## Math 8301 Fall 2018

Location and time: Vincent Hall 301, MWF 1:25pm-2:15pm

Text: Topology: A Geometric Approach by Terry Lawson.
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, another hour to be determined (or by appointment).

Course Content
I plan to cover most of the book in the fall semester.
Part I. A geometric introduction to topology
1. Basic point set topology (quick review)
2. Classification of surfaces (the notion of manifold introduced here and a very geometric approach to surface classification via handlebody)
3. The fundamental group and its application (the main idea of algebraic topology introduced here)
Part II. Covering spaces, CW complexes and Homology
4. Covering spaces
5. CW complexes
6. Homology

Homework
There will be 8 or 9 homework problem sets.

Homework 1, due September 17 (Monday).
Exercises in Chapter 1: 1.1, 1.2(c), 2.1, 2.4(b), 2.10, 4.4, 9.19, 6.7, 9.51, 7.1, 7.5, 7.6
Floating office hours:
Solutions to Hw1

Homework 2, due September 28 (Friday).
Exercises in Chapter 2: 1.1, 1.5, 1.6, 2.2(a), 2.3, 2.5, 3.3, 3.4, 4.1, 4.2, 4.3, 9.26
Floating office hours: 10:30-11:30 on Thursday, 9/27
Solutions to Hw2

Homework 3, due October 8 (Monday).
Exercises in Chapter 2: 5.8, 5.11, 6.2, 6.3, 6.4, 7.2, 7.5, 7.9, 9.31, 9.75, 9.45
Floating office hours: 10:30-11:30 on Thursday, 10/04, 10:30-11:30 on Monday, 10/08
Solutions to Hw3

Homework 4, due October 19 (Friday).
Exercises in Chapter 3: 1.10, 2.2, 3.2, 3.4, 4.1, 4.3, 4.7, 11.12, 11.16, 11.17, 11.19
Floating office hours: 10:30-11:30 on Thursday, 10/04, 10:30-11:30 on Monday, 10/08, 10:30-11:30 on Wednesday, 10/17.
Solutions to Hw4

Quiz 1, Friday, 10/26.
Test 1, Wednesday, 10/31.
Homework 5, due October 29 (Monday).
Exercises in Chapter 3: 8.3, 8.5, 8.8(a), 9.1, 9.4, 9.6, 9.7, 11.22, 11.23(a), 11.25, 11.39
Floating office hours: 10:30-11:30 on Monday, 10/29.
Solutions to Hw5

Homework 6, due November 19 (Monday).
Exercises in Chapter 4: 1.7, 1.14, 1.15, 1.21, 2.1, 2.9 (c, d, e, f), 2.19, 3.10, 3.17
Floating office hours: Wednesday, 10/31, 11/7. No office hour on Friday, 11/2.
Solutions to Hw6

Homework 7, due November 21 (Wednesday).
Exercises in Chapter 5: 1.3, 1.5, 1.6, 1.8, 1.11, 1.12, 2.7, 2.10, 2.11, 5.4
Floating office hours: 10:30-11:30 on Thursday, 11/15, Wednesday, 11/21. No office hour on Friday, 11/16.
Solutions to Hw7

Floating office hours: Wednesday, 11/28, Wednesday. 12/5.
Tests.
There are two in class tests and a take home final.
Test 1 on Wednesday, 10/31. It covers Chapter 3.
Test 2 on Friday, 11/30. It covers Chapters 4 and 5.

Take home final problems, due Monday, 12/17.

Exercises in Chapter 5: 1.13, 1.15, 1.16, 2.14, 2.16.
Exercises in Chapter 6: 2.4, 2.6, 4.1, 10.12, 10.13.
Find the 3rd double covering of S^1 x P.
Exercises in Chapter 3: 4.8. On the other hand, show that there are no coverings from K to T.
Which compact surfaces (without boundary) admit a double covering to the Klein bottle K? Show that K has a double covering by K itself (hint: K=P#P).
Exercises in Chapter 4: 1.9, 1.10.

Quizzes
There are two in class quizzes.

Quiz 1 on Friday, 10/26. It covers Chapters 1 and 2.
Chapter 1:
Definitions 1.4, 2.1, 2.3, 4.1, 5.3, 6.1, 6.2, 7.1
Lemmas 2.3, 2.4
Propositions 4.1, 4.2, 4.3, 4.4, 6.7, 6.9, 6.10, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6
Theorems 4.5, 5.2, 5.3, 6.5
Corollary 4.6
Chapter 2:
Definitions 1.1, 1.2, 1.3,2.1, 3.1, 4.1, 4.3, 4.4, boundary connected sum for handlebodies
Lemmas 2.1, 3.1, 3.2, 3.3, 3.4, 3.6, 5.2, 6.1, 6.3, 6.4
Propositions 3.5, 3.8
Theorems 3.7, 6.2, 6.5, 7.1

Quiz 2 on Monday, 11/19. It covers Chapters 3 and 4.
Chapter 3:
Definition of the fundamental group
Propositions 2.1, 8.1, 8.2, 8.3, 8.4, 9.3, 10.2
Theorems 3.1, 3.2, 3.3, 4.1, 4.2, 9.1, 9.4, 9.5
Definitions 4.1, 4.2, 8.1, 8.2
Corollaries 9.2
Lemmas 9.10
Chapter 4:
Definitions 1.1, 2.2, 3.1, 3.3, 3.4, 4.1, 4.2
Theorems 1.2, 1.3, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, 4.2