# CW Product via Whitehead map

Product CW-complexes are defined via characteristic maps rather than from attaching maps, so via maps from $\mathbb D^n$ rather than from $\mathbb S^{n-1}$, because we have the propriety that $\mathbb D^n\times\mathbb D^m=\mathbb D^{m+n}$. I want to define products in a synthetic way, only manipulating objetcs up to homotopy (so no discs, anywhere, even during the construction as a provisional step, because there are nothing more than trivial things up to homotopy) so I need to use only the spheres. So basically if $X$ and $Y$ are CW-complexes and $Z$ is the product, to attach one new cell to $Z_{p-1}$ means attaching a product of cells $\mathbb S^{n-1}\to X_{n-1}$ and $\mathbb S^{m-1}\to Y_{m-1}$, where $n+m=p$, and this should give a function $\mathbb S^{n+m-1}\to Z_{p-1}$. We actually already have such a map when the CW-complexes $X$ and $Y$ are the spheres $\mathbb S^{n}$ and $\mathbb S^{m}$, which is the Whitehead map. But note that there I mean the Whitehead product not defined as the attaching map of the product of two cells, because it would be a construction using discs in a implicit way. I take it defined from the homotopy pushout definition of the join of two spheres. Indeed we have then (for a CW product of spheres) $Z_{n+m-1}=\mathbb S^{n}\vee\mathbb S^{m}$ and this implies that the wanted attaching map is of type $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$, where by definition $\mathbb S^{n+m-1} = \mathbb S^{n-1} *\mathbb S^{m-1}$. Now I want this bit to be generalizable to every product of finite CW-complex (it can have just a finite number of cells if nedded).

So instead of a simple map $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$ want some function $\mathbb S^{n+m-1}\to Z_{p-1}$ where obviously $Z_{p-1}$ can be more complex than just a wedge. The definition of this map would look like that of the map $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$, so we should look a the pushout span $\mathbb S^{n-1}\leftarrow \mathbb S^{n-1}\times\mathbb S^{m-1}\to \mathbb S^{m-1}$ but then the attaching maps of the cells are not trivial like they were for products of sphere. This looks very closely like the Whitehead product but I guess there is some non trivial things to consider at this point and I feel like I can't find the good way to tackle the problem.

Thank you very much

• Could you revise your question? It's rather difficult to read. I've looked it over three times now, and I am not certain what you are looking for. – Ryan Budney Jul 12 at 3:13
• Sorry for that, I hope that this is better now. Thank you – elidiot Jul 12 at 10:28

The attaching map of the product of cells is sometimes described as an exterior join construction.

Let $F:D^n\to Cf\subseteq X$ be the characteristic map of an $n$-cell of $X$ with attaching map $f:S^{n-1}\to X_{n-1}$, and let $G:D^m\to Cg\subseteq Y$ be the characteristic map of an $m$-cell of $Y$ with attaching map $g:S^{m-1}\to Y_{m-1}$.

To describe the attaching map of the product of these cells, note that there is a homeomorphism $$S^{n+m-1} \approx D^n\times S^{m-1} \cup S^{n-1}\times D^m \subseteq D^n\times D^m.$$ Under this homeomorphism, the attaching map of the product cell is $$F\times g \cup f\times G: D^n\times S^{m-1} \cup S^{n-1}\times D^m\to Cf\times Y_{m-1} \cup X_{n-1}\times Cg,$$ the cofibre of which is $Cf\times Cg$.

The exterior join construction is perhaps not very well documented in the literature, but here are some references:

Baues, Hans Joachim, Iterierte Join-Konstruktionen, Math. Z. 131, 77-84 (1973). ZBL0244.55016.

Marcum, Howard J., Fibrations over double mapping cylinders, Ill. J. Math. 24, 344-358 (1980). ZBL0459.55009.

Stanley, Donald, On the Lusternik-Schnirelmann category of maps, Can. J. Math. 54, No. 3, 608-633 (2002). ZBL1002.55002.