Configuration Spaces

Frontmatter

On the structure of spaces of commuting elements in compact Lie groups

Abstract

In this note we study topological invariants of the spaces of homomorphisms Hom(π, G), where π is a finitely generated abelian group and G is a compact Lie group arising as an arbitrary finite product of the classical groups SU(r), U(q) and Sp(k).

Alejandro Adem, José Manuel Gómez

On the fundamental group of the complement of two real tangent conics and an arbitrary number of real tangent lines

Abstract

We compute the simplified presentations of the fundamental groups of the complements of the family of real conic-line arrangements with up to two conics which are tangent to each other at two points, with an arbitrary number of tangent lines to both conics. All the resulting groups turn out to be big.

Meirav Amram, David Garber, Mina Teicher

Intersection cohomology of a rank one local system on the complement of a hyperplane-like divisor

Abstract

Under a certain condition A we give a construction to calculate the intersection cohomology of a rank one local system on the complement to a hyper-plane-like divisor.

Dimitry Arinkin, Alexander Varchenko

A survey of some recent results concerning polyhedral products

Abstract

The purpose of this article is an exposition of recent results and applications arising from decompositions of suspensions of generalized moment-angle complexes also known as polyhedral products. Special examples of these spaces are complements of arrangements of complex coordinate planes as described in detail below.

The main features of this expository article are (1) a description of these decompositions together with (2) applications to the structure of the cohomology rings of these spaces. Definitions, examples, as well as one proof for a classical decomposition are included in this report.

Anthony P. Bahri, Martin Bendersky, Frederick R. Cohen, Samuel Gitler

Characters of fundamental groups of curve complements and orbifold pencils

Abstract

The present work is a user’s guide to the results of [7], where a description of the space of characters of a quasi-projective variety was given in terms of global quotient orbifold pencils.

Below we consider the case of plane curve complements. In particular, an infinite family of curves exhibiting characters of any torsion and depth 3 will be discussed. Also, in the context of line arrangements, it will be shown how geometric tools, such as the existence of orbifold pencils, can replace the group theoretical computations via fundamental groups when studying characters of finite order, specially order two. Finally, we revisit an Alexander-equivalent Zariski pair considered in the literature and show how the existence of such pencils distinguishes both curves.

Enrique Artal Bartolo, Jose Ignacio Cogolludo-Agustín, Anatoly Libgober

Analytic continuation of a parametric polytope and wall-crossing

Abstract

We define a set theoretic “analytic continuation” of a polytope defined by inequalities. For the regular values of the parameter, our construction coincides with the parallel transport of polytopes in a mirage introduced by Varchenko. We determine the set-theoretic variation when crossing a wall in the parameter space, and we relate this variation to Paradan’s wall-crossing formulas for integrals and discrete sums. As another application, we refine the theorem of Brion on generating functions of polytopes and their cones at vertices. We describe the relation of this work with the equivariant index of a line bundle over a toric variety and Morelli constructible support function.

Nicole Berline, Michèle Vergne

Embeddings of braid groups into mapping class groups and their homology

Abstract

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that the standard generators of the braid group are not mapped to Dehn twists.

Carl-Friedrich Bödigheimer, Ulrike Tillmann

The cohomology of the braid group B 3 and of SL 2(ℤ) with coefficients in a geometric representation

Abstract

This article is a short version of a paper which addresses the cohomology of the third braid group and of SL₂(ℤ) with coefficients in geometric representations. We give precise statements of the results, some tools and some proofs, avoiding very technical computations here.

Filippo Callegaro, Frederick R. Cohen, Mario Salvetti

Pure braid groups are not residually free

Abstract

We show that the Artin pure braid group P _n is not residually free for n ≥ 4. Our results also show that the corank of P _n is equal to 2 for n ≥ 3.

Daniel C. Cohen, Michael Falk, Richard Randell

Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements

Abstract

We investigate the interplay between the monodromy and the Deligne mixed Hodge structure on the Milnor fiber of a homogeneous polynomial. In the case of hyperplane arrangement Milnor fibers, we obtain a new result on the possible weights. For line arrangements, we prove in a new way the fact due to Budur and Saito that the spectrum is determined by the weak combinatorial data, and show that such a result fails for the Hodge-Deligne polynomials. In an appendix, we also establish a connection between the Hodge-Deligne polynomials and rational points over finite fields.

Alexandru Dimca, Gus Lehrer

The contravariant form on singular vectors of a projective arrangement

Abstract

We define the flag space and space of singular vectors for an arrangement A of hyperplanes in projective space equipped with a system of weights a: A → ℂ We show that the contravariant bilinear form of the corresponding weighted central arrangement induces a well-defined form on the space of singular vectors of the projectivization. If ∑_H∊A a(H) = 0, this form is naturally isomorphic to the restriction to the space of singular vectors of the contravariant form of any affine arrangement obtained from A by dehomogenizing with respect to one of its hyperplanes.

Michael J. Falk, Alexander N. Varchenko

Fox-Neuwirth cell structures and the cohomology of symmetric groups

Abstract

We use the Fox-Neuwirth cell structure for one-point compactifications of configuration spaces as the starting point for understanding our recent calculation of the mod-two cohomology of symmetric groups. We then use that calculation to give short proofs of classical results on this cohomology due to Nakaoka and to Madsen.

