Homology of gauge groups associated with special unitary groups

We study the homology of gauge groups associated with principal SU(n) bundles over the four-sphere. After computing the mod p homology of based gauge groups of SU(n) by combined use of the Serre spectral sequence and the Eilenberg-Moore spectral sequence, we compute the modp homology of gauge groups of SU(n) using the Serre spectral sequence for the evaluation fibration.     

Self homotopy groups with large nilpotency classes

We demonstrate that for any n>0 there exists a compact connected Lie group G such that the self homotopy group [G,G] has the nilpotency class greater than n, where [G,G] is a nilpotent group for a compact connected Lie group G.     

On the fundamental groups of manifolds with integral curvature bounds

In this paper, we give an upper bound on the growth of π₁(M) for a class of manifolds with integral Ricci curvature bounds. This generalizes the main theorem of [8] to the case where the negative part of Ricci curvature is small in an averaged L¹- sense.Received: 19 July 2004     

On the homotopy type of CW-complexes with aspherical fundamental group

This paper is concerned with the homotopy type distinction of finite CW-complexes. A (G,n)-complex is a finite n-dimensional CW-complex with fundamental-group G and vanishing higher homotopy-groups up to dimension n−1. In case G is an n-dimensional group there is a unique (up to homotopy) (G,n)-complex on the minimal Euler-characteristic level χmin(G,n). For every n we give examples of n-dimensional groups G for which there exist homotopically distinct (G,n)-complexes on the level χmin(G,n)+1. In the case where n=2 these examples are algebraic.     

Diagram rigidity for geometric amalgamations of free groups

In this paper, we show that a class of 2-dimensional locally CAT(-1) spaces is topologically rigid: isomorphism of the fundamental groups is equivalent to the spaces being homeomorphic. An immediate application of this result is a diagram rigidity theorem for certain amalgamations of free groups. The direct limits of two such amalgamations are isomorphic if and only if there is an isomorphism between the respective diagrams.     

Local indicability and relative presentations of groups

A relative presentation is a triple where A is a group, X is a set, and R is a set of words in the free product A∗F(X) where F(X) is the free group with basis X. Under certain hypotheses on the relative presentation , we show that (1) the group presented by is locally indicable; (2) the pre-aspherical model for is aspherical; (3) the Freiheitssatz holds for . The result has applications in the computation of cohomology of groups and the field of equations over groups.     

A separable manifold failing to have the homotopy type of a CW-complex

We show that (a variation of) the Prüfer surface, which is an example of a separable non-metrizable 2-manifold, does not have the homotopy type of a CW-complex. Received: 6 February 2007 Revised: 21 August 2007     

Cone-decompositions of the unitary groups

The Lusternik-Schnirelmann category of a space is a homotopy invariant. Cone-decompositions are used for giving upper-bound for Lusternik-Schnirelmann categories of topological spaces. Singhof has determined the Lusternik-Schnirelmann categories of the unitary groups. In this paper I give two cone-decompositions of each unitary group for alternative proofs of Singhof's result. One cone-decomposition is easy. The other is closely related to Miller's filtration and Yokota's cellular decomposition of the unitary groups.     

Homology of iterated loop spaces of homogeneous spaces associated with exceptional Lie groups

We study the mod 2 homology of the double and triple loop spaces of homogeneous spaces associated with exceptional Lie groups. The main computational tools are the Serre spectral sequence for fibrations Ωⁿ⁺¹G→Ωⁿ⁺¹(G/H)→ΩⁿH for n=1,2, and the Eilenberg-Moore spectral sequence associated with related fiber squares, which both converge to the same destination space H_∗(Ωⁿ(G/H);F₂). We also develop the generalized Bockstein lemma to determine the higher Bockstein actions.     

