Algebraic K-theory of groups wreath product with finite groups

The Farrell-Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example, for discrete subgroups of Lie groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249-297], virtually poly-infinite cyclic groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249-297], Artin braid groups [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515-526], a class of virtually poly-surface groups [S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press] and virtually solvable linear group [F.T. Farrell, P.A. Linnell, K-Theory of solvable groups, Proc. London Math. Soc. (3) 87 (2003) 309-336]. We extend these results in the sense that if G is a group from the above classes then we prove the conjecture for the wreath product G?H for H a finite group. The need for this kind of extension is already evident in [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515-526; S.K. Roushon, The Farrell-Jones isomorphism conjecture for 3-manifold groups, math.KT/0405211, K-Theory, in press; S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press]. We also prove the conjecture for some other classes of groups.     

A formula for the Whitehead group of a three-dimensional crystallographic group

We give an explicit formula for the Whitehead group of a three-dimensional crystallographic group Γ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of Γ.     

The proper L-S category of Whitehead manifolds

It is shown that the proper L-S category of an eventually end-irreducible, R²-irreducible Whitehead 3-manifold is 4. For this we prove, in the category of germs at infinity of proper maps, a partial analogue of the characterization by Eilenberg and Ganea of the L-S category of an aspherical space.     

JSJ-Decompositions of finitely presented groups and complexes of groups

A JSJ-splitting of a group G over a certain class of subgroups is a graph of groups decomposition of G which describes all possible decompositions of G as an amalgamated product or an HNN extension over subgroups lying in the given class. Such decompositions originated in 3-manifold topology. In this paper we generalize the JSJ-splitting constructions of Sela, Rips–Sela and Dunwoody–Sageev, and we construct a JSJ-splitting for any finitely presented group with respect to the class of all slender subgroups along which the group splits. Our approach relies on Haefliger’s theory of group actions on CAT(0) spaces. Submitted: October 2003 Revision: February 2005 Accepted: June 2005     

Vassiliev invariants and the Poincaré conjecture

This article examines the relationship between 3-manifold topology and knot invariants of finite type. We prove that in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants. If, on the other hand, Vassiliev invariants distinguish knots in each homotopy sphere, then the Poincaré conjecture is true (i.e. every homotopy 3-sphere is homeomorphic to the standard 3-sphere).     

Presentations of subgroups of the braid group generated by powers of band generators

According to the Tits conjecture proved by Crisp and Paris (2001) [4], the subgroups of the braid group generated by proper powers of the Artin elements σi are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups.The case of subgroups generated by powers of the band generators aij is more involved. We show that the groups are right-angled Artin groups again, if all generators are proper powers with exponent at least 3. We also give a presentation in cases at the other extreme, when all generators occur with exponent 1 or 2. Such presentations are distinctively more complicated than those of right-angled Artin groups.     

Residual properties of graph manifold groups

Let f:M→N be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen (1999) [16] showed that if f induces a homology equivalence on all finite covers, then f is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is finitely covered by a 3-manifold whose fundamental group is residually p for every prime p. We will show that this statement regarding graph manifold groups is not true in general, but we will show how to modify the argument of Perron and Shalen to recover their main result. As a by-product we will determine all semidirect products Z?Zd which are residually p for every prime p.     

Compact hyperbolic 4-manifolds of small volume

We prove the existence of a compact non-orientable hyperbolic 4-manifold of volume and a compact orientable hyperbolic 4-manifold of volume , obtainable from torsion-free subgroups of small index in the Coxeter group . At the time of writing these are the smallest volumes of any known compact hyperbolic 4-manifolds.

     

Subgroup coverings of some sporadic groups

A set of proper subgroups is a covering for a group if their union is the whole group. Determining the size of a smallest covering is an open problem for many simple groups. For some of the sporadic groups, we find subgroup coverings of minimal cardinality. For others we specify the isomorphism types of subgroups in a smallest covering and use graphs to calculate bounds for its size.     

Relative cohomology theory for profinite groups

In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer–Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the fundamental group of a compact aspherical 3-manifold is a profinite Poincaré duality group relative to the profinite completions of the fundamental groups of its boundary components.     

