MATH 562-TOPOLOGY II-Spring 2021

Syllabus

Course outline

References

Whitney topology: stability, genericity, Whitney embedding
(Lecture notes--in progress)

Lecture summaries and problem sets (last update 2/17)


W 1/20   C^r structure, C^r maps, C^r diffeomorphism
Lecture 1/20 (2 pages)


F 1/22  Tangent space, differential, tangent bundle (start)
Lecture 1/22 (5 pages)

M 1/25 differentiable structure on TM, vector bundles, nontriviality of TS^2
Lecture 1/25 (4 pages)


W 1/27  IFT, immersions (local form), embeddings, submanifolds
Lecture 1/27 (5 pages)


F 1/29  submersions, regular values, matrix groups
Lecture 1/29 (6 pages)

M  2/1 Lecture postponed (snow day)
Problem set 1: Lectures 1--5 (11 problems), see lecture summaries
problem 1
problem 2
problem 3
problem 4
problem 5
problem 6
problem 7
problem 8
problem 9
problem 11

W  2/3  Transversality (start)
Lecture 2/3 (3 pages)

F   2/5  Transversality of maps; paracompactness
Lecture 2/5 (5 pages)

M  2/8   Differentiable partitions of unity, first application
Lecture 2/8 (4 pages)

W 2/10  Partitions of unity: applications
Lecture 2/10 (4 pages)
Afternoon: problem session/ makeup

F 2/12 Embeddings in euclidean space
Lecture 2/12(3 pages)

M 2/15  Covering dimension of manifolds/ Riemannian metrics (start)
Lecture 2/15 (3 pages)

W 2/17 Riemannian metrics
Lecture 2/17 (4 pages)
Problem session (5:30--6:30) (Outline of problems  3, 7, 10)

F 2/19 Semicontinuous functions, Whitney topology
(Notes, p. 1--3)
Lecture 2/19 (3 pages)

M 2/22 W^1 Whitney topology/ continuity of composition/ openness of immersions
Lecture 2/22 (4 pages)

W 2/24 Openness of embeddings and diffeomorphisms
Lecture 2/24 (3 pages)

F 2/26 Sard's theorem (statement), stability of regular values, Whitney's embedding theorem
Lecture 2/26 (3 pages)

M 3/1 Manifolds with boundary
Lecture 3/1  (5 pages)

W 3/3 Transversality for manifolds with boundary/nonexistence of retractions/Brouwer fixed point
(for diff'ble maps, then for cont maps using Stone-Weierstrass--see[Milnor 1])
Lecture 3/3 (3 pages)
Read: Classification of 1-manifolds (see [Milnor 1])

F 3/5 Problem session
prob 12
prob 13
prob 14
prob 15
prob 16
prob 17
prob 18
prob 19
prob 20
prob 21
prob 22

M 3/8 Parametrized transversality and homotopy transversality theorems
Lecture 3/8 (4 pages)


W 3/10 Proof of Sard's theorem
Lecture 3/10 (6 pages)


F  3/12 Problem session/ Mod 2 intersection (start)
Lecture 3/12 (2 pages)


M 3/15 Boundary theorem/ Mod 2 degree
Applications: FTA, Brouwer fixed pt for cont maps
Lecture 3/15 (4 pages)

Suggested problems from G-P, sect. 4: 4, 5, 6, 9, 11, 12, 17, 19

Problem set 3 (discussion starts Wed. 3/17): Winding numbers and the Jordan-Brouwer separation theorem.

Problems 1-12 in section 5, chapter 2 of [G-P] (I'll do 12.)

