Exercise 2, § 3

Let $(X, a)$ and $(Y, b)$ be pointed topological spaces; suppose that the subspace $\{a\}$ of X is closed and that the pair $(X,\{a\})$ possesses the homotopy extension property.

a) Let $f: X\rightarrow Y$ be a continuous mapping and let $c$ be a path in Y. Prove that there exists a homotopy $\sigma : X\times \mathbf{I}\rightarrow Y$ such that $\sigma (a, t) =c(t)$ for all $t\in \mathbf{I}$ and $\sigma (x,0) =f(x)$ for all $x\in X$.

b) Prove that there exists a unique right action of the group $\pi_1(Y, b)$ on the set $[(X, a); (Y, b)]$ such that $[f]\cdot [c]$ is the class of the mapping $x\mapsto \sigma (x,1)$, for every pointed continuous mapping $f: (X, a)\rightarrow (Y, b)$, every loop in Y at $b$, and every homotopy $\sigma$ as above.

c) Suppose that Y is arcwise connected. Prove that every continuous mapping of X into Y is homotopic to a pointed continuous mapping of $(X, a)$ into $(Y, b)$.

d) Let $f$ and $g$ be pointed continuous mappings of $(X, a)$ into $(Y, b)$. In order that $f$ and $g$ be the origin and the term of a pointed homotopy, it is necessary and sufficient that there exists a loop class $\gamma \in \pi_1(Y, b)$ such that $[f]\cdot \gamma = [g]$.