MATH 664- SPRING 2026- A. FREIRE

Syllabus, references, outline

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/ Freudenthal suspension
Problem set 1

2/17 Eilenberg-McLane spaces/ Fibrations (start)