Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 20F / a / Solution

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

Solved by gpt-5.6-sol high.