Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 21F / b / Solution

Besides concatenation $\ast$, loops in $U(2)$ have pointwise multiplicationContinuity of matrix multiplication makes this a loop, and pointwise multiplication of homotopies shows that it is well-defined on homotopy classes. Both operations have the constant loop $e(t)=I$ as identity, and they satisfy the interchange law

For homotopy classes $a,b$, the interchange law givesIt then givesThus loop concatenation is commutative on homotopy classes, and