# Homotopy extension property and relative homotopy

I'm having troubles with this problem:

Let $$A$$ be a closed subset of $$X$$ and $$B$$ a subset of $$Y$$. Assume that $$(X,A)$$ has the homotopy extension property with respect to $$B$$ and that $$(X\times I,X\times \partial I\cup A\times I)$$ has the homotopy extension property with respect to $$Y$$. Prove that if $$f:(X,A)\rightarrow (Y,B)$$ is homotopic (as a map of pairs) to a map which sends all of $$X$$ to $$B$$, then it is homotopic relative to $$A$$ to such a map."

Now, my problem is that I can't see how to use the (HEP) with respect to $$B$$ and $$Y$$ to prove this homotopy must be relative to $$A$$.

I'm not looking for solution, hints or any kind of help I will appreciate it.

The reference used is Edwin H. Spanier Algebraic topology.

• Where precisely does Spanier state this? Oct 9, 2018 at 17:24
• P57 problem D.4 Oct 9, 2018 at 19:06

We have a homotopy of pairs $$H : (X,A) \times I = (X \times I, A \times I) \to (Y,B)$$ such that $$H_0 = f$$ and $$H_1(X) \subset B$$. Here, $$H_t(x) = H(x,t)$$.

We shall find a homotopy of pairs $$G : ((X,A) \times I) \times I = (X \times I \times I, A \times I \times I) \to (Y,B)$$ such that

(1) $$G_0 = H$$

(2) $$(G_s)_0 = f$$ for all $$s \in I$$

(3) $$G(X \times \{ 1 \} \times I) \subset B$$

(4) $$H' = G_1$$ is a homotopy such that $$H'_t \mid_A = f \mid_A$$ for all $$t \in I$$.

Here, $$G_s : (X,A) \times I \to (Y,B)$$ is the homotopy defined by $$G_s(x,t) = G(x,t,s)$$.

Then obviously $$H'$$ is the desired homotopy.

By the HEP with respect to $$Y$$, it suffices to find $$\phi : (X \times \partial I \cup A \times I) \times I \to Y$$ such that

(1) $$\phi(x,t,0) = H(x,t)$$ for all $$(x,t) \in X \times \partial I \cup A \times I$$

(2) $$\phi(x,0,s) = f(x)$$ for all $$x \in X, s \in I$$

(3) $$\phi(X \times \{ 1 \} \times I \cup A \times I \times I) \subset B$$

(4) $$\phi(a,t,1) = f(a)$$ for all $$(a,t) \in A \times I$$.

To get $$\phi$$, you have to proceed in three steps:

(a) Find a homotopy $$\phi^A : A \times I \times I \to B$$ such that $$\phi^A(a,t,0) = H(a,t)$$, $$\phi^A(a,t,1) = f(a)$$ for all $$(a,t) \in A \times I$$ and $$\phi^A(a,0,s) = f(a)$$ for all $$a \in A$$, $$s \in I$$.

(b) Use the HEP with respect to $$B$$ to find a homotopy $$\phi^1 : X \times I \to B$$ such that $$\phi^1(x,0) = H_1(x)$$ for all $$x \in X$$ and $$\phi^1(a,s) = \phi^A(a,1,s)$$ for all $$a \in A$$, $$s \in I$$.

(c) Use $$\phi^A$$ and $$\phi^1$$ to define $$\phi$$ and check continuity.