Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 1 / a / Solution

For a CW complex $X$, the cellular chain complex isthe free abelian group generated by the oriented $n$-cells. Its differential is the compositeNaturality of the long exact sequences of the triples $X^{n-2}\subset X^{n-1}\subset X^n$ makes two successive connecting maps compose to zero, so $d_{n-1}{d}_n=0$.

The cellular boundary formula says that the coefficient of an $(n-1)$-cell $e_{\alpha }^{n-1}$ in the boundary of $e_{\beta }^n$ is the degree of