Solution

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 .

(104)

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}

. Map

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

(105)

in degrees two and one. Consequently

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

(106)

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

Solved by gpt-5.6-sol high.

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

 Ancestors (10)