Higher homotopy classes form groups and are abelian above degree one

For any continuous $\varphi :I\to I$ fixing endpoints, $a((1-t)s+t\varphi (s),u)$ is a boundary-fixed homotopy from a to its reparametrization. The coordinate formula is jointly continuous, not merely continuous separately in u. Taking ${\varphi }_R(s)=\min (2s,1)$ and ${\varphi }_L(s)=\max (2s-1,0)$ gives $a\ast e\simeq a\simeq e\ast a$. These are the loop-law formulas of F2 with u retained as a parameter.

For $w=(a\ast b)\ast c$, set $\varphi (s)=s\mathrm{/}2$ on $[0,1\mathrm{/}2]$, $s-1\mathrm{/}4$ on $[1\mathrm{/}2,3\mathrm{/}4]$, and $2s-1$ on $[3\mathrm{/}4,1]$. The pieces agree and fix endpoints. Substitution gives $w(\varphi (s),u)=(a\ast (b\ast c))(s,u)$ on all three intervals. Step 1.1 therefore proves associativity on classes.

The map equal to $a(2s(1-t),u)$ for $s\le 1\mathrm{/}2$ and $a(2(1-s)(1-t),u)$ for $s\ge 1\mathrm{/}2$ pastes continuously. It fixes the exterior boundary, begins at $a\ast a^{-}$ and ends at e. Applying the same formula to $a^{-}$ contracts $a^{-}\ast a$. Together with steps 1.1–2.1 and F1 this verifies the group axioms of F3.

For $n\ge 2$ let $\ast$ and $\circ$ concatenate in coordinates 1 and 2. Each has the same two-sided unit by step 1.1. Pasting four quarter-cubes gives $(a\circ b)\ast (c\circ d)=(a\ast c)\circ (b\ast d)$ on representatives. Therefore on classes $a\ast b=(a\circ e)\ast (e\circ b)=(a\ast e)\circ (e\ast b)=a\circ b$, whereas $a\ast b=(e\circ a)\ast (b\circ e)=(e\ast b)\circ (a\ast e)=b\circ a$. Hence the common operation commutes. For n=1 there is no second coordinate, and no commutativity claim is made.