Taming 3-manifolds using scalar curvature

In this paper we address the issue of uniformly positive scalar curvature on noncompact 3-manifolds. In particular we show that the Whitehead manifold lacks such a metric, and in fact that \mathbbR³{\mathbb{R}^3} is the only contractible noncompact 3-manifold with a metric of uniformly positive scalar curvature. We also describe contractible noncompact manifolds of higher dimension exhibiting this curvature phenomenon. Lastly we characterize all connected oriented 3-manifolds with finitely generated fundamental group allowing such a metric.     

Generalizations of Fox Homotopy Groups,Whitehead Products,and Gottlieb Groups

In this paper, we redefine the Fox torus homotopy groups and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group τ = [Σ(V×W⋃ *), X] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of τ, thereby improving a result of Varadarajan. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 320–328, March, 2005.     

Algebraic K-theory of generalized free products and functors Nil

In this paper, we extend Waldhausen's results on algebraic K-theory of generalized free products in a more general setting and we give some properties of the Nil functors. As a consequence, we get new groups with trivial Whitehead groups.     

Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm

We define an infinite sequence of new invariants, δ_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the δ_n give obstructions to a 3-manifold fibering over S¹ and to a 3-manifold being Seifert fibered. Moreover, we show that the δ_n give computable algebraic obstructions to a 4-manifold of the form X×S¹ admitting a symplectic structure even when the obstructions given by the Seiberg-Witten invariants fail. There are also applications to the minimal ropelength and genera of knots and links in S³.     

On sse-embedded subgroups of finite groups

Abstract

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified.     

COMPLETENESS OF THE PARABOLIC SUBGROUPS OF PROJECTIVE ORTHOGONAL GROUPS

All parabolic subgroups and Borel subgroups of PΩ(2m 1, F) over a linear-able field F of characteristic 0 are shown to be complete groups, provided m > 3.     

s-Permutably embedded subgroups and p-supersolvable groups

Let G be a finite group,and H a subgroup of G.H is called s-permutably embedded in G if each Sylow subgroup of H is a Sylow subgroup of some s-permutable subgroup of G.In this paper,we use s-permutably embedding property of subgroups to characterize the p-supersolvability of finite groups,and obtain some interesting results which improve some recent results.     

Moves for standard skeleta of 3-manifolds with marked boundary

A 3-manifold with marked boundary is a pair (M, X), where M is a compact 3-manifold whose (possibly empty) boundary is made up of tori and Klein bottles, and X is a trivalent graph that is a spine of ?M. A standard skeleton of a 3-manifold with marked boundary (M, X) is a standard sub-polyhedron P of M such that P ?? ?M coincides with X and with ?P, and such that ${P \cup \partial M}$ is a spine of ${M\setminus B}$ (where B is a ball). In this paper, we will prove that the classical set of moves for standard spines of 3-manifolds (i.e. the MP-move and the V-move) does not suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary. We will also describe a condition on the 3-manifold with marked boundary that allows to establish whether the generalised set of moves, made up of the MP-move and the L-move, suffices to relate to each other any two standard skeleta of the 3-manifold with marked boundary. For the 3-manifolds with marked boundary that do not fulfil this condition, we give three other moves: the CR-move, the T₁-move and the T₂-move. The first one is local and, with the MP-move and the L-move, suffices to relate to each other any two standard skeleta of a 3-manifold with marked boundary fulfilling another condition. For the universal case, we will prove that the non-local T₁-move and T₂-move, with the MP-move and the L-move, suffice to relate to each other any two standard skeleta of a generic 3-manifold with marked boundary. As a corollary, we will get that disc-replacements suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary.     

External derivations of internal groupoids

If H is a G-crossed module, the set of derivations of G in H is a monoid under the Whitehead product of derivations. We interpret the Whitehead product using the correspondence between crossed modules and internal groupoids in the category of groups. Working in the general context of internal groupoids in a finitely complete category, we relate derivations to holomorphisms, translations, affine transformations, and to the embedding category of a groupoid.     

Positive open book decompositions of Stein fillable 3-manifolds along prescribed links

It is known by Loi and Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L^′ to L such that the link L∪L^′ is the binding of a positive open book decomposition of the Stein fillable 3-manifold. To prove the assertion, we will use the divide, which is a generalization of real morsification theory of complex plane curve singularities, and 2-handle attachings along Legendrian curves.