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.

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