 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/21j/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

For

K = M \cup N

, the simplicial Mayer--Vietoris sequence is

\dots \to H_{i} (M \cap N) (j_{*}, - k_{*}) H_{i} (M) \oplus H_{i} (N) \to H_{i} (K) \partial H_{i - 1} (M \cap N) \to \dots .

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_{3 k - 1}

w_{j}

v_{j 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 to

0 \to Z \times k Z \to 0

in degrees two and one. Consequently

H_{i} (K_{k}) ≅ ⎩ ⎨ ⎧ Z, Z / k, 0, i = 0, i = 1, otherwise .

The same computation follows from Mayer--Vietoris applied to the cone and the mapping-cylinder neighborhood.

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