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, Hopf invariant (ref.:
toplib, [Pontrjagin, paragraphs 2, 3, 4 in Chapter 2 and paragraph 2 in
Ch. 3]
Th 5/2 Classification of maps from the 3-sphere to the 2-sphere [Pontrjagin Ch. 4, paragraphs 1,2]
Notes on framed cobordism
(in progress, currrently 5 pages)
Presentations:
Sam: classification of maps from the (n+1)-sphere to the n-sphere. [Pontrjagin Ch.4, paragraph 3]
Amer: Stiefel-Whitney classes and immersions of manifolds into euclidean space.
Tu 5/7 (last day) Spin groups, presentation (Tariq): Spin structures and w_2.
Presentations: Ivy, Jared (afternoon)