MATH 664-ALGEBRAIC TOPOLOGY II-SPRING 2024-A. Freire


Course outline and syllabus

Lecture summaries

(These are summaries of idealized lectures, with references; what would be included in an actual lecture if I had say 90 min, instead of 75.)
Constantly being updated, and forward-looking, like a plan for the near future of the course.

Alternative proofs

Once in a while I'll be moved to make slight adaptations to proofs found in the canonical literature, and will record them here.

Student presentation topics


COURSE LOG

PART 0: Poincare duality
(overflow from the fall course)

Tu 1/23    Orientation of manifolds (review): local orientation, oriented and R-oriented manifolds; fundamental class.
                 Kronecker product and cup product (in singular homology/cohomology); adjoint of cup product, relative versions, naturality


Th 1/24  Direct limits of abelian groups/cohomology  with compact supports/Poincare duality for noncompact oriented manifolds (statement)/
              augmentation map and Kronecker product/ duality in the case of R^n


Tu 1/30  Mayer-Vietoris sequence in relative cohomology. Proof of Poincare duality

Th 2/1   Poincare duality in de Rham cohomology

Tu 2/6  Poincare duality in de Rham cohomology (conclusion)

Mayer-Vietoris sequences in de Rham cohomology and the proof of duality

PART 1: INTRO HOMOTOPY THEORY

Th 2/8 Homotopy groups/Homotopy exact sequence

Tu 2/13 weak homotopy equivalence (Whitehead)/compression principle, compression lemma

Th 2/15 mapping cylinders/ homotopy extension for  CW pairs/proof of Whitehead's theorem/spaces with same htpy gps, not htpy eq

Tu 2/20 Cellular approx of maps and htopies/weak htpy equiv and bijections of htopy sets/htopy eq of X, X/A (A contractible)
simplification of n-connected CW complexes/relative version.

Th 2/22 homotopy groups of products and wedges/ cones and suspensions/ Freudenthal suspension theorem (start)

Tu 2/27 Freudenthal suspension (end)/pi_n(S^n)/ Homotopy groups of spheres (stable homotopy, Whitehead products)/action of pi_1 on based maps, bijection of quotient and free homotopy sets.

Th 2/29 Action of pi_1(A) on pi_n(X,A)/case of pi_2(X,A)/The Hurewicz homomorphism/Homotopy effect of adding a cell

Tu 3/5 Proof of Hurewicz theorem (see the "alternative proofs" link, or [Hatcher, 4.2].)

Th 3/7 Eilenberg-MacLane spaces: existence/uniqueness. Fiber bundles (start): homotopy lifting, exact sequence.

Tu 3/12, Th 3/14: SPRING BREAK

PART 2: FIBER BUNDLES and CHARACTERISTIC CLASSES

Tu 3/19 locally trivial fibrations [Hatcher, Cohen 1.1 to 2.2, Milnor-Stasheff 2,3,5]

Th 3/21 Vector bundles and principal bundles: examples [Cohen 1.2, Milnor-Stasheff]

Tu 3/26 Classification of vector bundles [Milnor-Stasheff no.5, Husemoller pp. 28--34].

Th 3/28 NO CLASS DAY

Tu 4/2 Characteristic classes: Stiefel-Whitney classes [Cohen ch. 3, Milnor-Stasheff 4--11]

Th 4/4 Classification of principal bundles [ ref: Steenrod]

Tu 4/9 Applications: classifying spaces of G-principal bundles and vector bundles [ref. R. Cohen's notes: ch. 2, sections 2, 5.1, 5.2]

Th 4/11 Thom isomorphism, euler class, Gysin sequence [Cohen's notes Ch. 3 sec.2; Milnor-Stasheff 9,10, 14]

Tu 4/16 Def. of Chern classes/ approach via connections and curvature forms [ref: Bott-Tu, R. Cohen, Milnor-Stasheff]

Th 4/18: Chern classes and curvature (cont.)/ Intro to Cech cohomology


PART 3: FRAMED COBORDISM, OBSTRUCTION THEORY

Tu 4/23 Obstruction theory I: the obstruction cocycle and the difference cochain
Presentations:
George: Chern classes of line bundles via the curvature of a connection and Cech cohomology
Dalen: Hopf's theorem classifying maps from n-dimensional complexes to the n-sphere

Th 4/25  Obstruction theory II: Hopf-Whitney theorem, maps to K(pi, n) and cohomology
Notes on obstruction theory (8 pages)

Tu 4/30 Pontrjagin's framed cobordism (ref.: toplib)
Presentation: Amer (Wednesday 5/1)

Th 5/2 Spin tructures and w_2 [Lawson-Michelson, Friedrich]
Presentations: Sam, Tariq-afternoon

Tu 5/7 (last day)Framed cobordism,  Hopf invariant [toplib]
Presentations: Ivy, Jared (afternoon)