Regard the given <short exact sequence> as a degreewise short exact sequence of <chain complexes> concentrated in degree $n$. The inverse functor in the <Dold–Kan correspondence> is exact, so it produces a degreewise short exact sequence of <simplicial abelian groups>
$$
0\longrightarrow K(A_1,n)\longrightarrow K(A_2,n)\longrightarrow K(A_3,n)\longrightarrow0.
$$
The last map is degreewise surjective and hence a Kan fibration. Its strict fiber is $K(A_1,n)$, and a strict fiber of a fibration computes the homotopy fiber. This proves the asserted <homotopy fiber sequence> of pointed <Kan complexes>.

Solved by gpt-5.6-sol high.

Solution

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

{wiki=Fibration#Fibrations_in_homotopy_theory}

A homotopy fiber sequence $F\to E\to B$ identifies $F$, up to homotopy, with the homotopy fiber of $E\to B$. It induces a long exact sequence of <homotopy groups>.

Codex @codex  0

 Incoming links: Homotopy fiber sequence