The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$. These groups contain as special cases the odd dimensional general linear groups ${\mathrm{GL}}_{2n+1}(R)$ where R is any ring, the odd dimensional orthogonal and symplectic groups ${\mathrm{O}}_{2n+1}(R)$ and ${\mathrm{Sp}}_{2n+1}(R)$ where R is any commutative ring and further the first author's even dimensional unitary groups ${\mathrm{U}}_{2n}(R,\mathrm{\Lambda })$ where $(R,\mathrm{\Lambda })$ is any form ring. We classify the E-normal subgroups of the groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ (i.e. the subgroups which are normalized by the elementary subgroup ${\mathrm{EU}}_{2n+1}(R,\mathrm{\Delta })$), under the condition that R is either a semilocal or quasifinite ring with involution and $n\ge 3$. Further we investigate the action of ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ by conjugation on the set of all E-normal subgroups.     

Subnormal structure of non-stable unitary groups over rings

Let R be a commutative ring with identity in which 2 is invertible. Let H denote a subgroup of the unitary group U(2n,R,Λ) with n≥4. H is normalized by EU(2n,J,Γ^J) for some form ideal (J,Γ^J) of the form ring (R,Λ). The purpose of the paper is to prove that H satisfies a “sandwich” property, i.e. there exists a form ideal (I,Γ^I) such that

EU(2n,IJ⁸Γ^J,Γ)⊆H⊆CU(2n,I,Γ^I).     

On the homotopy groups of separable metric spaces

The aim of this paper is to discuss the homotopy properties of locally well-behaved spaces. First, we state a nerve theorem. It gives sufficient conditions under which there is a weak n-equivalence between the nerve of a good cover and its underlying space. Then we conclude that for any (n−1)-connected, locally (n−1)-connected compact metric space X which is also n-semilocally simply connected, the nth homotopy group of X, πn(X), is finitely presented. This result allows us to provide a new proof for a generalization of Shelah?s theorem (Shelah, 1988 [18]) to higher homotopy groups (Ghane and Hamed, 2009 [8]). Also, we clarify the relationship between two homotopy properties of a topological space X, the property of being n-homotopically Hausdorff and the property of being n-semilocally simply connected. Further, we give a way to recognize a nullhomotopic 2-loop in 2-dimensional spaces. This result will involve the concept of generalized dendrite which introduce here. Finally, we prove that each 2-loop is homotopic to a reduced 2-loop.     

Unions of slender groups

It is well-known that the class of slender groups is closed under extensions, arbitrary direct sums and subgroups. We show that it is also closed under taking unions of continuous well-ordered ascending chains of groups where each factor of consecutive groups is slender (Theorem 2.2). We also prove that the class of slender groups cannot be obtained from any set of slender groups by forming unions of chains of the mentioned kind and taking subgroups (Theorem 4.2). This generalizes a result by G?bel-Wald [11]. We observe that the same results hold if in the theorems ‘slender’ is replaced by ‘reduced torsion-free’ or ‘cotorsion-free’. Received: 30 June 2005     

Singular unitary dilations

We study the problem of determining which bounded linear operator on a Hilbert space can be dilated to a singular unitary operator. Some of the partial results we obtained are (1) every strict contraction has a diagonal unitary dilation, (2) everyC ₀ contraction has a singular unitary dilation, and (3) a contraction with one of its defect indices finite has a singular unitary dilation if and only if it is the direct sum of a singular unitary operator and aC ₀(N) contraction. Such results display a scenario which is in marked contrast to that of the classical case where we have the absolute continuity of the minimal unitary power dilation of any completely nonunitary contraction.     

On a class of hyperbolic 3-manifolds and groups with one defining relation

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, arbitrarily many of the same volume. The fundamental groups of these 3-manifolds are groups with one defining relation. Our main result is a classification of these manifolds up to homeomorphism, resp. isometry.     

On the existence of representations of finitely presented groups in compact Lie groups

Given a finite, connected 2-complex X   such that _b2(X)?1$b_2(X)?1$ we establish two existence results for representations of the fundamental group of X into compact connected Lie groups G  , with prescribed values on certain loops. If _b2(X)=1$b_2(X)=1$ we assume G=SO(3)$G=\mathrm{SO}(3)$ and that the cup product on H¹(X;Q)$H^1(X;\mathbb{Q})$ is non-zero.