3
$\begingroup$

This is part of a problem from Hatcher's book. Let $Y$ be the subspace of $\mathbb{R}^{2}$ shown in the picture. Let $Z$ be the zigzag space of $Y$ indicated by the heavier line. Show that there is a deformation in the weak sense (i.e. a homotopy $f_{t}:Y\rightarrow Y$ such that $f_{0}=id_{Y},f_{1}(Y)\subset Z$, and $f_{t}(Z)\subset Z$ for all $t$).

enter image description here

In an earlier part to the problem involving just one of the triangular pieces, I showed that it deformation retracts to any point in the base line, basically by contracting each of the thin lines down to its foot and then contracting the base line. Now when considering $Y$, does this same idea give a weak deformation retraction onto $Z$? One thing that is not so clear to me is continuity. Also, if the idea works, I don't know how the homotopy can be written down explicitly.

$\endgroup$

Your Answer

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

Browse other questions tagged or ask your own question.