论文标题
STEM延伸的第三个同源性和Whitehead的二次函数
The third homology of stem-extensions and Whitehead's quadratic functor
论文作者
论文摘要
令$ a \ rightarrowtail g \ twebeadRightArrow q $为词干延伸，让$ρ：a \ times g \ to g $为乘法映射。我们表明有一个自然的地图$φ：h_1（σ_2^ε，{\ rm tor} _1^{\ Mathbb {z}}}}}}（{} _ {2^\ infty} a，{} a，{} _ {2^\ infty} a）\ to h_3（g，\ mathbb {z}）/ρ_\ ast（a \ otimes _ {\ mathbb {z}} h_2（g，g，\ mathbb {z}））$，这样，$φ$的图像与自然图$ h_3（a s \ a_3（a，a s \ nath）的形象相结合h_3（g，\ mathbb {z}）/ρ_\ ast（a \ otimes _ {\ mathbb {z}} h_2（g，g，\ mathbb {z}））$。这里使用的重要工具是Whitehead的二次函数$γ$。作为主要结果证明的一部分，我们对自然地图$γ（a）\ \ otimes _ {\ mathbb {z}} a $，$γ（a）\ mapsto a \ otime a $的核心进行了精确的同源描述。
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 $Γ(A) \to A\otimes_{\mathbb{z}} A$, $γ(a)\mapsto a\otimes a$.
