The third homology of stem-extensions and Whitehead's quadratic functor

Abstract

Let $A \rightarrowtail G\twoheadrightarrow Q$ be a stem-extension and let $ρ: A\times G\to G$ be the multiplication map. We show that there is a natural map $φ: H_1(Σ_2^ε, {\rm Tor}_1^{\mathbb{Z}}({}_{2^\infty}A,{}_{2^\infty}A))\to H_3(G,\mathbb{Z})/ρ_\ast(A \otimes_{\mathbb{Z}} H_2(G,\mathbb{Z}))$ such that, the image of $φ$ coincides with the image of the natural map $H_3(A,\mathbb{Z})\to H_3(G,\mathbb{Z})/ρ_\ast(A \otimes_{\mathbb{Z}} H_2(G,\mathbb{Z}))$. An important tool used here is Whitehead's quadratic functor $Γ$. As part of our proof of the main result, we give a precise homological description of the kernel of the natural map $Γ(A) \to A\otimes_{\mathbb{z}} A$, $γ(a)\mapsto a\otimes a$.