Semi-Biplanes and Antiregular Generalied Quadrangles

A natural method to construct semi-biplanes from antiregular generalized quadrangles is introduced. Properties of the semi-biplanes constructed are discussed. In the finite case and in the topological case the semi-biplanes that arise bear a strong resemblance to semi-biplanes that arise in the natural way from projective planes admitting an involutory homology.     

Semi-biplanes on the cylinder

We construct semi-biplanes from 2-dimensional projective planes and 2-dimensional circle planes.Herrn Professor Dr. H. Salzmann zum 65. GeburtstagThis research was supported by a Feodor Lynen fellowship.     

The co-existence of elations and homologies of prime order

N. Krier and J. C. D. S. Yaqub have proved that if a projective plane admits an involutory homology and an involutory elation, then does not belong to the Lenz-Barlotti class I₁, and belongs to the class I₂. In this paper, we find the classification of projective planes having a homology of orderp and an elation of orderq, wherep andq are primes.This is based on a part of the doctoral dissertation of A. Solai Raju. The work was supported by a Senior Research Fellowship of the CSIR, India.     

Sur le tétraèdre régulier dans un espace projectif complexe

Résumé On montre l'existence, dans un espace projectif complexe, d'un tétraèdre régulier ayant un groupe de symétrie isomorphe à celui du tétraèdre régulier euclidien. On précise la classe d'homologie de dimension 2, qui contient le complexe simplicial qui lui est associé.

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We show the existence, in a complex projective space, of a regular tetrahedron which has its symmetry group isomorphic to that of the euclidean regular tetrahedron. We give precisely the two-dimensional homology class, containing the simplicial complexe associated to it.

     

Linear k-blocking Sets

We point out the relationship between normal spreads and the linear k-blocking sets introduced in [9]. We give a characterisation of linear k-blocking sets proving that the projections and the embeddings of a PG(kt,q) in are linear k-blocking sets of . Finally, we construct some new examples. Received December 19, 1997/Revised September 19, 2000 RID="*" ID="*" Partially supported by Italian M.U.R.S.T.     

Detecting K-Theory by Cyclic Homology

We discuss which part of the rationalized algebraic K-theoryof a group ring is detected via trace maps to Hochschild homology,cyclic homology, periodic cyclic or negative cyclic homology.2000 Mathematics Subject Classification 19D55.     

4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups

 This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit non-solvable automorphism groups. It is shown that such a plane is either semi-classical or a single plane admitting the group SL(2, ). Various characterizations of this single Laguerre plane are obtained. Received October 17 2000; in revised form April 23 2001 Published online August 5, 2002     

Metric homology

Metric homology is a homology theory constructed on semialgebraic (or compact subanalytic) sets with singularities. Metric homology is an invariant under semialgebraic (subanalytic) bi‐Lipschitz homeomorphisms and not a topological invariant. As in intersection homology theory, classes of admissible chains are defined using a semialgebraic stratification and a perversity function. In contrast to intersection homology, the perversity is a rational‐valued function. Instead of the topological dimension of the intersection of a chain with a stratum, we consider the so‐called volume‐growth number. This number is a sort of generalization of Hausdorff dimension. In the second part of the paper we describe one‐dimensional metric homology for spaces with isolated singularities and calculate some concrete examples. © 2000 John Wiley & Sons, Inc.     

The universality of equivariant complex bordism

We show that if A is an abelian compact Lie group, all A-equivariant complex vector bundles are orientable over a complex orientable equivariant cohomology theory. In the process, we calculate the complex orientable homology and cohomology of all complex Grassmannians. Received: 14 February 2000; in final form: 4 August 2000 / Published online: 19 October 2001     

Discrete approximations for complex Kac–Moody groups

We construct a map from the classifying space of a discrete Kac–Moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac–Moody group and prove that it is a homology equivalence at primes q different from p. This generalizes a classical result of Quillen, Friedlander and Mislin for Lie groups. As an application, we construct unstable Adams operations for general Kac–Moody groups compatible with the Frobenius homomorphism. Our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete Kac–Moody groups.     

