Tom Farrell, Peter Linnell
We first prove that the Whitehead group of a torsion-free virtually solvable linear group vanishes. Next we make a reduction of the fibered isomorphism conjecture from virtually solvable groups to a class of virtually solvable Q-linear groups. Finally we prove an L-theory analogue for elementary amenable groups.
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We give an explicit formula for Whitehead group of a three-dimensional crystallographic group in terms of the Whitehead groups of its virtually infinite cyclic subgroups.
Some criteria concerning the rational vanishing of Whitehead groups
该研究通过构造有限群G的例子，探讨张量积ℤ⊗_{ℤAut(G)}Wh(G)的秩是否为零，结合第一作者的定理，推导得出含G为正规子群的群的Whitehead群的有理性质。
By use of Siegel's method and the classical results of homotopy groups of spheres and Lie groups, we determine some Gottlieb groups of projective spaces or give the lower bounds of their orders. Furthermore, making use of the properties of Whitehead products, we determine some Whitehead center groups of projective spaces.
This paper will be concerned with proving that certain Whitehead groups of torsion-free elementary amenable groups are torsion groups and related results, and then applying these results to the Bass conjecture. In particular we shall establish the strong Bass conjecture for an arbitrary elementary amenable group.
Let Wh^w(G) be the K_1-group of square matrices over the integral group ring ZG which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let D(G) be the division closure of ZG in the algebra U(G) of operators affiliated to the group von Neumann algebra. Let C be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let G be a torsionfree group which belongs to C. Then we prove that Wh^w(G) is isomorphic to K_1(D(G)). Furthermore we show that D(G) is …
We compute the Whitehead groups of the associative rings in a class which includes (twisted) formal power series rings and the augmentation localizations of group rings and polynomial rings. For any associative ring A, we obtain an invariant of a pair (P,α) where P is a finitely generated projective A-module and α:P \to P is an endomorphism. This invariant determines (P,α) up to extensions, yielding a computation of the (reduced) endomorphism class group of A. We also refine the analysis by Pajitnov and Ranicki of the Whitehead group of the Novikov ring.
We combine results about Whitehead groups of finite groups with results about genetic bases of finite $p$-groups to compute the Whitehead groups of some metacyclic $p$-groups. Let $C_{p^n}$ denote a cyclic group of order $p^n$ for $p$ an odd prime and $n$ a positive integer. We present a conjecture for the torsion part of the Whitehead group of $C_{p^n}\times C_{p^n}$ and we show that torsion part of the Whitehead group of the semidirect product of $C_{p^{n-1}}$ by $C_p$ is isomorphic to $C_p^{(n-2)(p-1)}$. The techniques we use can be applied to any abelian $p$-group and to many other $p$-gro…
Reduced Unitary Whitehead Groups over Function Fields of p-adic Curves
研究p进曲线函数域上二次扩张的约化单位元Whitehead群的平凡性，核心方法为分析中心单代数的周期和involution结构，主要贡献是证明当p≠2时该群为平凡群。
Groups with Spanier-Whitehead duality
本文基于 Kasparov 的工作研究离散群的 Spanier-Whitehead K-对偶性，通过标准“gamma 元素”技术与 property gamma 循环方法构造对偶类，将其与 Baum-Connes 猜想比较，并证明了一大类群的对偶性。
Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P , the subgroup Cl\_1 (ZP) of SK\_1 (ZP), in terms of a genetic basis of P. We also introduce a deflation map Cl\_1 (ZP) $\rightarrow$ Cl\_1 (Z(P/N)) , for a normal subgroup N of P , and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of SK\_1 (ZP), when P is an elementary abelian p-group.
The seminal work of Waldhausen, Farrell and Jones, Igusa, and Weiss and Williams shows that the homotopy groups in low degrees of the space of homeomorphisms of a closed Riemannian manifold of negative sectional curvature can be expressed as a functor of the fundamental group of the manifold. To determine this functor, however, it remains to determine the homotopy groups of the topological Whitehead spectrum of the circle. The cyclotomic trace of B okstedt, Hsiang, and Madsen and a theorem of Dundas, in turn, lead to an expression for these homotopy groups in terms of the equivariant homotopy …
The Whitehead group and stably trivial $G$-smoothings
本文指出闭流形平滑结构的有限性与稳定双射性质在等变情形下不成立，并利用受控$h$-配边构造了$G$-流形的无限多个$G$-平滑结构，且证明它们在与$\mathbb{R}$取积后合痕。
We provide descriptions of the Whitehead groups, and the algebraic $K$-theory groups, of the fundamental group of a connected, oriented, closed $3$-manifold in terms of Whitehead groups of their finite subgroups and certain Nil-groups. The main tools we use are: the K-theoretic Farrell-Jones isomorphism conjecture, the construction of models for the universal space for the family of virtually cyclic subgroups in 3-manifold groups, and both the prime and JSJ-decompositions together with the well-known geometrization theorem.
We prove the A-theoretic Isomorphism Conjecture with coefficients and finite wreath products for solvable groups.
We give formulas for the Whitehead groups and the rational $K$-theory groups of the (integer group ring of the) Hilbert modular group in terms of its maximal finite subgroups.
We develop a version of $\mathrm{G}$-theory for an $\mathbb{F}_1$-algebra (i.e., the $\mathrm{K}$-theory of pointed $G$-sets for a pointed monoid $G$) and establish its first properties. We construct a Cartan assembly map to compare the Chu--Morava $\mathrm{K}$-theory for finite pointed groups with our $\mathrm{G}$-theory. We compute the $\mathrm{G}$-theory groups for finite pointed groups in terms of stable homotopy of some classifying spaces. We introduce certain Loday--Whitehead groups over $\mathbb{F}_1$ that admit functorial maps into classical Whitehead groups under some reasonable hypot…
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