CW structure on spaces obtained by attaching cells wildly - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:17:08Z http://mathoverflow.net/feeds/question/23415 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23415/cw-structure-on-spaces-obtained-by-attaching-cells-wildly CW structure on spaces obtained by attaching cells wildly Florian 2010-05-04T11:23:37Z 2010-05-04T18:29:20Z <p>Is there necessarily a CW structure on a space build out of cells without demanding them to be attached in "right" order?</p> <p>More precisely, let $X$ be a topological space such that the map $\emptyset\to X$ factorizes as a transfinite composition of inclusions $$\emptyset\to\ldots\to X_\beta\to X_{\beta+1}\to\ldots\to X$$ where every map $X_\beta\to X_{\beta+1}$ is a pushout $$\begin{array}{rcl} S^{n-1} &amp;\to&amp; X_\beta\\ \downarrow &amp;&amp; \downarrow\\ D^{n} &amp;\to&amp; X_{\beta+1} \end{array}$$ for some $n\geq 0$. Is $X$ necessarily a CW complex?</p> http://mathoverflow.net/questions/23415/cw-structure-on-spaces-obtained-by-attaching-cells-wildly/23425#23425 Answer by Tyler Lawson for CW structure on spaces obtained by attaching cells wildly Tyler Lawson 2010-05-04T13:48:28Z 2010-05-04T15:53:36Z <p>Fix an irrational number $\alpha$.</p> <p>Let <code>$X_2 = [0,1]$</code> (built from attaching a 1-cell to two 0-cells) and, for each larger $n$, let <code>$X_n$</code> be built from <code>$X_{n-1}$</code> with a new a 1-cell by attaching the ends to $0$ and the fractional part of $n \alpha$. Take $X$ to be the union.</p> <p>There are no embeddings from an open disc $D^n$ into $X$ for n greater than 1. If $X$ admitted a CW-complex structure, this would force it to be 1-dimensional. However, $X$ cannot be homeomorphic to a 1-dimensional CW-complex, for example because the set of points which have no neighborhood homeomorphic to $\mathbb{R}$ do not form a discrete subspace.</p> http://mathoverflow.net/questions/23415/cw-structure-on-spaces-obtained-by-attaching-cells-wildly/23466#23466 Answer by Dan Ramras for CW structure on spaces obtained by attaching cells wildly Dan Ramras 2010-05-04T18:29:20Z 2010-05-04T18:29:20Z <p>The space you get through such a process of cell attachments is at least homotopy equivalent to a CW complex, although maybe not homeomorphic (as Tyler's example shows).</p> <p>This comes up in Milnor's <a href="http://books.google.com/books?id=A9QZZ3S_QxwC&amp;printsec=frontcover&amp;dq=milnor+morse+theory&amp;source=bl&amp;ots=f0k2X8O2Gq&amp;sig=IV-kP6TaMqLXCsf88DPF2YmkaLg&amp;hl=en&amp;ei=7mbgS-38CovcsgOUg-2CBQ&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CA8Q6AEwAA#v=onepage&amp;q&amp;f=false" rel="nofollow">Morse Theory</a> (see pp. 21-24), where you build a CW complex inductively by attaching cells according to the critical points of a Morse function on a manifold. There's no reason that the indices have to be increasing, and this causes one to attach cells "in the wrong order," i.e. a cell of dimension n may be attached to existing cells of dimension n or greater. In his book on Morse Theory, Milnor shows that up to homotopy the resulting space is a CW complex. The basic point is that each attaching map can be deformed to the appropriate skeleton, and attaching a cell along two homotopic maps produces two homotopy equivalent spaces (this is a bit tricky, and Milnor attributes it to Hilton). Finally, an argument with homotopy colimits allows one to treat the case of countably many cell attachments.</p> <p>I suppose that this applies to Tyler's example, and shows that the space he builds is homotopy equivalent to an infinite wedge of circles.</p>