0
$\begingroup$

I am beginning to learn algebraic topology from Hatcher's text and I had a question about the definition of homotopy relative to a subspace. The passage in the text goes as follows:

A homotopy $f_{t}: X \rightarrow X$ that gives a deformation retraction of $X$ onto a subspace $A$ has the property that $f_{t}|_{A} = id$ for all $t$. In general, a homotopy $f_{t}: X \rightarrow Y$ whose restriction to a subspace $A \subset X$ is independent of $t$ is called a homotopy relative to $A$.

I do not understand exactly what is meant by "whose restriction to a subspace $A \subset X$ is independent of $t$." From what I gathered reading the wikipedia article, I am guessing this means that $f_{t}(a) = id(a) = a$ for all $a \in A$ and $t \in [0,1]$?

$\endgroup$
0
$\begingroup$

You are mixing the concept of a deformation retraction onto a subspace with the more general notion of homotopy relative to a subspace.

In the latter case, for each $a\in A$, you want the image of $a$ under $f_t$ to be the same for any $t$; but $f_t$ need not be the identity on the subspace. In fact, $f_t$ being the identity does not even make sense in general as $f_t$ is a map between two (possibly different) topological spaces $X$ and $Y$.

In the specific case of a deformation retraction, where $f_t$ is a map from $X\to X$ and restricts to the identity on the subspace $A\subset X$, we have $f_t(a)=id(a)=a$ as you have written.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.