Stable equivalence of real projective configurations

Khashin, Sergeĭ I.; Mazurovskii, V. F.

doi:10.1090/trans2/173/11

Cited by 4 publications

(13 citation statements)

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“…In other words (see [3]), the stable equivalence of configurations of half dimension coincides with the homology equivalence.…”

Section: E+(n K) and E_(n K)mentioning

confidence: 95%

“…Proposition [3]. Let n be an even natural number, let V be an oriented space of dimension 2n, and let q E S2V * be a nonsingular quadratic form of signature (n, n).…”

Section: 12mentioning

confidence: 99%

“…Definitions [3]. A nonsingular matriz configuration of type (n, k) is an ordered collection {m0, ..., rnk-1 E Mn} of k matrices such that det(rni -rnj) ~ 0 Vi,j, i ~ j.…”

Section: Matrix Configurationsmentioning

confidence: 99%

“…Theorem [3]. It is easy to check that the graph F which is constructed by the vector configuration {A0,... ,Ak} corresponding to a matrix configuration {m0,... , mk-t } has the form rid = sign (det(rni -mi) ) .…”

Section: Matrix Configurationsmentioning

confidence: 99%

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Projective graph theory and configurations of lines

Khashin¹

1998

J Math Sci

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The spaces of disjoint configurations of k-dimensional subspaces in NP 2k+1 (for ezample, lines in RP a) are studied. These spaces are modeled by various simplicial schemes, and the homology groups of the latter are computed in certain cases. We use the fact that every configuration can be assigned a so-called projective graph, which is a class of graphs with respect to a certain equivalence relation. Bibliography: 5 titles.

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“…In other words (see [3]), the stable equivalence of configurations of half dimension coincides with the homology equivalence.…”

Section: E+(n K) and E_(n K)mentioning

confidence: 95%

“…Proposition [3]. Let n be an even natural number, let V be an oriented space of dimension 2n, and let q E S2V * be a nonsingular quadratic form of signature (n, n).…”

Section: 12mentioning

confidence: 99%

“…Definitions [3]. A nonsingular matriz configuration of type (n, k) is an ordered collection {m0, ..., rnk-1 E Mn} of k matrices such that det(rni -rnj) ~ 0 Vi,j, i ~ j.…”

Section: Matrix Configurationsmentioning

confidence: 99%

Section: Matrix Configurationsmentioning

confidence: 99%

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Projective graph theory and configurations of lines

Khashin¹

1998

J Math Sci

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“…The associated switching class or homology equivalence class is independent of labelings and orientations. A result of Khashin and Mazurovskii, Theorem 3.2 in [7], states that homology-equivalent spindles (spindles defining the same switching class) are isotopic. We have thus: Isotopy classes of spindle-configurations have thus an easy combinatorial description and can be considered as "understood", either in terms of spindle-equivalence classes or in terms of switching classes, in contrast to the general case where no (provenly) complete invariants are available.…”

Section: Introductionmentioning

confidence: 99%

Spindle-configurations of skew lines

Bacher

Garber

2007

Geom. Topol.

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We prove a conjecture of Crapo and Penne which characterizes isotopy classes of skew configurations with spindlestructure. We use this result in order to define an invariant, spindle-genus, for spindle-configurations.We also slightly simplify 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 which constructs a spindle in a given switching class, or proves non-existence of such a spindle.Corresponding author.

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