Homotopy equivalence of pairs

Question

Let $ I = [0,1] \subseteq \mathbb{R}$. I want to prove that the pair $(I^n,\partial I^n)$ is homotopy equivalent to $(\mathbb{R}^n,\mathbb{R}^n\setminus\{ 0,0,...,0 \})$, but I have a problem with the definition itself.

Can someone please state the definition of homotopy equivalence of pairs? Apparently, I couldn't find it anywhere.

Answer 1

Let $A\subset X$ and $B\subset Y$ be CW-pairs (or any pairs of topological spaces such that the inclusions are cofibrations, see at the end).

The map $f:X\to Y$ is a homotopy equivalence between the pairs $(X,A)$ and $(Y,B)$ if there is a map $g:Y\to X$ such that:

1. $f(A)\subset B$
2. $g(B)\subset A$
3. There is a homotopy $H:X\times[0,1]\to X$ such that
   • $H(x,0)=x$ for all $x\in X$
   • $H(x,1)=g\circ f(x)$ for all $x\in X$
   • $H(x,t)\in A$ for all $x\in A, t\in [0,1]$
4. There is a homotopy $K:Y\times[0,1]\to Y$ such that
   • $K(y,0)=y$ for all $y\in Y$
   • $K(y,1)=f\circ g(y)$ for all $y\in Y$
   • $K(y,t)\in B$ for all $y\in B, t\in [0,1]$

You can find a broader definition, where the inclusion maps $i:A\to X$ and $j:B\to Y$ are not inclusions, but only cofibrations, here as part of Proposition 1.15.

Note that not all inclusions are cofibrations, but it is the case for CW-pairs.

Answer 2

Generally a map of pairs $f \colon (X,A) \to (Y,B)$ is a map $f \colon X \to Y$ with $f(A) \subseteq B$. (This gives you a category of pairs.) Two maps $f_0, f_1$ of pairs are homotopic if they are linked by a homotopy $f_t$ with $f_t(A) \subseteq B$ for each $t$.

So just like the definition of homotopy equivalence of spaces, two pairs $(X,A)$ and $(Y,B)$ are homotopy equivalent if there are maps $f \colon (X,A) \to (Y,B)$ and $g \colon (Y,B) \to (X,A)$ such that the composites $g \circ f \colon (X,A) \to (X,A)$ and $f \circ g \colon (Y,B) \to (Y,B)$ are homotopic (as maps of pairs) to the identity maps.