Chad Giusti, Dev Sinha

Basic questions on Artin-Tits groups

Abstract

This paper is a short survey on four basic questions on Artin-Tits groups: the torsion, the center, the word problem, and the cohomology (K(π, 1) problem). It is also an opportunity to prove three new results concerning these questions: (1) if all free of infinity Artin-Tits groups are torsion free, then all Artin-Tits groups will be torsion free; (2) If all free of infinity irreducible non-spherical type Artin-Tits groups have a trivial center then all irreducible non-spherical type Artin-Tits groups will have a trivial center; (3) if all free of infinity Artin-Tits groups have solutions to the word problem, then all Artin-Tits groups will have solutions to the word problem. Recall that an Artin-Tits group is free of infinity if its Coxeter graph has no edge labeled by ∞.

Eddy Godelle, Luis Paris

Rational cohomology of the real Coxeter toric variety of type A

Abstract

The toric variety corresponding to the Coxeter fan of type A can also be described as a De Concini-Procesi wonderful model. Using a general result of Rains which relates cohomology of real De Concini-Procesi models to poset homology, we give formulas for the Betti numbers of the real toric variety, and the symmetric group representations on the rational cohomologies. We also show that the rational cohomology ring is not generated in degree 1.

Anthony Henderson

Arrangements stable under the Coxeter groups

Abstract

Let B be a real hyperplane arrangement which is stable under the action of a Coxeter group W. Then W acts naturally on the set of chambers of B. We assume that B is disjoint from the Coxeter arrangement A = A (W) of W. In this paper, we show that the W-orbits of the set of chambers of B are in one-to-one correspondence with the chambers of C = B ∪ B which are contained in an arbitrarily fixed chamber of A. From this fact, we find that the number of W-orbits of the set of chambers of B is given by the number of chambers of C divided by the order of W. We will also study the set of chambers of C which are contained in a chamber b of B. We prove that the cardinality of this set is equal to the order of the isotropy subgroup W _b of b. We illustrate these results with some examples, and solve an open problem in [H. Kamiya, A. Takemura, H. Terao, Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. 47 (2011) 379–400] by using our results.

Hidehiko Kamiya, Akimichi Takemura, Hiroaki Terao

Quantum and homological representations of braid groups

Abstract

By means of a description of the solutions of the KZ equation using hypergeometric integrals we show that the homological representations of the braid groups studied by Lawrence, Krammer and Bigelow are equivalent at generic complex values to the monodromy of the KZ equation with values in the space of null vectors in the tensor product of Verma modules of sl₂(C).

Toshitake Kohno

Cohomology of the complement to an elliptic arrangement

Abstract

We consider the complement to an arrangement of hyperplanes in a cartesian power of an elliptic curve and describe its cohomology with coefficients in a nontrivial rank one local system.

Andrey Levin, Alexander Varchenko

Residual nilpotence for generalizations of pure braid groups

Abstract

It is known that the pure braid groups are residually torsion-free nilpotent. This property is however widely open for the most obvious generalizations of these groups, like pure Artin groups and like fundamental groups of hyperplane complements (even reflection ones). In this paper we relate this problem to the faithfulness of linear representations, and prove the residual torsion-free nilpotence for a few other groups.

Ivan Marin

Some topological problems on the configuration spaces of Artin and Coxeter groups

Abstract

In the first part we review some topological and algebraic aspects in the theory of Artin and Coxeter groups, both in the finite and infinite case (but still, finitely generated). In the following parts, among other things, we compute the Schwartz genus of the covering associated to the orbit space for all affine Artin groups. We also give a partial computation of the cohomology of the braid group with non-abelian coefficients coming from geometric representations. We introduce an interesting class of “sheaves over posets”, which we call “weighted sheaves over posets”, and use them for explicit computations.

Davide Moroni, Mario Salvetti, Andrea Villa

Chromatic quasisymmetric functions and Hessenberg varieties

Abstract

We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian polynomials, the one in symmetric function theory deals with a refinement of the chromatic symmetric functions of Stanley, and the one in algebraic geometry deals with Tymoczko’s representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some remarkable connections between these topics.

John Shareshian, Michelle L. Wachs

Geometric and homological finiteness in free abelian covers

Abstract

We describe some of the connections between the Bieri-Neumann-Strebel-Renz invariants, the Dwyer-Fried invariants, and the cohomology support loci of a space X. Under suitable hypotheses, the geometric and homological finiteness properties of regular, free abelian covers of X can be expressed in terms of the resonance varieties, extracted from the cohomology ring of X. In general, though, translated components in the characteristic varieties affect the answer. We illustrate this theory in the setting of toric complexes, as well as smooth, complex projective and quasi-projective varieties, with special emphasis on configuration spaces of Riemann surfaces and complements of hyperplane arrangements.

Alexander I. Suciu

Minimal stratifications for line arrangements and positive homogeneous presentations for fundamental groups

Abstract

The complement of a complex hyperplane arrangement is known to be homotopic to a minimal CW complex. There are several approaches to the minimality. In this paper, we restrict our attention to real two dimensional cases, and introduce the “dual” objects so called minimal stratifications. The strata are explicitly described as semialgebraic sets. The stratification induces a partition of the complement into a disjoint union of contractible spaces, which is minimal in the sense that the number of codimension k pieces equals the k-th Betti number.

We also discuss presentations for the fundamental group associated to the minimal stratification. In particular, we show that the fundamental groups of complements of a real arrangements have positive homogeneous presentations.

Masahiko Yoshinaga

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