Wonderful models of subspace arrangements

Concini, Corrado De; Procesi, Claudio

doi:10.1007/bf01589496

Cited by 283 publications

(470 citation statements)

References 6 publications

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“…The second task is to define correctly the residues which will be integrals over suitable cycles. Since we are in dimension > 1 we are faced with the problem that the poles are on divisors with a complicated intersection pattern, this implies that we need to use a model where the divisor has normal crossings as in [8]. This is done in §6.…”

Section: Theorem 52 the Subalgebra Of The Algebra Of Differential Fomentioning

confidence: 99%

On the geometry of toric arrangements

Concini

Procesi

2005

Transformation Groups

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127

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Section: Theorem 52 the Subalgebra Of The Algebra Of Differential Fomentioning

confidence: 99%

On the geometry of toric arrangements

Concini

Procesi

2005

Transformation Groups

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127

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“…⊲ Using rational models De Concini and Procesi derived that the multiplicative structure of the rational cohomology in the case of complex arrangements is determined by combinatorial data: intersection lattice and dimension function [CP95]. Their techniques were applied by Yuzvinsky to give an explicit description for the rational cohomology ring for complex arrangements [Yuz98].…”

Section: Introductionmentioning

confidence: 99%

The cohomology rings of complements of subspace arrangements

Longueville¹,

Schultz²

2001

Math Ann

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Abstract. The ring structure of the integral cohomology of complements of real linear subspace arrangements is considered. While the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of Goresky and MacPherson, we describe the multiplicative structure in terms of the intersection poset, the dimension function and orientations of the participating subspaces for the class of arrangements without pairs of intersections of codimension one. In particular, this yields a description of the integral cohomology ring of complex arrangements conjectured by Yuzvinsky. For general real arrangements a weaker result is obtained. The approach is geometric and the methods are elementary.

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“…In the case of rational coe¯-cients, this was proved in De Concini & Procesi (1995), and, in the more complicated integral case, in Deligne et al (2000) and de Longueville & Schultz (2001) with the help of some ideas from Yuzvinsky (1998).…”

Section: (E) Multiplication In Cohomologymentioning

confidence: 91%

Homology of spaces of knots in any dimensions

Vassiliev

2001

Philosophical Transactions of the Royal Society of London. Seri

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I shall describe the recent progress in the study of cohomology rings of spaces of knots in R n , H ¤ (fknots in R n g), with arbitrary n > 3.`Any dimensions' in the title can be read as dimensions n of spaces R n , as dimensions i of the cohomology groups H i , and also as a parameter for di¬erent generalizations of the notion of a knot.An important subproblem is the study of knot invariants. In our context, they appear as zero-dimensional cohomology classes of the space of knots in R 3 . It turns out that our more general problem is never less beautiful. In particular, nice algebraic structures arising in the related homological calculations have equally (or maybe even more) compact description, of which the classical`zero-dimensional' part can be obtained by easy factorization.There are many good expositions of the theory of related knot invariants. Therefore, I shall deal almost completely with results in higher (or arbitrary) dimensions.Keyword s: op erads; order complexes; simplicial resolutions; combinatorial formulae; sp ectral sequences Main constructionWe consider both the standard compact knots, i.e. smooth embeddings S 1 ! R n , and the long knots, i.e. embeddings R 1 ! R n coinciding with a standard linear embedding outside some compact subset in R 1 (see gure 1). The study of the latter space is more essential, because the algebraic structure of the cohomology ring of the space of standard knots is built from that of the similar ring for long knots (here playing the role of the`coe¯cient ring'), the topological non-triviality of the circle S 1 , and a certain family of its con guration spaces.Let us denote by K the space of all smooth maps S 1 ! R n (respectively, of maps R 1 ! R n with such boundary conditions). This is a linear (respectively, an a¯ne) space. The discriminant § » K is the set of all maps which are not smooth embeddings, i.e. have either self-intersections or singular points. The space of knots is the di¬erence K n § . (a) Arnold's reductionIt is convenient to study the cohomology group of the space of knots by a sort of Alexander duality,The bar in the notation · H ¤ means that we consider Borel{Moore homology, i.e. the homology group of the one point compacti cation, and n1 is the notation for

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Wonderful models of subspace arrangements

Cited by 283 publications

References 6 publications

On the geometry of toric arrangements

On the geometry of toric arrangements

The cohomology rings of complements of subspace arrangements

Homology of spaces of knots in any dimensions

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