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Hall invariants, homology of subgroups, and characteristic varieties
 International Mathematics Research Notices
, 2002
"... Given a finitelygenerated group G, and a finite group Gamma, Philip Hall defined delta_Gamma to be the number of factor groups of G that are isomorphic to Gamma. We show how to compute the Hall invariants by cohomological and combinatorial methods, when G is finitelypresented, and Gamma belongs to ..."
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Given a finitelygenerated group G, and a finite group Gamma, Philip Hall defined delta_Gamma to be the number of factor groups of G that are isomorphic to Gamma. We show how to compute the Hall invariants by cohomological and combinatorial methods, when G is finitelypresented, and Gamma belongs to a certain class of metabelian groups. Key to this approach is the stratification of the character variety by the jumping loci of the cohomology of G, with coefficients in rank 1 local systems over a suitably chosen field K. Counting relevant torsion points on these "characteristic" subvarieties gives delta_Gamma(G). In the process, we compute the distribution of primeindex, normal subgroups K of G according to the dimension of the the first homology group of K with K coefficients, provided charK does not divide the index of K in G. In turn, we use this distribution to count lowindex subgroups of G. We illustrate these techniques in the case when G is the fundamental group of the complement of an arrangement of either affine lines in C^2, or transverse planes in R^4.
Configurations Of Skew Lines
 LENINGRAD MATHEMATICS JOURNAL
, 1990
"... This article is a survey of results on projective configurations of subspaces in general position. It is written in the form of a popular introduction to the subject, with much of the material accessible to advanced high school students. However, in the part of the survey concerning configuration ..."
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This article is a survey of results on projective configurations of subspaces in general position. It is written in the form of a popular introduction to the subject, with much of the material accessible to advanced high school students. However, in the part of the survey concerning configurations of lines in general position in threedimensional space we give a complete exposition.
CONFIGURATIONS OF SKEW LINES JULIA VIRO AND OLEG VIRO
, 2006
"... Abstract. This article is a survey of results on projective configurations of subspaces in general position. It is written in the form of introduction to the subject, with much of the material accessible to advanced high school students. However, in the part of the survey concerning configurations o ..."
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Abstract. This article is a survey of results on projective configurations of subspaces in general position. It is written in the form of introduction to the subject, with much of the material accessible to advanced high school students. However, in the part of the survey concerning configurations of lines in general position in threedimensional space we give a detailed exposition. The first version of this paper was written as an elementary introductory text for highschool students. It was published [3] in the the journal “Kvant”, the third issue of 1988, but in a shortened form. Then we expanded the article in order to encompass or at least mention some related questions. However we decided to keep the style of [3], in the hope that it would also be appreciated by a professional mathematician. We apologized to a reader, who would find the style irritating, and we mentioned that the material in the first twothirds of the article (through the section on “sets of five lines”) was announced in the note [2], while the final third of the article is written in a more traditional style. The expanded version [10] of [3] was published in the first volume of a Russian
Spindle configurations of skew lines
, 2002
"... Abstract. We simplify slightly the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration. Finally, we describe an algorithm constructing a spindle in a given switching class, provided such a spindle exists ..."
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Abstract. We simplify slightly the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration. Finally, we describe an algorithm constructing a spindle in a given switching class, provided such a spindle exists. 1.
Spindleconfigurations of skew lines
"... This paper is a contribution to the classification of configurations of skew lines, as studied mainly by Viro and his collaborators. We prove an improvement of a conjecture made by Crapo and Penne which characterizes isotopy classes of skew configurations with spindlestructure. By this result we ca ..."
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This paper is a contribution to the classification of configurations of skew lines, as studied mainly by Viro and his collaborators. We prove an improvement of a conjecture made by Crapo and Penne which characterizes isotopy classes of skew configurations with spindlestructure. By this result we can define an invariant, spindlegenus, for spindleconfigurations. 57M25, 57Q37, 57Q45; 51H10, 57M27 1
HOMOTOPY TYPES OF COMPLEMENTS OF 2ARRANGEMENTS IN R 4
 TOPOLOGY
, 2000
"... We study the homotopy types of complements of arrangements of n transverse planes in R 4, obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2arrangement in R^4 is not determined by the cohom ..."
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We study the homotopy types of complements of arrangements of n transverse planes in R 4, obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2arrangement in R^4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.
Homotopy types of complements of 2arrangements in R^4
 Topology
, 2000
"... We study the homotopy types of complements of arrangements of n transverse planes in R^4, obtaining a complete classification for n <= 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2arrangement in R^4 is not determined by the co ..."
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We study the homotopy types of complements of arrangements of n transverse planes in R^4, obtaining a complete classification for n <= 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2arrangement in R^4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.