Lecture material
| Date | Topics
|
|---|
| Feb. 7 | The homotopy category, homotopy groups
|
| Feb. 9 | Action of the fundamental group, relative homotopy groups
|
| Feb. 12 | Covering spaces, disc lifting, relation with
homotopy groups
|
| Feb. 14 | Fibrations and the long exact sequence (PS1 due)
|
| Feb. 16 | Cofibrations, homotopy extension, mapping cylinders
|
| Feb. 19 | Holiday, no classes
|
| Feb. 20 | Compactly generated spaces, function spaces, loop spaces
|
| Feb. 21 | Cellular maps, the Whitehead theorem (PS2 due)
|
| Feb. 23 | Cellular maps, homotopy fibers, statement of major
theorems
|
| Feb. 26 | Approximating maps by cell inclusions, homotopy
groups of wedges
|
| Feb. 28 | Homotopy excision, Freudenthal suspension (PS3 due)
|
| Mar. 2 | The Hurewicz theorem, the homology Whitehead theorem
|
| Mar. 5 | Postnikov factorizations and Eilenberg-Maclane spaces
|
| Mar. 7 | Introduction to spectral sequences (PS4 due)
|
| Mar. 9 | The Leray-Serre spectral sequence, some examples
|
| Mar. 12 | Proofs: filtered chain complexes, exact couples
|
| Mar. 14 | Deriving the Serre spectral sequence (PS5 due)
|
| Mar. 16 | Attempted brief overview of simplicial sets
|
| Mar. 19 | Vector bundles, principal G-bundles
|
| Mar. 21 | Principal G-bundles, classifying spaces (PS6 due)
|
| Mar. 23 | Characteristic classes, calculations for line bundles
|
| Mar. 26-30 | Spring break, no classes
|
| Apr. 2 | Recap, cohomology of BU(n)
|
| Apr. 4 | Chern classes, Stiefel-Whitney classes, Pontriagin classes
|
| Apr. 6 | Stiefel manifolds and Grassmannians(PS7 due)
|
| Apr. 9 | Projectivizations, polynomials satisfied by characteristic classes
|
| Apr. 11 | The addition formula for characteristic classes(PS8 due)
|
| Apr. 13 | Symmetric polynomials, the Chern character, and the Grothendieck group K(X)
|
| Apr. 16 | Holiday, no classes
|
| Apr. 18 | KO(X), some calculations (PS9 due)
|
| Apr. 20 | Representability of K and KO, statements and consequences of Bott periodicity
|
| Apr. 23 | The tangent bundle of RP^n in KO(RP^n), classifying spaces
|
| Apr. 25 | EG -> BG is a classifying space, some omissions (PS10 due)
|
| Apr. 27 | Representability of cohomology by Eilenberg-Maclane spaces, cohomology operations
|
| Apr. 30 | Stable cohomology operations, Steenrod squares
|
| May. 2 | Cup-i products and definition of Steenrod squares, basic formulas(PS11 due)
|
| May. 4 | Steenrod squares are stable, "space of multiplications is contractible", independence from choices
|
| May. 7 | Digression about the Steenrod algebra & stable homotopy groups, admissible monomials
|
| May. 9 | Excess, admissible monomials form a basis(PS12 due)
|
| May. 11 | Apology, the dual Steenrod algebra, transgressions and Steenrod operations, cohomology of Eilenberg-MacLane spaces
|
| May. 14 | Cohomology operations and K-theory
|
| May. 16 | Audience Q&A about random further topics
|