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.     

Über projektive Moduln und Endlichkeitshindernisse bei Transformationsgruppen

We study projective modules in the category of functors from homogeneous spaces into abelian groups. Such functors have been considered by Bredon [1]. We show that protective functors are determined by a set of ordinary projective modules over suitable group rings. The general notions are applied to give a quick proof for the product formula of the finiteness obstruction for transformation groups. These finiteness obstructions are straightforward extensions of the Swan-Wall obstructions (see e. g. Quinn [7]). They are important in the study of homotopy representations (tom Dieck — Petrie [3], [4]). This work is also related to Rothenberg [8].     

The Deformation Spaces of Projective Structures on 3-Dimensional Coxeter Orbifolds

A real projective structure on a 3-orbifold is given by locally modeling the orbifold by real projective geometry. We present some methodology to study Coxeter groups which are fundamental groups of 3-orbifolds with representations in and deformation spaces. There are related examples by Benoist. These examples give us nontrivial deformation spaces of projective structures.     

A Criterion for Projective Modules

In case G is a finite group, there is a well-known criterion for projective modules: A ? G-module M is projective if and only if it is ? -free and has finite projective dimension. We first investigate whether only finite groups satisfy the above criterion. In the class of groups L H 𝔉, we conclude that this is true. Secondly, we consider the problem when a stably flat Γ-module is projective, where Γ is an arbitrary group. We show that if Γ is an L H 𝔉-group, then every stably flat cofibrant ? Γ-module is projective.     

On the equivalence of Lie symmetries and group representations

We consider families of linear, parabolic PDEs in n dimensions which possess Lie symmetry groups of dimension at least four. We identify the Lie symmetry groups of these equations with the (2n+1)-dimensional Heisenberg group and SL(2,R). We then show that for PDEs of this type, the Lie symmetries may be regarded as global projective representations of the symmetry group. We construct explicit intertwining operators between the symmetries and certain classical projective representations of the symmetry groups. Banach algebras of symmetries are introduced.     

On projective representations for compact quantum groups

We study actions of compact quantum groups on type I-factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz?s results concerning Peter-Weyl theory for compact quantum groups. The main new phenomenon is that for general compact quantum groups (more precisely, those which are not of Kac type), not all irreducible projective representations have to be finite-dimensional. As applications, we consider the theory of projective representations for the compact quantum groups associated with group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).     

Geometric families of 4-dimensional Galois representations with generically large images

We study compatible families of four-dimensional Galois representations constructed in the étale cohomology of a smooth projective variety. We prove a theorem asserting that the images will be generically large if certain conditions are satisfied. We only consider representations with coefficients in an imaginary quadratic field. We apply our result to an example constructed by Jasper Scholten (A non-selfdual 4-dimensional Galois representation, , 1999), obtaining a family of linear groups and one of unitary groups as Galois groups over . Research partially supported by MEC grant MTM2006-04895.     

42-arcs in PG(2, q) left invariant by PSL(2, 7)

For an odd prime p?≠ 7, let q be a power of p such that ${q^3\equiv1 \pmod 7}$ . It is known that the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to PSL(2, 7). For such a projective group Γ, we investigate the geometric properties of the (unique) Γ-orbit Ω of size 42 such that the 1-point stabilizer of Γ in Ω is a cyclic group of order 4. We present a computational approach to prove that Ω is a 42-arc provided that q?≥ 53 and q?≠ 37³, 11⁶, 5⁶, 3⁶. We discuss the case q?=?53 in more detail showing the completeness of Ω for q?=?53.     

Der Verband der zusammenhängenden Untergruppen einer kommutativen algebraischen Gruppe

We classify the connected commutative algebraic groups having as lattice of connected closed subgroups a projective space and determine the coordinate fields of these projective spaces. Furthermore we determine the not necessarily commutative connected algebraic groups with a distributive or boolean lattice of connected closed subgroups.

Received: 3 December 2004     

Continuous planar functions

This paper deals with continuous planar functions and their associated topological affine and projective planes. These associated (affine and projective) planes are the so-called shift planes and in addition to these, in the case of planar partition functions, the underlying (affine and projective) translation planes. We introduce a method that allows us to combine two continuous planar functions ? → ? into a continuous planar function ?² → ?². We prove various extension and embedding results for the associated affine and projective planes and their collineation groups. Furthermore, we construct topological ovals and various kinds of polarities in the associated topological projective planes.     

PSL（2，F）的一个嵌入定理及其应用

设Ｆ是任意域，Ｇ代表ＳＬ（２，Ｆ）或ＰＳＬ（２，Ｆ）．本文的主要结果是：设Ｋ是Ｆ的子域，则Ｇ中同构于ＳＬ（２，Ｋ）或ＰＳＬ（２，Ｋ）的子群在Ｇ的自同构的作用下彼此共轭，利用这一结果，本文明确确定了A₁^[1]型的仿射Ｋａｃ－Ｍｏｏｄｙ群的一类极大正规子群．     

Ovoids and translation planes revisited

Kantor has previously described the translation planes which may be obtained by projecting sections of ovoids in ⁺(8, q)-spaces to ovoids in corresponding ⁺(6, q)-spaces. Since the Klein correspondence associates spreads in 4-dimensional vector spaces with ovoids in ⁺(6, q)-spaces, there are corresponding translation planes of order q ² and kernel containing GF(q). In this article, we revisit some of these translation planes and give some presentations of the spreads. Motivated by various properties of the planes, we study, in general, translation planes which admit certain homology groups and/or elation groups. In particular, we develop new constructions of projective planes of Lenz-Barlotti class II-1.Finally, we show how certain projective planes of order q ² of Lenz-Barlotti class II-1 may be considered equivalent to flocks of quadratic cones in PG(3, q).This work was partially supported by NSF grant DMS-8800843.     

