MATH 664- SPRING 2026- A. FREIRE

Syllabus, references, outline

Ralph Cohen's Stanford Lecture Notes

1/20  Mapping cylinder, homotopy extension and other problems

1/23 Euclidean neighborhood retracts/proof of Tietze's theorem on homotopy extension for CW pairs
review of degree of maps

1/27 (snow day) Hopf's theorem on classification of maps from n-manifolds to the n-sphere, up to homotopy
Hopf degree theorem

1/29 Homotopy groups; definition, first properties. [ref: Hatcher, ch.4]

2/3 Relative homotopy groups, long exact sequence, htopy groups of a product

2/5 Htopy groups of a wedge, action of fundamental group, free homotopy classes/
Whitehead's theorem (a map inducing isomorphisms of all homotopy groups is a homotopy eq.)

2/10 n- connected cell complex htopy eq to no cells of dim 1,...n /Hurewicz isomorphism (start)

2/12 Hurewicz iso (conclusion), corollaries (Whitehead's thm 2)

Problem list for the course
(Sources: Hatcher, Fomenko-Fuchs, Cohen)

2/17 Freudenthal's suspension theorem/Whitehead product (def.)
Ref: [Fomenko-Fuchs 10.1]

2/19 Eilenberg-McLane spaces; Existence, examples, uniqueness up to htopy type
Ref: [Hatcher p.365-366; Fomenko-Fuchs 11.7, 11.8]

2/24 Fibrations and fiber bundles: homotopy lifting, homotopy exact sequence of a fibration
Ref: [S-T Hu Ch III, p.61-65] 

2/26 Fibrations: examples. Hopf fibrations, K(Z,2), Homotopy groups of spheres, path space fibration (loop space fiber), any map is
homotopy equivalent to a fibration (comparison with cofibrations via the mapping cylinder)
Ref: [Cohen, ch. 4; Fomenko-Fuchs 9.4 to 9.7 and 10.4]

3/3 Obstruction theory: obstruction cocycle and deformation cochain
Obstruction theory notes

3/5 Obstruction theory: Hopf-Whitney theorem/ Homotopy classes to K(G,n) and nth cohomology with G coefficients.

3/10, 3/12: SPRING BREAK

3/17: vector bundles, principal bundles: first definitions, examples
[Cohen 1.1, 1.2/Milnor-Stasheff #2]

3/19: Changing the fiber/from principal to vector bundles and back/transition functions and structure group
[Cohen 1.3]

3/24: pullback bundle, product bundle, Whitney sum, orthogonal complement/tangent bundle of RP^n/homotopy invariance of the pullback

3/26 classification of principal bundles; universal principal bundles (aspherical total space suffices)
Classification of principal bundles and vector bundles

3/31 examples of universal bundles; homotopy groups of G and G-principal bundles over spheres. Classifying spaces BG.
4/2 SPRING RECESS (no class)

4/7 Characteristic classes of vector bundles (start):  classifying spaces and maps, Kuenneth formula, Leray-Hirsch theorem.

4/9 Projectivized vector bundle; construction of SW and Chern classes via Leray-Hirsch [cp. Husemoller]

4/14 Axioms for SW classes; examples [cp. Milnor-Stasheff]: Whitney duality, SW classes of spheres and projective spaces,

4/16 SW numbers and cobordism/ verification of the axioms/splitting maps and Whitney sum formula

4/21 Vanishing of w_1 and c_1; oriented bundles, euler class.