Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 21F (Algebraic Topology)

A based loop is a continuous map $\alpha :[0,1]\to X$ with $\alpha (0)=\alpha (1)=x_0$. Two loops are equivalent when they are joined by a homotopy that keeps both endpoints at $x_0$. The fundamental group ${\pi }_1(X,x_0)$ is the set of these based-homotopy classes.

For loops $\alpha$ and $\beta$, define concatenation byIf $H$ and $K$ are based homotopies from $\alpha$ to ${\alpha }^{\prime }$ and from $\beta$ to ${\beta }^{\prime }$, concatenating $H(s,-)$ and $K(s,-)$ gives a based homotopy from $\alpha \ast \beta$ to ${\alpha }^{\prime }\ast {\beta }^{\prime }$. Thus multiplication $[\alpha ][\beta ]=[\alpha \ast \beta ]$ is well-defined.

Concatenating three paths with different breakpoints changes only the speed of traversal. Linear interpolation between the two increasing piecewise-linear parametrizations gives a based homotopy, proving associativity on classes. The constant loop $c(t)=x_0$ is an identity, since deleting its stationary half is another endpoint-fixing reparametrization. The inverse of $[\alpha ]$ is represented by $\bar{\alpha }(t)=\alpha (1-t)$. Indeed, $\alpha \ast \bar{\alpha }$ contracts throughand the analogous contraction handles $\bar{\alpha }\ast \alpha$. Hence the operation satisfies all group axioms.

Use the continuous determinant mapIt induces a homomorphism on fundamental groups. Under the standard identificationthe class of $t\mapsto e^{2\pi it}$ corresponds to its winding number $1$. Sincewe obtainFor $k\ne 0$ this is not the identity, so $[\gamma {]}^k$ could not have been the identity in ${\pi }_1(U(2),I)$.

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