16-Dimensional Smooth Projective Planes with Large Collineation Groups

Smooth projective planes are projective planes defined on smooth manifolds (i.e. the set of points and the set of lines are smooth manifolds) such that the geometric operations of join and intersection are smooth. A systematic study of such planes and of their collineation groups can be found in previous works of the author. We prove in this paper that a 16-dimensional smooth projective plane which admits a collineation group of dimension d 39 is isomorphic to the octonion projective plane P₂ O. For topological compact projective planes this is true if d 41. Note that there are nonclassical topological planes with a collineation group of dimension 40.     

Immersions of the projective plane with one triple point

We consider C^∞ generic immersions of the projective plane into the 3-sphere. Pinkall has shown that every immersion of the projective plane is homotopic through immersions to Boy's immersion, or its mirror. There is another lesser-known immersion of the projective plane with self-intersection set equivalent to Boy's but whose image is not homeomorphic to Boy's. We show that any C^∞ generic immersion of the projective plane whose self-intersection set in the 3-sphere is connected and has a single triple point is ambiently isotopic to precisely one of these two models, or their mirrors. We further show that any generic immersion of the projective plane with one triple point can be obtained by a sequence of toral and spherical surgical modifications of these models. Finally we present some simple applications of the theorem regarding discrete ambient automorphism groups; image-homology of immersions with one triple point; and almost tight ambient isotopy classes.     

Fixed points of projectivities of prime order

This work begins with a review of the classical results for fixed points of projectivities in a projective plane over a general commutative field. The second section of this work features all the material necessary to prove the main result, which is presented in Theorem 2.8. It is shown that, in a finite projective plane of order q, there exists a projectivity g? of prime order p?>?3 if and only if p divides exactly one of the integers q ? 1, q, q?+?1, q ² + q + 1. Theorem 2.8 establishes a correspondence between the possible structures of points fixed by g?, as presented in Theorem 1.3, and the integer that is divisible by p. The special case of p = 2 is handled in Sect. 2.1, where it is shown that every involution is a harmonic homology for q odd and an elation for q even. The special case of p?=?3 is handled in Sect. 2.2, and Theorem 2.8 is adapted for p?=?3 and presented as Theorem 2.15. An application of Theorems 2.8 and 2.15 is determining the sizes of (n, r)-arcs that are stabilized by projectivities of prime order p in the finite projective plane of order q; in Sect. 3, this application is presented in Propositions 3.2 and 3.3.     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript> and rings over which the class of finitely generated strongly gorenstein projective modules is closed under extensions

In this paper, we consider the rings over which the class of finitely generated strongly Gorenstein projective modules is closed under extensions (called fs-closed rings). We give a characterization about the Grothendieck groups of the category of the finitely generated strongly Gorenstein projective R-modules and the category of the finitely generated R-modules with finite strongly Gorenstein projective dimensions for any left Noetherian fs-closed ring R.     

Some subsets of points in the plane associated to truncated Reed-Muller codes with good parameters

We gives some examples of subsets of points in the projective plane associated to truncated generalized projective Reed-Muller codes with good parameters, of dimensions 6 and 10 over GF(7), GF(8) and GF(9).     

Splitting kernels into small summands

Let λ be a regular cardinal. An epimorphism between abelian groups is λ -pure if it is projective with respect to abelian groups of size less than λ. We show that cotorsion groups A have λ-pure projective dimension greater than 1 for all uncountable λ ≤ |A/tA|, where tA denotes the torsion subgroup of A. For λ > |A/tA|, cotorsion groups A are λ-pure projective.     

A Theorem Concerning Nets Arising from Generalized Quadrangles with a Regular Point

Suppose is a generalized quadrangle (GQ) of order , with a regular point. Then there is a net which arises from this regular point. We prove that if such a net has a proper subnet with the same degree as the net, then it must be an affine plane of order t. Also, this affine plane induces a proper subquadrangle of order t containing the regular point, and we necessarily have that . This result has many applications, of which we give one example. Suppose is an elation generalized quadrangle (EGQ) of order , with elation point p. Then is called a skew translation generalized quadrangle (STGQ) with base-point p if there is a full group of symmetries about p of order t which is contained in the elation group. We show that a GQ of order s is an STGQ with base-point p if and only if p is an elation point which is regular.     

4-Dimensionale projektive Ebenen mit grosser abelscher Kollineationsgruppe

We show that a 4-dimensional connected abelian group can act in exactly five different ways as a collineation group of a compact 4-dimensional projective plane. Furthermore the complex projective plane is characterized as the only compact 4-dimensional projective plane which admits two different 4-dimensional abelian collineation groups.

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Herrn Professor Dr. Eelmut Karzel zum 60. Geburtstag