Manifolds admitting CW-structure with single n-cell - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:01:20Z http://mathoverflow.net/feeds/question/120799 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120799/manifolds-admitting-cw-structure-with-single-n-cell Manifolds admitting CW-structure with single n-cell Chris Gerig 2013-02-04T20:31:18Z 2013-02-08T20:21:46Z <p>Let $M$ be a topological $n$-manifold, closed and connected (not necessarily oriented):<br> <strong>When does $M$</strong> <em>not</em> <strong>admit (up to homotopy-type) a CW-structure with a single $n$-cell?</strong></p> <p>By classification of surfaces we assume $n>2$. By existence of smooth structures we assume $n>3$. In particular, if $M$ is smoothable then Morse theory provides us the desired structure.</p> <p>[]: To put this question into context, we have various ways of showing that $H_{n-1}(M)$ has either $0$ or $\mathbb{Z}_2$ as its torsion subgroup depending on orientability. One way, when $M$ is such a CW-complex, is to quickly look at the chain-complex differential $d:C_n(M)\cong\mathbb{Z}\to C_{n-1}(M)$ and note that $H_n(M)\cong\mathbb{Z}$ for $M$ orientable and $H_n(M;\mathbb{Z}_2)\cong\mathbb{Z}_2$ otherwise. So I would like to see how large of a class of manifolds this argument holds for.</p> <p>[[Addendum]]: After chatting with Allen Hatcher and Rob Kirby, who reaffirm the comments below, here are their resulting thoughts:<br> <strong>1)</strong> We should be careful with the Kirby theorem of $M$ being homotopy-equivalent to a finite complex, because this complex is obtained by first embedding $M$ into $\mathbb{R}^N$ and then wiggling the boundary of a tubular neighborhood ($M\times D^{N-n}$) of $M$ to be PL, and so the resulting complex could have $i$-cells with $i>n$.<br> <strong>2)</strong> When $\dim M\ne 4$ there is a handlebody-decomposition, and this can be arranged to have a single 0-handle (canceling the other 0-handles with available 1-handles -- we can do this because there are no smoothing obstructions in a neighborhood of the 3-skeleton). Taking the dual handlebody, we have a decomposition with a single n-handle. Passing from the handlebody-decomposition to the CW-decomposition (shrinking everything to their cores), we obtain the desired CW-complex with a single n-cell.<br> <strong>3)</strong> When $\dim M=4$ then a handlebody-decomposition exists if and only if $M$ is smoothable. So when $M$ is smoothable we can apply the argument in (2).<br> <strong>4)</strong> But even when $M$ is not smooth we get some positive results, in particular for the $E_8$ manifold. We build $E_8$ using Kirby calculus on an 8-link diagram, giving a decomposition of $E_8$ into a 0-handle plus eight 2-handles plus a contractible piece (without the contractible piece we get a space with boundary being a homology 3-sphere, namely the Poincare-sphere $S^3/G$ with $G=$ binary icosahedral group). In particular, flipping this structure over we see that $E_8$ is homotopy-equivalent to a CW-complex with a single 4-cell. <em>Furthermore, Lennart Meier's remark gets us all other simply-connected 4-manifolds.</em></p> <p><strong>We are thus left with the scenario that $M$ (up to homotopy) is a closed connected non-simply-connected non-smoothable 4-manifold.</strong> (which the comments below assert)</p>