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

For a simplicial abelian group $A$, its normalized chain complex of a simplicial abelian group isThe 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 ${\sum }_i(-1{)}^i{d}_i$.

The Dold–Kan correspondence says thatis an equivalence from simplicial abelian groups to nonnegatively graded chain complexes of abelian groups. The restriction of the right adjoint $K$ to ${\mathbf{Ch}}_{\ge 0}(\mathbf{Ab})$ is a quasi-inverse: both the unit $C_{\ast }\to N_{\ast }K(C_{\ast })$ and counit $K(N_{\ast }A)\to A$ are natural isomorphisms.

Under the Dold–Kan correspondence, a $1$-simplex of $K(C_{\ast })$ is a pair $(a,x)\in C_0\oplus C_1$ withThe horn ${\Lambda }_1^2$ consists of the edges $01$ and $12$, joined at vertex $1$. A map from it is therefore a pair of composable $1$-simplices, equivalently a triplewhere the first edge is $(a,x)$ and the second is $(a+{\partial }_{1}x,y)$. Thus

Let $A[n]$ be the nonnegative chain complex having $A$ in degree $n$, zero in every other degree, and zero differential. The simplicial Eilenberg–MacLane space isIts underlying simplicial set is a Kan complex, with ${\pi }_{n}K(A,n)\cong A$ and all other positive homotopy groups zero.

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 groupsThe 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.

Products of Eilenberg–MacLane spaces satisfywhere the last isomorphism uses the Chinese remainder theorem. This space is $2$-connected, so the Hurewicz theorem identifiesThe assumed surjection on $H_3$ is an isomorphism because the group is finite. Hence $f$ is an isomorphism on ${\pi }_3$; all other homotopy groups of the source and target vanish. Thus $f$ is a weak homotopy equivalence, and the Whitehead theorem for Kan complexes makes it a homotopy equivalence.