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Hatcher describes the Whitehead product as the composition of maps $S^{k+l-1} \to S^k \vee S^l \overset{f \vee g}{\to} X$. where the first map is the attaching map of the $(k+l)$-cell of $S^k \times S^l$ with its usual CW structure.

I can't find a description for the canonical attaching map for the CW structure of $S^k\times S^l$. The closest resembling case is that of a surface of genus $g$ where the $2$-cell is attached via the commutator of the generators of the $S^1$ summands in the wedge sum.

What is the canonical attaching map for the $CW$ structure $S^k \times S^l$?

user7090
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    Thinking of $S^k$ and $S^l$ as equipped with their minimal cell structures, then $S^k\times S^l$ inherits a product cell structure with a $(k+l)$-cell whose characteristic map is the product of the characteristic maps for the $k$-cell in $S^k$ and the $l$-cell in $S^l$. Your attaching map is then the restriction to the boundary $S^{k+l-1}$ – iwriteonbananas Apr 04 '16 at 20:04

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I went and asked to an explanation of this and the map is given by viewing the $(k+l)$-disk as $D^k \times D^l$ and then looking at the boundary, $$\partial(D^k \times D^l) = S^{k-1} \times D^l \cup D^k \times S^{l-1}$$

where $S^{k-1} \hookrightarrow S^k$ is the inclusion and $D^{l} \to S^l$ is given by the quotient mapping $(D^l, \partial D^l)$ for the first portion of the union. Similarly, $S^{l-1} \hookrightarrow S^l$ is the inclusion of the equator and again we use the quotient $(D^k, \partial D^k)$.

user7090
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