I'm having trouble picturing the homotopy group operation of concatenation between two pointed spaces.
For $n$-spheres, we have for $f,g: S^n \to X$
$$(f * g)(s_1,\ldots, s_n) = \begin{cases} f(s_1,\ldots, 2s_n) & \text{for }s_n \le 1/2,\\ g(s_1,\ldots, 2s_n-1) & \text{for }s_n \ge 1/2 \end{cases}$$
I understand the case with $S^1$, but when $n>1$ above, why is only one coordinate changed and not all the the others? i.e, only $s_n$ has been changed, but all the other coordinates are the same.
Not sure why I'm having trouble viewing this. Thanks.
From Essential Topology by Martin Crossley.
I like to think of the concatenation of two sphere maps as first shrinking the equator of the sphere to a point (so we get two spheres wedged together) then mapping the top sphere to $X$ according to $f$ and then mapping the bottom sphere to $X$ according to $g$. Because $f$ and $g$ are base point preserving maps, they agree at the wedge point of the two spheres so this is a well defined process.
This is really just the map you've described, but given in a more geometric way, instead of using symbols.