Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 21J / Solution

For $K=M\cup N$, the simplicial Mayer--Vietoris sequence is

Construct a simplicial mapping cone of a degree-$k$ map of a circle. Explicitly, let the target circle have vertices $v_0,v_1,v_2$, and let the source circle have vertices $w_0,\dots ,w_{3k-1}$. Map $w_j$ to $v_{j\mathrm{mod}3}$, triangulate each quadrilateral of its mapping cylinder, and cone the source circle to one new vertex. The resulting finite simplicial complex is the mapping cone $K_k$.

Its reduced cellular, or equivalently simplicial, chain complex collapses toin degrees two and one. ConsequentlyThe same computation follows from Mayer--Vietoris applied to the cone and the mapping-cylinder neighborhood.