On the Homotopy Theory of Arrangements

Falk, Michael; Randell, Richard

doi:10.2969/aspm/00810101

Cited by 46 publications

(76 citation statements)

References 12 publications

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“…An arrangement A in C n is called a K(π, 1) arrangement if the complement C n − H∈A H is a K(π, 1) space. All K(π, 1) arrangements are formal [7]. It would be interesting to know which (if any) discriminantal arrangements are K(π, 1).…”

Section: Rfmentioning

confidence: 99%

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Bayer

Brandt

1997

Journal of Algebraic Combinatorics

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Abstract. Manin and Schechtman defined the discriminantal arrangement of a generic hyperplane arrangement as a generalization of the braid arrangement. This paper shows their construction is dual to the fiber zonotope construction of Billera and Sturmfels, and thus makes sense even when the base arrangement is not generic. The hyperplanes, face lattices and intersection lattices of discriminantal arrangements are studied. The discriminantal arrangement over a generic arrangement is shown to be formal (and in some cases 3-formal), though it is in general not free. An example of a free discriminantal arrangement over a generic arrangement is given.

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Section: Rfmentioning

confidence: 99%

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Bayer

Brandt

1997

Journal of Algebraic Combinatorics

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“…Such "nice" groups include nilpotent groups, free groups, and nilpotent extensions involving these groups. However, there are arrangements [5] with K(ir, 1) complements which are not rational K(ir, 1). It remains to be seen whether the results of this paper have any bearing on the various conjectures [5] concerning K(ir, 1) arrangements.…”

Section: Introductionmentioning

confidence: 99%

“…However, there are arrangements [5] with K(ir, 1) complements which are not rational K(ir, 1). It remains to be seen whether the results of this paper have any bearing on the various conjectures [5] concerning K(ir, 1) arrangements. In particular, the conjecture "rational K(-k, 1) implies K(rr, 1)" of [5] has not been resolved.…”

Section: Introductionmentioning

confidence: 99%

“…The connections between the topology of M and the combinatorial geometry of A are the source of much current research in this area. The most successful investigations concern the cohomology of M [13,15], whereas the most difficult unsolved problems involve the homotopy groups of M [5]. 1 n this paper we study the link between cohomology and homotopy provided by Sullivan's theory of minimal models [14].…”

Section: Introductionmentioning

confidence: 99%

“…Because the sequence <p"(M) is related to the 1-minimal model 5? of M, we conjecture in [5] that the LCS formula holds precisely when S? determines H*(M).…”

Section: Introductionmentioning

confidence: 99%

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The Minimal Model of the Complement of an Arrangement of Hyperplanes

Falk¹

1988

Transactions of the American Mathematical Society

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ABSTRACT. In this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory. Let A be a finite collection of hyperplanes in C', and let M = C' -UHeA H. We say A is a rational K(ir, 1) arrangement if the rational completion of M is aspherical. For these arrangements an identity (the LCS formula) is established relating the lower central series of tt\ (M) to the cohomology of M. This identity was established by group-theoretic means for the class of fiber-type arrangements in previous work. We reproduce this result by showing that the class of rational K(ir, 1) arrangements contains all fiber-type arrangements. This class includes the reflection arrangements of types A/ and B[. There is much interest in arrangements for which Af is a K(ir, 1) space. The methods developed here do not apply directly because M is rarely a nilpotent space. We give examples of K(ir, 1) arrangements which are not rational K(tt,1) for which the LCS formula fails, and K(tt, 1) arrangements which are not rational K(tt, 1) where the LCS formula holds. It remains an open question whether rational K(it, 1) arrangements are necessarily K(ir, 1).

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Line-Closed Matroids, Quadratic Algebras, and Formal Arrangments

Falk

2002

Advances in Applied Mathematics

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Let G be a matroid on ground set . The Orlik-Solomon algebra A G is the quotient of the exterior algebra on by the ideal generated by circuit boundaries. The quadratic closure A G of A G is the quotient of by the ideal generated by the degree-two component of . We introduce the notion of the nbb set in G, determined by a linear order on , and show that the corresponding monomials are linearly independent in the quadratic closure A G . As a consequence, A G is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. ]. These results generalize to the degree r closure of G .The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for to be free and for the complement M of to be a K π 1 space. Formality of is also necessary for A G to be a quadratic algebra. We clarify the relationship between formality, lineclosure, and other matroidal conditions related to formality. We give examples to show that line-closure of G is not necessary or sufficient for M to be a K π 1 or for to be free.  2002 Elsevier Science (USA) 250 0196-8858/02 $35.00  2002 Elsevier Science (USA) All rights reserved.

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On the Homotopy Theory of Arrangements

Abstract: In this paper an arrangement d is a finite collection of hyperplanes {HI, .. " Hn} through the origin in ct. We wish to examine the complementary space M = c! -U~=lHt from a topological point of view. More

Cited by 46 publications

References 12 publications

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The Minimal Model of the Complement of an Arrangement of Hyperplanes

Line-Closed Matroids, Quadratic Algebras, and Formal Arrangments

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