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51
Isomorphism conjectures in algebraic Ktheory
 J. Amer. Math. Soc
, 1993
"... 1.1 The Isomorphism Conjecture in algebraic Ktheory........ 2 1.2 Main Results and Corollaries.................... 4 1.3 A brief outline............................ 6 ..."
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Cited by 110 (12 self)
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1.1 The Isomorphism Conjecture in algebraic Ktheory........ 2 1.2 Main Results and Corollaries.................... 4 1.3 A brief outline............................ 6
Topology of homology manifolds
 Ann. of Math
, 1996
"... The study of the localglobal geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satis ed by manifolds [25, 57]. Bing's work on decomposition space theory ..."
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Cited by 52 (13 self)
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The study of the localglobal geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satis ed by manifolds [25, 57]. Bing's work on decomposition space theory opened new perspectives. He constructed important examples of 3dimensional homology manifolds with nonmanifold points, which led to the study of other structural properties of these spaces, and also established his shrinking criterion that can be used to determine when homology manifolds obtained as decomposition spaces of manifolds are manifolds [4]. In the 1970s, the fundamental work of Cannon and Edwards on the double suspension problem led Cannon to propose a conjecture on the nature of manifolds, and generated a program that culminated with the EdwardsQuinn characterization of higherdimensional topological manifolds as ENR homology manifolds satisfying a weak general position property known as the disjoint disks property [17, 26,23]. Starting with the work of Quinn [45, 47], a new viewpoint has emerged. Recent advances [10] use techniques of controlled topology to produce a wealth of previously
Spaces over a Category and Assembly Maps in Isomorphism Conjectures in Kand LTheory
"... : We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory of reduced group C algebras. The approach is through spectra over the orbit category of a discrete ..."
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Cited by 49 (12 self)
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: We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory of reduced group C algebras. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and describe such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and the study of the associated generalized homology and cohomology theories and homotopy limits. Key words: Algebraic K and Ltheory, BaumConnes Conjecture, assembly maps, spaces and spectra over a category AMSclassification number: 57 Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G=H, considered as left Gsets, and morphisms are Gmaps. This is a useful construct for o...
The Ktheoretic FarrellJones Conjecture for hyperbolic groups
 Invent. Math
"... Abstract. We prove the Ktheoretic FarrellJones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit. ..."
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Cited by 24 (17 self)
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Abstract. We prove the Ktheoretic FarrellJones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.
Neighborhoods in stratified spaces with two strata
, 1998
"... We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applic ..."
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Cited by 14 (8 self)
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We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the construction of exotic stratifications, a multiparameter isotopy extension theorem and an h–cobordism extension theorem.
The FarrellJones isomorphism conjecture for 3manifold groups
"... Abstract. We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for π1(M) for a large class of 3manifold M. We also prove that if the FIC is true for irreducible 3manifold groups then it is true for all 3ma ..."
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Cited by 10 (6 self)
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Abstract. We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for π1(M) for a large class of 3manifold M. We also prove that if the FIC is true for irreducible 3manifold groups then it is true for all 3manifold groups. In fact, this follows from a more general result we prove here, namely we show that if the FIC is true for each vertex group of a graph of groups with trivial edge groups then the FIC is true for the fundamental group of the graph of groups. This result is part of a program to prove FIC for the fundamental group of a graph of groups where all the vertex and edge groups satisfy FIC. A consequence of the first result gives a partial solution to a problem in the problem list of R. Kirby. We also deduce that the FIC is true for a class of virtually PD3groups. Another main aspect of this article is to prove the FIC for all Haken 3manifold groups assuming that the FIC is true for Bgroups. By definition Bgroups are the fundamental groups of ‘compact irreducible 3manifolds with incompressible nonempty boundary so that each boundary component is a surface of genus ≥ 2’. We also prove the FIC for a large class of Bgroups. This implies that the FIC is true for all 3manifold groups provided Thurston’s Geometrization conjecture is true and the FIC is true for Bgroups. We also prove using a recent result of L.E. Jones that the surjective part of the FIC is true for any Bgroup.
KTHEORY HOMOLOGY OF SPACES
"... Let KR be a nonconnective spectrum whose homotopy groups give the algebraic Ktheory of the ring R. We give a description of the associated homology theory KR∗(X) associated to KR. We also show that the various constructions of KR in the literature are homotopy equivalent, and so give the same homol ..."
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Cited by 9 (0 self)
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Let KR be a nonconnective spectrum whose homotopy groups give the algebraic Ktheory of the ring R. We give a description of the associated homology theory KR∗(X) associated to KR. We also show that the various constructions of KR in the literature are homotopy equivalent, and so give the same homology theory.
Fibered isomorphism conjecture for complex manifolds
, 2002
"... Abstract. In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective ..."
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Cited by 7 (6 self)
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Abstract. In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative Kgroups of the fundamental groups of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds including a large class of 3manifolds which has a finite sheeted cover fibering over the circle.
On the exponent of the cokernel of the forgetcontrol map on K0groups
 MATH
"... For groups that satisfy the Isomorphism Conjecture in lower Ktheory, we show that the cokernel of the forgetcontrol K0groups is composed by the NK0groups of the finite subgroups. Using this information, we can calculate the exponent of each element in the cokernel in terms of the torsion of the ..."
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Cited by 7 (1 self)
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For groups that satisfy the Isomorphism Conjecture in lower Ktheory, we show that the cokernel of the forgetcontrol K0groups is composed by the NK0groups of the finite subgroups. Using this information, we can calculate the exponent of each element in the cokernel in terms of the torsion of the group.
Relative hyperbolicity, classifying spaces, and lower algebraic Ktheory
, 2007
"... For Γ a relatively hyperbolic group, we construct a model for the universal space among Γspaces with isotropy on the family VC of virtually cyclic subgroups of Γ. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in O + (n, 1) ..."
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Cited by 6 (5 self)
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For Γ a relatively hyperbolic group, we construct a model for the universal space among Γspaces with isotropy on the family VC of virtually cyclic subgroups of Γ. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in O + (n, 1) = Isom(H n). We use the information we obtain to explicitly compute the lower algebraic Ktheory of the Coxeter group Γ3 (a nonuniform lattice in O + (3, 1)). Part of this computation involves calculating certain Waldhausen Nilgroups for Z[D2], Z[D3].