Hochishild-Serre谱组在同系条件下的$d_3$微分表达式
Expression de la differéntielle $d_3$ de la suite spectrale de Hochishild-Serre en cohomologie bornée réelle
摘要：
对于离散群，我们构造了两个系数在约化实$\ell_1$ -同调的第二空间中的有界上同调类。准确地说，我们关联到任何离散群$G$一个有界上同调类的程度2注明$\frak{g}_2\in H_b^2(G, \bar{H}_2^{\ell_1}(G, \mathbb R))$。对于$G$和$Π$组和$θ: Π\rightarrow Out(G)$任何同态，我们关联一个有界的上同调类的程度三注意$[θ]\in H_b^3(Π, \bar{H}_2^{\ell_1}(G, \mathbb R))$。当外同态$θ: Π\rightarrow Out(G)$诱导$G$由$Π$扩展时，我们证明了类$\frak{g}_2$是$Π$不变的，并且Hochschild-Serre谱序列的差分$d_3$在类$[θ]$：$d_3(\frak{g}_2)=[θ]$上发送类$\frak{g}_2$.证明了对任意整数$n\geq 0$，Hochschild-Serre谱序列在实有界上同调下的微分$d_3 : E_3^{n, 2}\rightarrow E_3^{n+3, 0}$是$[θ]$类的杯积.
英文摘要：
For discrete groups, we construct two bounded cohomology classes with coefficients in the second space of the reduced real $\ell_1$-homology. Precisely, we associate to any discrete group $G$ a bounded cohomology class of degree two noted $\frak{g}_2\in H_b^2(G, \bar{H}_2^{\ell_1}(G, \mathbb R))$. For $G$ and $Π$ groups and $θ: Π\rightarrow Out(G)$ any homomorphism we associate a bounded cohomology class of degree three noted $[θ]\in H_b^3(Π, \bar{H}_2^{\ell_1}(G, \mathbb R))$. When the outer homomorphism $θ: Π\rightarrow Out(G)$ induces an extension of $G$ by $Π$ we show that the class $\frak{g}_2$ is $Π$-invariant and that the differential $d_3$ of Hochschild-Serre spectral sequence sends the class $\frak{g}_2$ on the class $[θ]$ : $d_3(\frak{g}_2)=[θ]$. Moreover, we show that for any integer $n\geq 0$ the differential $d_3 : E_3^{n, 2}\rightarrow E_3^{n+3, 0}$ of Hochschild-Serre spectral sequence in real bounded cohomology is given as a cup-product by the class $[θ]$.
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