Volume 23, Issue 3

The generalized conjugacy problem (has g a conjugate in K for K rational?) is solved for finitely generated (f.g.) virtually free groups with constraints that go beyond the context-free level, a new result for the free group itself. Moldavanskii's theorem on simultaneous conjugacy of f.g. subgroups of a free group is also generalized for virtually free groups and this wider class of constraints. The solution set of the equation x –1 g φ ( x ) ∈ K in the free group ( φ a virtually inner automorphism, K rational) is shown to be rational and effectively constructible, and a similar result is proved for the equation xgx –1 ∈ K in a f.g. virtually free group. The twisted conjugacy problem with context-free constraints is also proved to be decidable for the free group.

The Kreck monoids l 2 q +1 (ℤ[ π ]) detect s -cobordisms amongst certain bordisms between stably diffeomorphic 2 q -dimensional manifolds and generalise the Wall surgery obstruction groups, . In this paper we identify l 2 q +1 (ℤ[ π ]) as the edge set of a directed graph with vertices a set of equivalence classes of quadratic forms on finitely generated free ℤ[ π ] modules. Our main theorem computes the set of edges l 2 q +1 ( υ, υ ′) ⊂ l 2 q +1 (ℤ[ π ]) between the classes of the forms υ and υ ′ via an exact sequence Here sbIso( υ, υ ′) denotes the set of “stable boundary isomorphisms” between the algebraic boundaries of υ and υ ′. As a consequence we deduce new classification results for stably diffeomorphic manifolds.

We consider the Banach space LUC( G ) (RUC( G )) of the bounded left (right) uniformly continuous functions on a locally compact group G , the Banach space UC( G ) = LUC( G )∩RUC( G ) of the bounded uniformly continuous functions on G and the quotient Banach space LUC( G )/UC( G ). If G is a sin group, that is, the right and the left uniformities coincide, then the quotient LUC( G )/UC( G ) is trivial, so we are concerned with non- sin groups. We bring up the topological structure of G to the Banach space structure of the quotient by showing first that when the compact covering number k ( G ) is greater than or equal to the local weight b ( G ), there is a linear isometric copy of in LUC( G )/UC( G ). Then, using the generalised Kakutani–Kodaira theorem due to Zhiguo Hu, we deduce that in the case of a general non- sin group, there exists a compact normal subgroup N of G such that ℵ 0 ≤ b ( G / N ) ≤ k ( G ) and LUC( G )/UC( G ) contains a linear isometric copy of .

The notion of Igusa–Todorov classes is introduced in connection with the finitistic dimension conjecture. As application we consider conditions on special ideals which imply the Igusa–Todorov and other finiteness conditions on modules proving the finitistic dimension conjecture and related conjectures in those cases.

In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Čech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups possess a natural smooth structure. In the last section we provide some explicit examples.

We describe flat covers and cotorsion envelopes over formal triangular matrix rings. To describe the flat covers we use Quillen factorizations of linear maps relative to the classical cotorsion pair. Using flat covers over formal triangular matrix rings we prove the existence and minimality of Quillen factorization.

Let be the nilpotent Lie group of all unipotent upper triangular complex n × n matrices and let I n denote the maximal order of the holonomy groups of all infranilmanifolds with -geometry. By analyzing the algebraic structure of and studying its isometry group, we prove that I 3 = 24.

In this paper, we study the category of algebraic Bol loops over an algebraically closed field of definition. On the one hand, we apply techniques from the theory of algebraic groups in order to prove structural theorems for this category. On the other hand, we present some examples showing that these loops lack some nice properties of algebraic groups; for example, we construct local algebraic Bol loops which are not birationally equivalent to global algebraic loops.