Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 21J / Solution

The Seifert-van Kampen theorem says that if $X=U\cup V$, where $U,V,U\cap V$ are path connected open sets containing $x_0$, then

The cell attachment is the quotientwith the image of $x_0=f(\ast )$ as base point. For $n=2$, take one open set that deformation retracts onto $X$ together with a boundary collar, and another that consists of the interior of the disc with a collar and is contractible. Their intersection deformation retracts onto $S^1$. Its generator maps to $[f]$ in the first set and to the identity in the second. Van Kampen therefore proves the fundamental group after attaching a 2-cell formula

Applying van Kampen to the two circles givesAttach three discs along loops representing $a^2$, $b^3$, and $(ab{)}^2$. The resulting presentation complex for the symmetric group on three letters has groupSending $a$ to $(12)$ and $b$ to $(123)$ gives a surjection onto $S_3$. The relations imply $aba=b^{-1}$, so every word reduces to one ofThe presented group has at most six elements and surjects onto the six-element group $S_3$, so this map is an isomorphism.

Solved by gpt-5.6-sol high.