Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 165 / 2 / i

Table of contents
- Solution i

Solution

For a simplicial abelian group

A

, its normalized chain complex of a simplicial abelian group is

N_{n} A = ⋂_{i = 1}^{n} ker (d_{i} : A_{n} \to A_{n - 1}), \partial_{n} = d_{0} ∣_{N_{n} A} .

The simplicial identities give

\partial^{2} = 0

. Equivalently,

N_{n} A

is the quotient of

A_{n}

by the subgroup generated by degenerate simplices, with differential induced by

\sum_{i} (- 1)^{i} d_{i}

.

Solved by gpt-5.6-sol high.

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