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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 I need to use 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. Indeed we have then $Z_{n+m-1}=\mathbb S^{n}\vee\mathbb S^{m}$ and this implies that the function is of type $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$. 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).

We want some function $\mathbb S^{n+m-1}\to Z_{p-1}$ and we would like to do the same thing as in the construction of the map $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$, so look a the pushout $\mathbb S^{n-1}\leftarrow \mathbb S^{n-1}\times\mathbb S^{m-1}\to \mathbb S^{m-1}$ 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.

I would like to know if some work about this had already be done, I'd be very glad that you have some synthetic reference about this problem.

Thank you very much

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.

I would like to know if some work about this had already be done (in a synthetic "up-to-homotopy" way), I'd be very glad that you have some reference about this problem.

Thank you very much

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 I need to use 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 have the Whitehead product gives such a map when the CW-complexes $X$ and $Y$ are the spheres $\mathbb S^{n}$ and $\mathbb S^{m}$, indeed we have $Z_{n+m-1}=\mathbb S^{n}\vee\mathbb S^{m}$ and so we want a function $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$ (which is the Whitehead map). 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).

We want some function $\mathbb S^{n+m-1}\to Z_{p-1}$ and we would like to do the same thing as in the construction of the map $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$, so look a the pushout $\mathbb S^{n-1}\leftarrow \mathbb S^{n-1}\times\mathbb S^{m-1}\to \mathbb S^{m-1}$ 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.

I would like to know if some work about this had already be done, I'd be very glad that you have some synthetic reference about this problem.

Thank you very much

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 I need to use 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. Indeed we have then $Z_{n+m-1}=\mathbb S^{n}\vee\mathbb S^{m}$ and this implies that the function is of type $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$. 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).

We want some function $\mathbb S^{n+m-1}\to Z_{p-1}$ and we would like to do the same thing as in the construction of the map $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$, so look a the pushout $\mathbb S^{n-1}\leftarrow \mathbb S^{n-1}\times\mathbb S^{m-1}\to \mathbb S^{m-1}$ 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.

I would like to know if some work about this had already be done, I'd be very glad that you have some synthetic reference about this problem.

Thank you very much