Some remarks on the homology of nilpotent groups
In this article we study the homology of nilpotent groups. In particular a certain vanishing result for the homology and cohomology of nilpotent groups is proved.
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arXiv subjects
Fatemeh Yeganeh Mokari
Publications and source records attributed to Fatemeh Yeganeh Mokari.
The third homology of stem-extensions and Whitehead's quadratic functor
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 $Γ(A) \to A\otimes_{\mathbb{z}} A$, $γ(a)\mapsto a\otimes a$.
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Third homology of perfect central extensions
For a central perfect extension of groups $A \rightarrowtail G\twoheadrightarrow Q$, first we study the natural image of $H_3(A,\mathbb{Z})$ in $H_3(G, \mathbb{Z})$. As a particular case, we show that if the extension is universal this image is 2-torsion. Moreover when the plus-construction of the classifying space of $Q$ is an $H$-space, we also study the kernel of the surjective homomorphism $H_3(G,\mathbb{Z}) \to H_3(Q, \mathbb{Z})$.
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The homology of $\mathrm{SL}_2$ of discrete valuation rings
Let $A$ be a discrete valuation ring with field of fractions $F$ and (sufficiently large) residue field $k$. We prove that there is a natural exact sequence $H_3(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}]) \to H_3(\mathrm{SL}_2(F),\mathbb{Z}[\frac{1}{2}])\to \mathcal{RP}_1(k)[\frac{1}{2}]\to 0$, where $\mathcal{RP}_1(k)$ is the refined scissors congruence group of $k$. Let $Γ_0(\mathfrak{m}_A)$ denote the congruence subgroup consisting of matrices in $\mathrm{SL}_2(A)$ whose lower off-diagonal entry lies in the maximal ideal $\mathfrak{m}_A$. We also prove that there is an exact sequence $0\to \overline{\mathcal{P}}(k)[\frac{1}{2}]\to H_2(Γ_0(\mathfrak{m}_A),\mathbb{Z}[\frac{1}{2}])\to H_2(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}])\to I^2(k)[\frac{1}{2}]\to 0$, where $I^2(k)$ is the second power of the fundamental ideal of the Grothendieck-Witt ring $\mathrm{GW}(k)$ and $\overline{\mathcal{P}}(k)$ is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) $\mathcal{P}(k)$ of $k$.
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Virtual rational Betti numbers of nilpotent-by-abelian groups
In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian group $G$, where the abelianization $N/N'$ of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N'$ is $2(c(n-1)-1)$-tame as a $G/N$-module, $c$ the nilpotency class of $N$, then $\mathrm{vb}_j(G):=\sup_{M\in\mathcal{A}_G}\dim_\mathbb{Q} H_j(M,\mathbb{Q})$ is finite for all $0\leq j\leq n$, where $\mathcal{A}_G$ is the set of all finite index subgroups of $G$.
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