0
$\begingroup$

Does any CW complex paired with a contractible space satisfy the homotopy extension property? I am asking this question because I want to answer the question given here and I was told that it is just proposition 0.17 in hatcher, but the statement said that the pair must satisfy HEP

How to use the homology version of Whitehead theorem to prove this question?

$\endgroup$
1
  • 4
    $\begingroup$ A pair $(X,A)$ where $A$ is a sub-CW-complex of $X$ (more generally, if $(X,A)$ is a relative CW-complex) always has the HEP $\endgroup$ Dec 12 '19 at 7:25
1
$\begingroup$

No, take a circle relative the complement of one point. You can embed the circle into $\mathbb{R}$ at time 0, and then homotope the complement into an open line segment. This cannot be extended to a homotopy of the circle since you cannot even extend the function at time 1 to the whole circle.

One requirement of the HEP is to be closed.

$\endgroup$

Your Answer

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