Welcome to MathOverflow

Question

5

Let X be CW complex. I'm trying to prove (using Zorn's lemma) that there is maximal contractible subcomplex. Problem is that I'm not able to show that increasing union of contractible subcomplexes has to be contractible itself.

flag	asked Oct 4 2011 at 17:08 Mate 85●4

Neil Strickland · Accepted Answer · 2011-10-04 17:51:52Z UTC

13

By various standard lemmas, a CW complex $X$ is contractible if and only if every map $u:S^{n-1}\to X$ (for any $n>0$) can be extended over $B^n$. In this context $u(S^{n-1})$ is compact and therefore (by another standard lemma) contained in some subcomplex with only finitely many cells. If $X$ is the union of some totally ordered family of subcomplexes $X_\alpha$, it follows that $u(S^{n-1})\subseteq X_\alpha$ for some $\alpha$. This is enough to prove what you want.

answered Oct 4 2011 at 17:51

Neil Strickland
13.3k●1●28●66

1

Calling Whitehead's theorem a standard lemma is a slight understating :) – Mariano Suárez-Alvarez Oct 4 2011 at 18:13

Saying "Whitehead's theorem" is almost as misleading (I believe Henry had proved at least two reasonably well-known results). – Igor Rivin Oct 4 2011 at 20:42

So for short: $\pi_n$ commutes with increasing unions in consideration and detect contractibility by Whitehead. – Martin Brandenburg Oct 4 2011 at 21:54

Increasing union of contractible CW complexes

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.