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 ${\sum }_i(-1{)}^i{d}_i$.

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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.

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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

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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.

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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.

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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.

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