 8 |
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