2
$\begingroup$

In this proof of the Lemma 2.34 of Algebraic Topology I don't understand 2 things. enter image description here

$(1)$ Hatcher defines $Y_i:=T\cup (X\times[i,\infty])$, then I think that it's obvius the fact that $Y_i$ deformation retracts onto $Y_{i+1}$, since $[i,\infty]$ deformation retracts onto $[i+1,\infty]$.

$(2)$ With the retractions of $Y_i$ in $Y_{i+1}$ how I can obtain a deformation retract of $X\times [0,\infty)$ onto $T$?

$\endgroup$
2
  • 1
    $\begingroup$ I've edited your title a bit, since it doesn't seem like you're actually claiming that something is wrong with Hatcher's proof; you're asking for clarification on two points. Feel free to rollback the edit if you see fit. $\endgroup$ Feb 5 '17 at 21:25
  • $\begingroup$ Thanks :) Now...Can you help me with the problem? I'm desperate! xD $\endgroup$ Feb 5 '17 at 21:27
0
$\begingroup$

Concerning (1), we cannot just expect a deformation retraction of $X\times \lbrack i,\infty )$ to $X\times \lbrack i+1,\infty )$ to extend to a deformation retraction of the larger space $T\cup X\times\lbrack i,\infty )$. For instance, $S^1$ is the union of its northern and southern (closed) hemispheres. Both hemispheres deformation retract to a point, but there is no deformation retraction of $T$ which, e.g., is the identity on the southern hemisphere while deforming the northern hemisphere to a point.

As for (2), the point is to compress each $Y_i$ in increasingly short time intervals. The resulting homotopy is well-defined and continuous, essentially due to the "pasting lemma" from elementary point-set topology; the intervals $\lbrack 1 - 1/2^i, 1 - 1/2^{i+1}\rbrack$ form a closed cover of the unit interval and the partial homotopies agree on the intersections $\lbrace 1/2^i\rbrace$ by construction. This allows us to eventually compress every $Y_i$ (this is no weirder than Achilles catching up to the turtle).

$\endgroup$

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.