W 3/17 Problem session: Mod 2 winding numbers, Jordan-Brouwer separation theorem
[G-P sect 3.5, Munkres 1 61, 63, 65]
prob 1     prob 4   prob 2   prob 5
prob 3     prob 6   prob 8   prob 7
prob 9     prob 11

F  3/19 Problem session (cont'd)
Lecture 3/19 (3 pages)
(Solutions to prob 7, 8 (alt), 10, 12)


M 3/22 Borsuk-Ulam theorem [G-P sect 3.6, Munkres 1, 57]
Spheres are simply connected [Munkres 1, 59]
Lecture 3/22
Review problems: [G-P] p.95 no 3, Munkres 1 p.359: 2, 4(d)

W  3/24 Retracts and deformation retracts/ fund group of products/
nullhomotopy and extension to the ball/Applns: matrices with positive entries, nonvanishing vector fields on the ball
Review: p. 353, 1
Lecture 3/24

F 3/26 Appln to dimension theory/ Homotopy equivalence. Example: pi_1 (fig eight) not abelian.
Review problems: p. 375: 3,5 p. 366: 4
Lecture 3/26 (6 pages)

Next: Seifert-van Kampen theorem [Munkres 1: 70, 71]

In preparation, read: Group theory facts [Munkres 1:  67, 68, 69] esp.:
finitely presented groups, free groups, free products of groups

M 3/29 Seifert-van Kampen theorem: statement, proof (start)
Lecture 3/29 (4 pages)

W 3/31 S-vK proof (conclusion)--examples
Lecture 3/31 (4 pages)

F 4/2: NO LECTURE (University holiday)
Take-home midterm    

solutions

M 4/5 Fundamental group: adding a cell/compact surfaces
Lecture 4/5 (5 pages)

W 4/7 conjugacy class of a covering and classification; regular coverings
(Munkres 1, no. 79, 81)
Lecture 4/7 (5 pages)
Review: p. 483--1, 2, 4

F 4/9 lifting of maps over a covering/homomorphisms of coverings are covering maps
Lecture 4/9 (4 pages)

Homework set 4
(Turn in 2 solutions from 1-5, 2 from 6-10; due Monday 4/19.)
solutions
(problems 2b, 3, 5, 6, 8)
Proper Covering Maps (3 pages)
(Motivation for problems 1 and 2 in HW set 4.)

M 4/12 Covering automorphisms and group actions
Lecture 4/12

W 4/14 Properly discontinuous actions and regular covers
Lecture 4/14
Non-Hausdorff example (includes two problems)

F 4/16 Existence of a simply-connected covering space
Lecture 4/16
Exercises: Munkres p 499/500, 1--5 (proof that the fund group of a compact manifold
is finitely generated.)

M 4/19 Orientable manifolds, oriented double cover
Lecture 4/19 (5 pages)
Ref: G-P 3.2, Milnor 1, no. 5

W 4/21 Brouwer degree and applications
Ref: Milnor 1, no.5
Lecture 4-21 (3 pages)
Homework set 5 (due 4/26)

F 4/23 Index of vector field singularities, Poincare-Hopf theorem
Ref. Milnor 1, no.6
Lecture 4/23 (5 pages)

M  4/26: HW 5 due/Hopf's theorem on maps to the sphere (problems 4 to 9 in HW5)
Lecture 4/26 (3 pages)
(Summary and slight reorganization of the proof in G-P.)
prob 2  prob 3  prob 4  prob 5
prob 6  prob 8  prob 9

W  4/28 Vector field singularities and Euler characteristic.
Lecture 4-28 (3 pages)  (based on [G-P, p. 146 and p. 148-150])
prob 10

F  4/30: optional problem session
Review problems on orientation and degree   (8 problems)

Final exam Tuesday May 4, 10:30--12:30--Ayres 121

Included in final (material since 4/5):

Seifert/v-Kampen theorem/ effect of attaching a cell (Munkres 70, 72)
Lifting of maps/covering homomorphisms and automorphisms/regular coverings (Munkres)
Properly discontinuous group actions (Munkres)
Existence of the universal cover (Munkres)
Orientable manifolds (Guillemin-Pollack 3.2, Milnor no.5)
Brouwer degree and applications (Milnor no. 5)
Vector field singularities, Poincare-Hopf (Milnor no. 6, G-P 3.5, 3.7)
Hopf's theorem on dgree and homotopy of maps to S^n (G-P 3.6)

final     solutions (final exam)