MATH 664-ALGEBRAIC TOPOLOGY II- A. FREIRE

Topics and references

Syllabus

COURSE LOG

M 1/24   homotopy groups, invariance under covering maps, relative homotopy groups
(Hatcher 4.1,  F-F sect. 8)

W  1/26  homotopy exact sequence, compression principle

F 1/28 Whitehead's theorem  (weak htpy equiv implies htpy equiv for CW)-proof of the subspace case

Problems (Hatcher 4.1): 2, 6.

Prove: if (X,A) is a CW pair, Y a CW complex, f a cellular map from A to Y, then the space obtained by attaching X to Y via f
is a CW complex. As a consequence, show that the mapping cylinder of a cellular map of CW complexes has a CW complex structure.
(OK to assume all CW complexes in question are connected and finite.)

M 1/31 Whitehead's theorem (end of proof.) More on weak htpy equiv.
[F-F 9.5, 11.4, 11.5] [Hatcher p 356-357]

W 2/2  Weak homotopy equivalence: action on homology, cohomology
Freudenthatal suspension theorem: surjectivity

F 2/4 Unlinking of submanifolds, Freudenthal suspension: surjectivity

M 2/7 Freudenthal suspension: application to htopy groups of spheres, inv. of degree under suspension
Thom-Pontrjagin theory, start: framed submanifolds, framed cobordism

W 2/9 From maps of spheres to framed submanifolds

F 2/11 From framed submanifolds to maps of spheres (ref: Pontrjagin)

M 2/14 The framed cobordism group. Freudenthal suspension again

W 2/16 Freudenthal suspension via cobordism (end)

F 2/18 Fiber bundles and homotopy lifting

M 2/21  Homotopy lifting, homotopy sequence for Serre fibrations (start)

W 2/23 Homotopy sequence, examples; Hopf fibrations, Stiefel and Grassman manifolds

F 2/25 Stiefel and Grasssman manifolds: loc trivial fiber bundles, homotopy groups
Ref: Milnor-Stasheff, Hatcher

M 2/28 Tautological n-dim vector bundles over G_n /infinite Grassmannian, classification of n-vector bundles
Ref: Milnor-Stasheff sect. 5

W  3/2 cellular approximation: n-connected CW models/ Hurewicz theorem (homology of n-connected spacess.)
(Hatcher p.353-354, p. 366-7)

F  3/4 Htopy groups of a bouquet of spheres/capping with (n+1)-cells/Hurewicz isomoprhism
(F-F p. 132-3, Hatcher Ex 4.29 and p. 367)

M  3/7  Relative Hurewicz theorem/ homotopy addition theorem
(Ref: Bredon VII.10, VII.9)

W  3/9 Homotopy/homology ladder/Whitehead's theorem II (maps induced on homotopy and homology)/
Effect of attaching a cell (Ref. Bredon VII.11)

F  3/11  Existence & uniqueness of K (G,n)/Postnikov towers/ Borel conjecture
(Ref: Hatcher p. 365-6; Bredon VII.11)

Problems from Hatcher:
4.1:  6, 11, 12, 15, 20
4.2: 6, 8, 12, 19, 31, 32

4.1.11, 4.1.12: Contractible CW complexes (Utley)
4.1.20: finiteness of homotopy classes of maps (Valdeon Sauza)

M 3/14, W 3/15, F 3/18: SPRING BREAK

Next topics:

1. Obstruction theory (F-F 18.1, 18.2/Steenrod 32--34)
2. Homotopy construction of cohomology and applications (F-F 18.3, 18.4)
3. Mappings from n-manifolds to the n-sphere (Pontrjagin III.1)/ mappings from (n+1)-manifolds to n-sphere [Rengaswami]
4. Hopf invariant (Pontrjagin III.2)
5. Vector bundles, constructions (Milnor-Stasheff #3) [ Islam]
6. Stiefel-Whitney classes (M-S #4)
7. Grassmann manifolds, cohomology ring (M-S #6,#7) [N Burns]
8. Stiefel-Whitney classes: existence (M-S #8)
9. Oriented bundles, euler class (M-S #9)
10.Thom isomorphism theorem (M-S #10)
11. Applications to smooth manifolds (M-S #11)
12. Spin structures on vector bundles (Lawson-Michelson II.1)

M 3/21 Obstruction theory: the obstruction cocycle

W 3/23 Obstruction theory: the difference cochain, main lemmas

F 3/25 Obstruction theory: main extension theorem, relative version.
Fundamental class of a K(G,n)

M 3/28 Applications of obstr theory: free htopy classes and cohomology, Hopf theorem

W 3/30 geom approach to Hopf theorem/model for K(Z,2)/ axioms for reduced ciohomology (Hatcher p. 202)

F  4/1 reduced suspension and loop space/ reduced cohomology from pointed homotopy classes to K(G,n) (Hatcher p. 394-402)

M 4/4 Vector bundles, constructions (Tariq)

W 4/6 Stiefel-Whitney classes: axiomatic definition, first examples

F 4/8 Stiefel-Whitney classes: example, total SW class of real projective space, applications

M 4/11 Hopf invariant of maps S^(2k+1) to S^(k+1)  [Patrick]

W  4/13 Universal bundles, Grassmann manifolds

F no class  (U holiday)

M 4/18  Cohomology ring of Grassman manifolds, applications [Nathan]

W 4/20 Mapings from (n+1)-manifolds to the n-sphere [Sathya]

F 4/22 Problems on homotopy and CW complexes [Jeffrey, Guillermo]

M 4/25 Obstruction theory and SW classes

W 4/27  Obstruction theory and SW classes (cont'd)

F 4/29 Oriented bundles, euler class

M 5/2 Bott periodicity [Bryan]

W 5/4 Geometric construction of SW classes as obstructions: w_1 and orientability, Stiefel's theorem

Framing 3-manifolds with bare hands, by Benedetti and Lisca (ArXiv 2018)

This recent paper includes several more or less "bare hands" proofs of Stiefel's thm, and a kind of survey of the topic.

Exercise: complete the outline of proof of Prop. 2 (p.6 of the paper) to conclude w_2(M)=0. Recall that via
obstruction theory, this gives a proof of Stiefel's thm using only basic facts about SW classes.

F 5/6 Spin group and spin structure

M 5/9 Spin structure on TM and w_2
Notes on spin structures and w_2 (6 pages)