Please turn in the Seifert-van Kampen problems from last week (Section 1.2 # 3, 4, 8, 19).

In addition, do

Section 1.3 # 1, 3, and 4a (just the \(S^2 \cup \mathrm{diameter}\) part).

and

A. Think of \(S^1 = \{z\in\mathbb{C}: |z| = 1 \} \). Let \(p:S^1 \to S^1\) be given by \(p(z) = z^n\). For any \(x \in S^1\), give an explicit evenly covered neighborhood \(U\) of \(x\), i.e. \( p^{-1}(U) = \sqcup V_a \) where \( p|_{V_a} :V_a \to U\) is a homeomorphism.

Think about the following problems, but do turn them in:
Section 1.2 # 7, 9, 10, 11, 20.

This homework assignment is due Thursday, January 31, in class.

Hatcher Section 1.1, p.38 # 1, 5, 8, 11, 16. Section 1.2 # 3, 4, 8, 19.

Then

A. Let \( f: (X,x) \to (Y,y) \) be a continuous map.  Show that the induced map \( f_\ast: \pi_1(X, x) \to \pi_1(Y,y) \) is a group homomorphism.  (After some comments made in class, it suffices to prove that for loops \(\phi\) and \(\psi\), we have the equality of maps/loops \( f(\phi\cdot\psi) = f(\phi)\cdot f(\psi) \)

B. Let \( (X,x) \xrightarrow{f} (Y,y) \xrightarrow{g} (Z,z) \) be maps.  Show that the induced maps \( \pi_1(X,x) \xrightarrow{f_\ast} \pi_1(Y,y) \xrightarrow{g_\ast} \pi_1(Z,z) \) satisfy \(g_\ast \circ f_\ast = (g\circ f)_\ast \)

C. For any pointed space \((X,x)\), show that \((\mathrm{id}_X)_\ast = \mathrm{id}_{\pi_1(X,x)} \)

D. Classify the letters A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z up to homotopy type.  Little justification is needed.

Due Thursday, 24 January, in class.

This assignment consists of problems from Chapter 0 and problem A, below.

p 18: #1, 2 (give a formula for the deformation retraction), 3, 9, 10, 14

A. Let \( X, Y \) be CW complexes with \( A \) a subcomplex of \( Y \). Given a homotopy \(F: A \times [0,1] \to X \) between gluing maps \( f,g: A \to X \), prove that \( X\sqcup_F \left(Y \times [0,1] \right) \) deformation retracts onto \( X \sqcup_f Y \).