Higher homotopy basepoint transport and moving homotopies

Use the centered cube $[-1,1{]}^n$, r=||z||∞. For a cube a based at x1, set $T_{\gamma }a(z)=a(2z)$ for r≤1/2 and $\gamma (2-2r)$ for r≥1/2. At r=1/2 both give x1, and at r=1 the value is x0. Closed pasting proves continuity. The identical formula with a homotopy of a or an endpoint-fixed homotopy of γ proves independence of both representatives.

Two successive transports have a core of radius 1/4 and two nested shells tracing η then γ as the radius increases. A positive piecewise-linear change of radial variable matches these breakpoints with those for transport by γ*η, and interpolation of that change with the identity gives a boundary-fixed homotopy; its core is scaled by the same positive factor. A constant shell can similarly be shrunk to width zero, since all its values equal the boundary value. Finally γ followed by its reverse contracts rel endpoints by the explicit retracing formula $\gamma (2s(1-t))$ on the first half and $\gamma (2(1-s)(1-t))$ on the second. Applying step 1.1 to this path homotopy gives ${\beta }_{\bar{\gamma }}{\beta }_{\gamma }=\mathrm{i}\mathrm{d}$ and the reverse identity.

More generally let $F(z,t)$ be a homotopy of cubes whose boundary value is γ(t). Define $K(z,t)=F(2z,t)$ on r≤1/2 and $K(z,t)=\gamma (t(2-2r))$ on r≥1/2. The seam values are γ(t) and the outer boundary is x0. Thus K is a based homotopy from F(-,0) with a constant shell to $T_{\gamma }F(-,1)$. Removing the constant shell by step 2.1 proves $[F(-,0)]={\beta }_{\gamma }[F(-,1)]$. Apply this to F(z,t)=H(a(z),t) to obtain $f_{\ast }={\beta }_{\gamma }{g}_{\ast }$.

For a based at x1, the homotopy with fixed core a(2z) and shell $\gamma (1-t+t(2-2r))$ has moving boundary $\bar{\gamma }(t)$, starts at a with a constant shell and ends at $T_{\gamma }a$. Paste this homotopy for a and b on the two coordinate-1 half-cubes: their common face has the same value $\bar{\gamma }(t)$, so it is a continuous moving-boundary homotopy from a*b to $T_{\gamma }a\ast T_{\gamma }b$, up to the removable constant shells. Step 3.1 gives $[a\ast b]={\beta }_{\bar{\gamma }}([T_{\gamma }a\ast T_{\gamma }b])$; applying the inverse identity of step 2.1 proves multiplicativity of βγ. Hence it is a group isomorphism.

In dimension one, increasing the centered coordinate from -1 to 1 traverses the left shell from γ(0) to γ(1), then a, then the right shell from γ(1) to γ(0). Positive reparametrization gives exactly $[\gamma \ast a\ast \bar{\gamma }]$. This verifies the first-loop-first convention.