The Space of Cellular Maps - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:22:01Z http://mathoverflow.net/feeds/question/45584 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45584/the-space-of-cellular-maps The Space of Cellular Maps Jeff Strom 2010-11-10T18:44:56Z 2010-11-11T14:17:36Z <p>Let $X$ and $Y$ be CW complexes.</p> <p>Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular Approximation Theorem tells us that the inclusion $\mathrm{CW}(X,Y)\to \mathrm{map}(X,Y)$ induces an isomorphism on $\pi_0$ (in both the pointed and unpointed contexts).</p> <p>My question: is the inclusion a weak equivalence? (Feel free to use any reasonable topology.)</p> <p>EDIT: The answer is NO. I'm leaving this here since some interesting ideas are percolating in the comments.</p>