Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 21F (Algebraic Topology)

The map $p$ is the quotient of ${\mathbb{R}}^2$ by integer translations, andGiven a based loop $\gamma$ at $x_0$, the path lifting theorem gives a unique lift $\tilde{\gamma }$ beginning at $0$. Its endpoint lies in ${\mathbb{Z}}^2$. The homotopy lifting property shows that this endpoint depends only on $[\gamma ]$, so defineLifting a concatenation after translating the second lift by the endpoint of the first shows that $\Phi$ is a group homomorphism. It is surjective because, for every $m\in {\mathbb{Z}}^2$, the path $r\mapsto p(rm)$ is a loop whose lift ends at $m$. If a lift ends at zero, it is a loop in the simply connected space ${\mathbb{R}}^2$ and contracts there; projecting the contraction proves that its original loop is null-homotopic. Thus $\Phi$ is injective, and the lift-endpoint description of the fundamental group of the torus gives

Let $A\in G{L}_2(\mathbb{Z})$. If $r-r^{\prime }\in {\mathbb{Z}}^2$, then $A(r-r^{\prime })\in {\mathbb{Z}}^2$, sodefines a continuous map on $X={\mathbb{R}}^2/{\mathbb{Z}}^2$. Since $\det A=\pm 1$, the integral general linear group contains $A^{-1}$ as another integer matrix, and $f_{A^{-1}}$ is the inverse of $f_A$. Hence $f_A$ is a homeomorphism, and $f_A(x_0)=x_0$.

The map $A:{\mathbb{R}}^2\to {\mathbb{R}}^2$ is the lift of $f_A$ fixing the origin. A loop whose lift ends at $m\in {\mathbb{Z}}^2$ is sent to a loop whose lift ends at $Am$. Under the isomorphism in part (a), the induced map is thereforeThis is precisely the integral linear automorphism of the torus associated with $A$.

Choose ${\hat{x}}_i\in p_i^{-1}(x_0)$. By the classification of connected covering spaces and the degree of a connected covering, each cover corresponds to an index-two subgroupAn index-two subgroup is the kernel of a nonzero homomorphism ${\mathbb{Z}}^2\to \mathbb{Z}/2\mathbb{Z}$. There are three:The group $G{L}_2({\mathbb{F}}_2)$ acts transitively on the three nonzero linear functionals on ${\mathbb{F}}_2^2$, and its elementary matrices lift to the integral general linear group. Hence there is $A\in G{L}_2(\mathbb{Z})$ such that

Take $\psi =f_A$. Since ${\psi }_{\ast }(H_1)=H_2$, the lifting criterion for a covering space gives a based liftsuch thatApplying the same argument to $A^{-1}$ gives a lift in the reverse direction. The composites are based lifts of the appropriate identity maps; uniqueness of based lifts makes them identities. Thus $\varphi$ and $\psi$ are homeomorphisms, and the required square commutes. This proves the equivalence of connected double covers of the torus.