Homework 1: Due Wednesday, September 4

1) Show that path homotopy is an equivalence relation.

2) Compute the first chain group of the directed graph here. Fully justify your reasoning.

3) Compute H_1 of the directed graph in Problem #2. You may use a calculator for matrix operations. You do not need to show your matrix work.

Extra credit: make a Frayer model of any concept we have learned in class so far.

Homework 2: Due Monday, September 16

1) Complete the proof showing that \(H_n\) is a functor by showing that \(H_n(f)\) is a homomorphism, \(H_n(1_X) = 1_{H_N(X)}\), and that if \(g \colon Y \to Y'\) is a continuous map, then \(H_n(gf) = H_n(g)H_n(f)\) (see Rotman Theorem 1.25).

2) Suppose that \(A \subset X\). Prove that if the inclusion map induces an isomorphism \(H_n(A) \cong H_n(X)\) then \(H_n(X,A) = 0\) for all \(n \geq 0\).

3) Hatcher Section 2.1 #27 (page 133).

Extra credit: make a Frayer model of any concept we have learned in class since Monday, August 26.

Homework 3: Due Monday, September 23

1) Hatcher Section 2.2 #9ab (Hint for part a: Use Hatcher Example 0.8)

2) Hatcher Section 2.2 #31

3) Hatcher Section 2.2 #36 (Hint: you may use the sentence starting with "From the Splitting Lemma" on pg. 148 without proof.)

Extra credit: make a Frayer model of any concept we have learned in class since the last homework was due.

Homework 4: Due Monday, October 7

1) Hatcher Section 2.2 #40 only the first sentence. The second sentence is worth 4 points of extra credit.

2) Read and fully understand the proof of the Universal Coefficient Theorem for cohomology. You may use any reference you like. If you have questions, write them down and turn them in. If you don't, just write down which reference you used. As long as you write something down, you will get credit (honor system).

3) Hatcher Section 3.1 #1, just the part for fixed G. Note that the morphisms Ext(f, G) are the induced maps on cohomology. The part for fixed H is worth 4 points of extra credit.

Extra credit: make a Frayer model of any concept we have learned in class since the last homework was due.

Homework 5: Due Monday, October 21

1) We saw in class that the functor \(-\otimes G\) is right-exact. Give an example showing that it is not left-exact.

2) Compute \(H_1(T^2; \mathbb{Z}/4\mathbb{Z})\) using the Universal Coefficient Theorem. For two points extra credit, compute \(H_2(T^2; \mathbb{Z}/4\mathbb{Z})\).

3) Hatcher Section 3.3 #25. You may use without proof the fact that the groups \(H_{k-1}(M;\mathbb{Z})\) are finitely generated. (Discussion of this fact is on Hatcher p. 527)

Extra credit: make a Frayer model of any concept we have learned in class since the last homework was due.

Extra extra credit: are CW-complexes manifolds? Are simplicial complexes manifolds? If yes, prove it. If no, give a counterexample. (2 pts each)

Homework 6: Due Monday, November 4

1) Go to the DONUT database and read a paper on an application of persistent homology. Write down which paper you chose along with a 3-4 sentence summary.

2) Draw the persistence diagram for the filtration on the top of page 66 in CTDA. Ignore the red/blue coloring.

3) Prove Proposition 3.9 in CTDA (page 78).

Extra credit: make a Frayer model of any concept we have learned in class since the last homework was due.

Homework 7: Due Monday, November 11

If you have a laptop, install Python and Jupyter notebook. If you are unfamiliar with Python, I think that installing Anaconda is the easiest way to do this. If you do not have a laptop, get a Google colab account. To get your points, show me on Monday or earlier that you are ready to go.

Homework 8: Due Monday, November 25

Spend two hours working in Python. Choose your own adventure, depending on your level. Write an email with a few sentences about what you did.

Beginner: Work problems at Project Euler. They aren't related to topology, but will help you get comfortable with coding.

Intermediate: Finish writing the function to compute the ECT (feel free to look at the solutions on Github if you get stuck).

Advanced: Write a function to compute a Cech filtration on point cloud data and its persistence diagram (this might take more than two hours, so just work on it for two hours).

Extra credit homework: Due Friday, December 13 via email

Do three problems from Dr. Vassilev's website that were not assigned during the semester.