Exercise 23, § 1

Let $X$ and $Y$ be topological spaces. One says that a homotopy $\sigma : X\times$ $\mathbf{I}\rightarrow Y$ is stationary if there exists a real number $\delta >0$ such that $\sigma (x, t) =\sigma (x,0)$ for all $x\in X$ and all $t\in [0, \delta ]$. Let A be a closed subspace of X; one denotes by $j$ the canonical injection of A into X. One says that the couple $(X,A)$ possesses the property of extension of stationary homotopies if, for every topological space Y, every continuous mapping $f: X\rightarrow Y$ and every stationary homotopy $\sigma : A\times \mathbf{I}\rightarrow Y$ whose origin is equal to $f|A$, there exists a homotopy $\widetilde{\sigma}: X\times \mathbf{I}\rightarrow Y$ of origin $f$ which extends $\sigma$.

a) Prove that the following properties are equivalent:

(i) The couple $(X,A)$ has the property of extension of stationary homotopies ;

(ii) The couples $(X,A)$ and (Cyl($j$)$, \alpha_j(A\times \{0\})$ are homotopic ;

(iii) There exists a continuous mapping $\varphi : X\rightarrow \mathbf{I}$ such that $\varphi (a) = 0$ for all $a\in A$ and a homotopy $\sigma : X\times \mathbf{I}\rightarrow X$ such that $\sigma (x,0) =x$ for $x\in X$, $\sigma (a, t) =a$ for $(a, t)\in A\times \mathbf{I}$, and $\sigma (x,1)\in A$ if $\varphi (x)<1$.

(iv) There exists a continuous mapping $\varphi : X\rightarrow \mathbf{I}$, a subspace V of X and a homotopy $\sigma : V\times \mathbf{I}\rightarrow X$ such that $\varphi (x) = 1$ if $x\notin V,\varphi (a) = 0$ for $a\in A,\sigma (x,0) =x$ for $x\in V,\sigma (a, t) =a$ for $(a, t)\in A\times \mathbf{I},\sigma (x,1)\in A$ if $\varphi (x)<1$.

b) Suppose that the couple $(X,A)$ possesses the property of extension of stationary homotopies and that there exists a continuous function $u: X\rightarrow \mathbf{I}$ such that $\overset{-1}{u}(0) = A$. Prove that the the couple $(X,A)$ possesses the property of extension of homotopies.