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)