Mathematics 611, Fall 2019

Algebraic Topology I

Mondays, Wednesdays 4:40-5:55pm, Physics 119 (changed from 227)
Instructor: Lenny Ng
Office hours: Mondays 2-3, Tuesdays 10:30-11:30 (changed from 11-12)

Homework assignments

Unless otherwise specified, all problems are from Hatcher.

Remember that you're encouraged to work on the homework problems with your classmates, but you must write up your problem sets on your own. I am aware that some problems from Hatcher are discussed in various online places; it is not acceptable to seek out solutions there.

Problem set Due date Problems Solutions (outline)

1
September 4
Chapter 0: 4, 5, 6, 9, 13
Section 1.1: 6, 7, 16 (a,b,c,f), 20.
(Remark: for 1.1 #20, I don't think you really need to use Lemma 1.19.) HW 1
2
September 11
Section 1.2: 1, 4, 6, 8, 9, 11.
HW 2

3
September 18
Section 1.2: 12, 14, 22.
Extra problem: Use the Wirtinger presentation to calculate the fundamental group of the complement of the trefoil knot. (See here for a picture of the trefoil.) If all goes well, we'll see in class another way to calculate this fundamental group, resulting in the group with generators x, y and relation x²=y³. Prove that your answer algebraically agrees with this group by finding an explicit isomorphism between the two groups. HW 3

4
September 25
Section 1.3: 4, 5, 7, 8.
For #8, there's a typo: "Exercise 10 in Chapter 0" should be "Exercise 11 in Chapter 0." If you do use Exercise 11, please provide a proof. HW 4

5
October 2
Section 1.3: 10, 11, 14.
To clarify: in #14, "Find all the connected covering spaces" is in the same sense as #10. HW 5

6
October 16
Section 1.3: 16, 18, 19, 20.
For #19, please disregard the final sentence. (You can certainly think about it if you like, but I think it could use some clarification and I've decided it isn't worth it.) HW 6

7
October 23
Section 2.1: 2, 4, 5, 7, 8. HW 7

8
November 6
Section 2.1: 14, 15, 16, 17, 26.
For this problem set, please note that you can assume that simplicial and singular homology are isomorphic.
For #26, you may want to follow the idea of Example 1.25 to prove that relative H₁ of X/A is uncountable. (Don't just cite Example 1.25, although you can adapt its argument -- your argument should involve H₁ rather than the fundamental group.)
HW 8

9
November 13
Section 2.1: 20, 22, 27.
(For #20, see pp. 8-9 for the definitions of suspension and cone. Also, after you've solved #20, please think about #21 but you don't need to turn it in.)
For these problems, you can use anything in section 2.1. (In particular, you can use the Five Lemma on p. 129; I'll get to this soon in class, but feel free to use it in the meantime.)
HW 9

10
November 20
HW 10
HW 10

11
December 4
Note: please do these problems but don't turn them in. I've posted solutions (to the right).
Section 2.2: 2, 3, 11, 12, 14, 20, 22, 23.
HW 11

Take-home final exam is due on Friday December 13 at noon.

Course information

Course syllabus

Hatcher Algebraic Topology: link to online version of the textbook

Course lecture notes part 1
Course lecture notes part 2

Note!: Don't treat these notes as a definitive reference; they certainly contain mistakes and may not correspond precisely to what we actually cover in class.

Problem set	Due date	Problems	Solutions (outline)
1	September 4	Chapter 0: 4, 5, 6, 9, 13 Section 1.1: 6, 7, 16 (a,b,c,f), 20. (Remark: for 1.1 #20, I don't think you really need to use Lemma 1.19.)	HW 1
2	September 11	Section 1.2: 1, 4, 6, 8, 9, 11.	HW 2
3	September 18	Section 1.2: 12, 14, 22. Extra problem: Use the Wirtinger presentation to calculate the fundamental group of the complement of the trefoil knot. (See here for a picture of the trefoil.) If all goes well, we'll see in class another way to calculate this fundamental group, resulting in the group with generators x, y and relation x²=y³. Prove that your answer algebraically agrees with this group by finding an explicit isomorphism between the two groups.	HW 3
4	September 25	Section 1.3: 4, 5, 7, 8. For #8, there's a typo: "Exercise 10 in Chapter 0" should be "Exercise 11 in Chapter 0." If you do use Exercise 11, please provide a proof.	HW 4
5	October 2	Section 1.3: 10, 11, 14. To clarify: in #14, "Find all the connected covering spaces" is in the same sense as #10.	HW 5
6	October 16	Section 1.3: 16, 18, 19, 20. For #19, please disregard the final sentence. (You can certainly think about it if you like, but I think it could use some clarification and I've decided it isn't worth it.)	HW 6
7	October 23	Section 2.1: 2, 4, 5, 7, 8.	HW 7
8	November 6	Section 2.1: 14, 15, 16, 17, 26. For this problem set, please note that you can assume that simplicial and singular homology are isomorphic. For #26, you may want to follow the idea of Example 1.25 to prove that relative H₁ of X/A is uncountable. (Don't just cite Example 1.25, although you can adapt its argument -- your argument should involve H₁ rather than the fundamental group.)	HW 8
9	November 13	Section 2.1: 20, 22, 27. (For #20, see pp. 8-9 for the definitions of suspension and cone. Also, after you've solved #20, please think about #21 but you don't need to turn it in.) For these problems, you can use anything in section 2.1. (In particular, you can use the Five Lemma on p. 129; I'll get to this soon in class, but feel free to use it in the meantime.)	HW 9
10	November 20	HW 10	HW 10
11	December 4	Note: please do these problems but don't turn them in. I've posted solutions (to the right). Section 2.2: 2, 3, 11, 12, 14, 20, 22, 23.	HW 11
		Take-home final exam is due on Friday December 13 at noon.