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

Abstract

Let $A\rightarrowtail G\twoheadrightarrow Q$ be a stem-extension and let $\rho :A\times G\to G$ be the multiplication map. We show that there is a natural map $\phi :H_1({\Sigma }_2^{\epsilon },{\mathrm{Tor}}_1^{\mathbb{Z}}({}_{2^{\infty }}A,{}_{2^{\infty }}A))\to H_3(G,\mathbb{Z})/{\rho }_{\ast }(A{\otimes }_{\mathbb{Z}}{H}_2(G,\mathbb{Z}))$ such that, the image of $\phi$ coincides with the image of the natural map $H_3(A,\mathbb{Z})\to H_3(G,\mathbb{Z})/{\rho }_{\ast }(A{\otimes }_{\mathbb{Z}}{H}_2(G,\mathbb{Z}))$. An important tool used here is Whitehead's quadratic functor $\Gamma$. As part of our proof of the main result, we give a precise homological description of the kernel of the natural map $\Gamma (A)\to A{\otimes }_{z}A$, $\gamma (a)\mapsto a\otimes a$.

11 pages