Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 21F (Algebraic Topology)

Let $i:Y\hookrightarrow M_f$ be the canonical inclusion and defineThis is well-defined because $[0,x]=f(x)$ in the mapping cylinder, and continuity follows from the universal property of the quotient topology. Clearly $r\circ i={\mathrm{id}}_Y$.

DefineThe formula respects the gluing relation at $t=0$, so it descends from the disjoint union to a continuous homotopy. It satisfies $H_0={\mathrm{id}}_{M_f}$ and $H_1=i\circ r$. Thusindeed $Y$ is a deformation retract of $M_f$.

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

Let $F_2=⟨a,b⟩={\pi }_1(S^1\vee S^1)$ and take the connected two-sheeted covering graph corresponding to the kernel ofIt has two vertices, a $b$-loop at each vertex, and two lifted $a$-edges joining the vertices. Choose one lifted $a$-edge as a spanning tree. The other three edges give the based loopsCollapsing the spanning tree identifies the cover with a wedge of three circles, so these loops freely generate its fundamental group. Hence they generate an explicit subgroupisomorphic to $F_3$, in agreement with the Reidemeister–Schreier theorem.