Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 6 / b

Table of contents
- Solution b

Solution

Apply the Snake lemma degree by degree to a short exact sequence of complexes

0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0.

If

[c] \in H_{n} (C)

, lift a cycle

c

to

b \in B_{n}

. Its boundary maps to zero in

C_{n - 1}

, so it comes from a cycle

a \in A_{n - 1}

\partial [c] = [a]

. The Snake-lemma exactness and independence checks yield

\dots \to H_{n} (A) \to H_{n} (B) \to H_{n} (C) \partial H_{n - 1} (A) \to H_{n - 1} (B) \to \dots .

This is the algebraic Mayer-Vietoris theorem for homology objects.

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