MATH 562-TOPOLOGY II-SPRING 2023-A. FREIRE

Tu-Th, 11:20 AM--12:35 PM, Ayres G-013
Office hours: Thursday 4:00--4:45, or by appointment


Content: Basic topics in algebraic and differential topology (continuation of Topology I)

Grading: based on problem sets and a comprehensive in-class final exam.

References and topics



Tu 1/24  Homotopy, homotopy equivalence: examples

Th 1/26 Euclidean neighborhood retracts and applications

Tu 1/31 Consequences of Borsuk's homotopy extension/Tubular neighborhood theorem (compact)

Homework set 1: problems 5, 11, 12, 19, 20 (E. Lima's text, chapter 1, p. 21)
(due Tuesday Feb. 7)
Some solutions

Th 2/2  Tubular nbd thm (noncompact)/ covering maps: surjectve local homeos are finite to one coverings iff proper (start)/
closed maps and stability of preimages

Tubular neighborhoods of submanifolds (4 pages, 1 problem)

Tu 2/7  closed maps with compact preimages are proper/discont. group actions by homeos

Th 2/9 prop. discont groups and coverings/ unique lifting of paths
Homework set 2: problems from 6.6 p.147 in [Lima]: 1(a)(b), 5, 8, 14, AND the problem in the handout "tubular neighborhoods" (p.3)
(due Thursday Feb 16.)
Solution to a problem

Tu 2/14 unique lifting of homotopies/ differentiable coverings

Th 2/16 Some applications of homotopy lifting/fundamental group

Tu 2/21 Fundamental group: change of basepoint, induced homomorphism

Th 2/23 Free homotopy and fundamental group

Tu 2/28 Fund. group is homotopy type invariant

Th 3/2 Lifting correspondence, fund group of D^1, subgroup of Pi_1 assoc. to a cover
Homework set 3
(due first Thursday after spring break.)
HW3-solutions

Tu 3/7 regular coverings/ coverings of a given space: morphisms/
basic lifting criterion

Th 3/9 class postponed

Tu 3/14, Th 3/16: SPRING BREAK

Tu 3/21: main lifiting theorem, covering automorphisms, applications

Th 3/24: existence of coverings; regular coverings and prop. disc. actions

Tu 3/28: Seifert-v. Kampen theorem

Th 3/30  2.3 Transversality thm/transversality homotopy thm/ extension thm

Tu 4/4 Intersection number mod 2 (2.4)
Note on the mod 2 degree

HW set 4: problems from Guillemin-Pollack, due 4/13
 p. 74; 4,5, p.82: Problems 4, 5, 6, 9, 11, 12

Th 4/6 "No class day" (UTK)

Tu 4/11 Mod 2 degree, smoothh Jordan-Brouwer theorem (start, 2.5)

Th 4/13 Smooth Jordan-Brouwer

Tu 4/18 Borsuk-Ulam theorem

Th 4/20 Oriented manifolds

Tu 4/25, Th 4/27:  oriented intersection, oriented degree
Homework set 5
(6 problems/due Th 5/4)
Hw 5 solutions

Tu 5/2 Euler characteristic, Lefschetz numbers, Poincare'-Hopf theorem

Th 5/4 Local Lefschetz numbers, indices of vector field singularities, Poincare-Hopf theorem

Tu 5/9 (last day) Hopf degree theorem

Review problems (from [G-P]:

p. 130: 6, 7, 8
p. 138; 4, 5, 18

FINAL EXAM: Monday, May 15, 3:30--6:00 (in the usual classroom)
final exam
solutions

Comprehensive, but emphasis on problems from the five homework sets, the review problems from Ch. 3 of G-P posted above, and
the problems on p. 144-146 of [G-P] (=proof of the Hopf degree theorem, presented in class.)