Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 20F (Algebraic Topology)

One form of the Seifert-van Kampen theorem says that if $X=U\cup V$, with $U,V,U\cap V$ open and path-connected and containing the base point, then ${\pi }_1(X)$ is the pushout of

For the fundamental group after attaching a 2-cell, take $U$ to be $X$ together with a collar of the attached disc and $V$ to be a slightly enlarged open disc. Then $U\simeq X$, $V$ is simply connected, and $U\cap V\simeq S^1$; its generator maps to $[\alpha ]$ in ${\pi }_1(X)$. Van Kampen therefore gives

Take $X=S_a^1\vee S_b^1$, whose fundamental group is the free group $F(a,b)$, and attach one 2-cell along the loop represented by $a^2{b}^{-3}$. The resulting connected presentation complex $Y$ satisfies

A covering space is a map $p:\tilde{X}\to X$ such that every $x\in X$ has an open neighbourhood $U$ for whichwith every restriction $p:U_{\lambda }\to U$ a homeomorphism.

Let $N=⟨⟨a⟩⟩$ in
${\pi }_1(Y)=⟨a,b\mid a^2{b}^{-3}⟩$. ThenThe subgroup is normal of index three, so the degree of a connected covering shows that every fibre $p^{-1}(y)$ has exactly three points.

The cell complex of a covering from a coset graph gives an explicit model. Take vertices $v_0,v_1,v_2$ with indices modulo three; put an $a$-loop $A_i$ at every $v_i$, and a directed $b$-edge $B_i:v_i\to v_{i+1}$. Attach a 2-cell at each $v_i$ alongMap every $v_i$ to $y_0$, every $A_i$ to the $a$-cell, every $B_i$ to the $b$-cell, and each 2-cell homeomorphically to the unique 2-cell of $Y$. Its attaching word is the lift of $a^2{b}^{-3}$ beginning at $v_i$. This defines the required connected three-sheeted covering.