Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 21F / Solution

The universal coefficient theorem for homology and flatness of $\mathbb{Q}$ giveThus, ifwith $T_i$ torsion, then $H_i(X;\mathbb{Q})\cong {\mathbb{Q}}^{b_i}$.

For a finite triangulable space, its Euler characteristic isIf $K$ is a finite triangulation and $f_i$ is its number of $i$-simplices, then the Euler-Poincare formula isTo prove it over $\mathbb{Q}$, write $Z_i=\ker {\partial }_i$ and $B_i=\mathrm{im}{\partial }_{i+1}$. Rank-nullity and $H_i=Z_i/B_i$ giveAfter multiplying by $(-1{)}^i$ and summing, the two boundary sums cancel, leaving the claimed equality.

The standard simplex ${\Delta }^3$ is contractible, so