CW structure on spaces obtained by attaching cells wildly - MathOverflow

CW structure on spaces obtained by attaching cells wildly

2010-05-04T11:23:37Z

Is there necessarily a CW structure on a space build out of cells without demanding them to be attached in "right" order?

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 $$ \begin{array}{rcl} S^{n-1} &\to& X_\beta\\ \downarrow && \downarrow\\ D^{n} &\to& X_{\beta+1} \end{array} $$ for some $n\geq 0$. Is $X$ necessarily a CW complex?

Answer by Tyler Lawson for CW structure on spaces obtained by attaching cells wildly

2010-05-04T13:48:28Z

Fix an irrational number $\alpha$.

Let $X_2 = [0,1]$ (built from attaching a 1-cell to two 0-cells) and, for each larger $n$, let $X_n$ be built from $X_{n-1}$ 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.

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.

Answer by Dan Ramras for CW structure on spaces obtained by attaching cells wildly

2010-05-04T18:29:20Z

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).

This comes up in Milnor's Morse Theory (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.

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.