## Math 8302

Spring 2019

**Location and time: **Vincent Hall 209, MWF
1:25pm-2:15pm

**Text: Topology: A Geometric Approach ***
* by Terry Lawson.

**Text: Differential Forms in Algebraic Topology ***
* by R. Bott and L. Tu.

Link to download the book

* http://users.metu.edu.tr/serge/courses/422-2014/supplementary/TGeometric.pdf
*

**Lecturer: **Tian-Jun Li, Vincent
Hall 260,
(612) 625-2036

Email: *lixxx248@umn.edu
*

URL:
*http://www.math.umn.edu/~lixxx248*

Office hours:
10:30-11:30 on Fridays, 11-12 on Mondays.

**Course Content
**

Sections 5, 6, 7, 12, 15, 16, 17 in Chapter 6 and sections 5, 6, 7 in Chapter 3 of the book by Lawson.

Chapter I, Chapter II and Section 15 of the book by Bott-Tu.
Gauss-Bonnet Theorem, Frobenius Theorem.

**
Homework
**

There will be 7 or 8 homework problem sets.

Homework 1, due Feb 8 (Friday).

Exercises in Chapter 6: 5.12, 6.6, 6.7, 6.11, 16.10, 17.15, 7.3, 12.3, 12.19, 12.20

This week's Friday office hour: 11-12.

Solutions to Hw1

Homework 2, due Feb 18 (Monday).

Exercises in Chapter 6: 15.5, 15.12, 15.14.

Exercises in Chapter 3: 5.5, 6.1, 7.1, 7.8, 7.13, 6.10

Solutions to Hw2

Homework 3, due March 4 (Monday).

From the book of Bott-Tu:

Exercises 2.1.1, 3.6, 4.3.1, 4.5, 4.8

Solutions to Hw3 (by Bruno Poggi)

Homework 4, due March 25 (Monday).

From the book of Bott-Tu:

Exercises 5.12, 5.16, 6.2, 6.10, 6.14, 6.20

Solutions to Hw4 (by Bruno Poggi)

Homework 5, due April 8 (Monday).

From the book of Bott-Tu, Chapter 6:

Exercises 32, 36, 43, 44, 45, 46

Special office hour: Friday (4/4), 2:30-3:30.

Solutions to Hw5 (by Lilly Webster)

Note: For 6.45, the integral of the Euler class should be -1.
Solutions to Exercise 6.45 (by Bruno Poggi)

**Tests.
**

There are two in class tests and a take home final.

Test 1 on Wednesday, 2/27. It covers the sections of HW1 and HW2.

Test 2 on Monday, 4/22. It covers Sections 1-6 of Bott-Tu.

Take home final problems (5 problems), due Friday, 5/10.

1. Exercises 8.4.

2 (a) Verify that the homotopy operator K given by (8.6) is well-defined. Specifically, show that each summand on the right is a differential form
on the intersection of the open sets U_{\alpha_i}, i=0, ..., p-1.

2 (b) Verify that the homotopy operator in Exercise 12.12.1 is well-defined. Specifically, show that each summand
on the right is a compactly supported differential form
on the intersection of the open sets U_{\alpha_i}, i=0, ..., p+1.

3. Exercises 11.19.

4. Exercise 11.26 (a), (b). No part (c)

For part (a), you could just prove (the equivalent statement) that the integral of the Poincare dual of
the diagonal over the graph of f is L(f) up to sign, where the sign is (-1)^{dim M}. The first part of Lemma 11.22 is useful here.

For part (b), Proposition 6.25 is useful here.

5. Use Exercise 11.26 (b) to prove that the Euler number of any compact, connected Lie group is zero.

**Grading
**

Each test accounts for %15 and the take home final accounts for %20, and homework accounts for %50.