Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 20H / Solution

Cover the circle by two open arcs whose intersection has two components, and take their products with $X$. Both members of this cover retract onto $X$, while their intersection retracts onto $X\bigsqcup X$. The Mayer–Vietoris sequence has the mapafter consistent identifications of the two overlap components. Its kernel and cokernel are each $H_j(X)$. Exactness therefore suppliesThis splits because the last group is free abelian: lift a basis and extend linearly. With $H_{-1}=0$, the result isStarting from a point and iterating this decomposition, Pascal's identity yields