Homotopy and Fundamental Groups

due by Tuesday, Sep 17, 2019

Most of the following problems are from Allen Hatcher’s Algebraic Topology, specifically the online version.

1. Hatcher, Chapter 0, Problem 4.
2. Hatcher, Chapter 0, Problem 5.
3. Hatcher, Chapter 0, Problem 6.
4. Hatcher, Chapter 1, Problem 5.
5. Hatcher, Chapter 1, Problem 7.
6. Let $(X, x_0)$ and $(Y, y_0)$ be based spaces and let $p_x : X \times Y\to X$ and $p_y: (X, Y) \to Y$ be the projection maps. Show that we have an isomorphism $\varphi: \pi_1(X\times Y, (x_0, y_0))\to \pi_1(X, x_0) \times \pi_1(Y, y_0)$ given by $\varphi(\xi) = ((p_x)_*(\xi), (p_y)_*(\xi))$.
7. Show that the Möbius band does not retract onto its boundary circle.