Cubical and spherical models of higher homotopy agree

After centering and doubling the cube, its interior is $(-1,1{)}^n$. The coordinate map $v_i\mapsto v_i\mathrm{/}(1-\mathrm{\mid }{v}_i\mathrm{\mid })$ is a homeomorphism to ${\mathbb{R}}^n$, with inverse $w_i\mapsto w_i\mathrm{/}(1+\mathrm{\mid }{w}_i\mathrm{\mid })$. Approaching the cube boundary sends the Euclidean norm to infinity, and conversely bounded images stay away from that boundary. Thus the map extends to a homeomorphism of the collapsed-boundary cube with ${\mathbb{R}}^n\cup \{\mathrm{\infty }\}$. Inverse stereographic projection $w\mapsto (2w\mathrm{/}(1+\mathrm{\parallel }w{\mathrm{\parallel }}^2),(\mathrm{\parallel }w{\mathrm{\parallel }}^2-1)\mathrm{/}(1+\mathrm{\parallel }w{\mathrm{\parallel }}^2))$ identifies this compactification with $S^n$, taking the quotient point $[\mathrm{\partial }{I}^n]$ to the north pole. Take that north pole as $s_0$ and choose the sphere orientation to agree with the cube interior; this gives the required based $h$.

Every boundary-constant map descends uniquely by F2, and every based spherical map pulls back to a cubical map. For a boundary-fixed homotopy, F3 makes its descent across $(I^n\to I^n\mathrm{/}\mathrm{\partial }{I}^n)\times {\mathrm{i}\mathrm{d}}_I$ continuous. Pullback is the inverse and preserves endpoint maps. The bijections on maps therefore induce inverse bijections on homotopy classes.

Collapse also the middle face $s_1=1\mathrm{/}2$. Each resulting half-cube with its boundary collapsed is an oriented based copy of the same sphere, using the positive affine rescalings $2s_1$ and $2s_1-1$ and the based homeomorphism $h$. The composite of this transported pinch with $a$ on the first copy and $b$ on the second pulls back to the defining formula for $a\ast b$. Hence the operations agree.