Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 21F / b / Solution

Cover the attached disc by a slightly enlarged interior and a collar of its boundary, and apply the Seifert-van Kampen theorem. The collar intersection has fundamental group generated by the boundary loop, whose image in $X$ is $[\alpha ]$, while that loop becomes trivial in the disc. Van Kampen therefore quotients by the smallest normal subgroup containing $[\alpha ]$:This is the general fundamental group after attaching a 2-cell.

Start with a wedge of two circles, whose fundamental group is the free groupAttach a disc along the loop represented by $a^3{b}^{-7}$. The displayed quotient gives a connected two-dimensional CW complex $X$ with