Translation planes admitting a pair of index three homology groups

It is shown that a translation plane of order which admits two homology groups of order must in fact admit symmetric homology groups of this order. It is further shown that a plane admitting such symmetric index 3 homology groups is, with a finite number of exceptions, a generalized André plane. A list of the possibly exceptional orders is determined. Received 20 March 2000.     

Quasi-isomorphisms of Koszul complexes

Let be a surjective homomorphism of noetherian local commutative rings that induces an isomorphism between the first Koszul homology modules and an epimorphism between the second Koszul homology modules. Then induces isomorphisms between Koszul homology modules in all dimensions.

     

Existence,unicité et homogénéité de certains immeubles hyperboliques

Résumé. Dans ce travail, nous introduisons une classe d'immeubles hyperboliques de dimension 2 à allure locale prescrite. Nous prouvons l'existence et l'unicité de ces immeubles. Nous montrons que leur groupe d'automorphismes est transitif sur les chambres. Ces résultats permettent d'identifier des immeubles hyperboliques construits par les diverses méthodes utilisées jusqu'ici: “blueprints” de Ronan-Tits et immeubles de Kac-Moody, revêtements ramifiés d'immeubles euclidiens, polygones de groupes etc... Nos méthodes s'appliquent en fait à des complexes polygonaux hyperboliques beaucoup plus généraux que les seuls immeubles. Received: 3 April 2000; in final form: 25 September 2000 / Published online: 23 July 2001     

Homologies associées aux variétés de Poisson

In this article, we obtain results on the cyclic homology of Poisson manifolds as defined by Kassel. In particular, we compute the cyclic homology of any symplectic variety. We also show that there exists a family of new operators of the type of the S operator of Connes, defined on the cyclic homology of a Poisson manifold, and describe the associated exact triangles associated to these new operators.

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Received February 23, 1998 / Published online July 20, 2000     

A multiplication in cyclic homology

We define a multiplication on the cyclic homology of a commutative, cocommutative bialgebra with ``superproduct.' In the case when is a field of characteristic zero the cyclic homology becomes a polynomial algebra in one generator. (The Loday-Quillen multiplication is trivial in that case.)

     

The Gergonne and Nagel centers of a tetrahedron

The Gergonne center of a triangle is the intersection of the cevians through the points where the incircle touches the sides. This does not admit a direct generalization to tetrahedra since the cevians of a tetrahedron through the points where the insphere touches the faces are not necessarily concurrent. This article introduces an alternative definition of the Gergonne center that coincides with the previous definition for the triangle and that admits a generalization to tetrahedra. The same is done for the Nagel center. Received 4 February 2000; revised 1 May 2000.     

The Neumann-Siebenmann invariant and Seifert surgery

We discuss some relations between the invariant originated in Fukumoto-Furuta and the Neumann-Siebenmann invariant for the Seifert rational homology 3-spheres. We give certain constraints on Seifert 3-manifolds to be obtained by surgery on knots in homology 3-spheres in terms of these invariants.Mathematics Subject Classification (2000): 57M27, 57N13, 57N10Dedicated to Professor Yukio Matsumoto for his 60^th birthday     

Maximal Baer groups in translation planes and compatibility with homology groups

The translation planes with spreads in PG(3, q) that admit at least two Baer groups of order q–1 are classified.     

Minimum distance of orthogonal line-Grassmann codes in even characteristic

In this paper we determine the minimum distance of orthogonal line-Grassmann codes for q even. The case q odd was solved in [3]. For $n\ne 3$ we also determine their second smallest distance. Furthermore, we show that for q even all minimum weight codewords are equivalent and that the symplectic line-Grassmann codes are proper subcodes of codimension 2n of the orthogonal ones.