arXiv subjects

Emanuele Delucchi

Publications and source records attributed to Emanuele Delucchi.

36 records · Page 2Linked to original sources

arXiv2017

Group actions on semimatroids

We initiate the study of group actions on (possibly infinite) semimatroids and geometric semilattices. To every such action is naturally associated an orbit-counting function, a two-variable "Tutte" polynomial and a poset which, in the realizable case, coincides with the poset of connected components of intersections of the associated toric arrangement. In this structural framework we recover and strongly generalize many enumerative results about arithmetic matroids, arithmetic Tutte polynomials and toric arrangements by finding new combinatorial interpretations beyond the realizable case. In particular, we thus find a class of natural examples of nonrealizable arithmetic matroids. Moreover, under additional conditions these actions give rise to a matroid over the ring of integers. As a stepping stone toward our results we also prove an extension of the cryptomorphism between semimatroids and geometric semilattices to the infinite case.

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arXiv2016

Modular elimination in matroids and oriented matroids

We introduce a new axiomatization of matroid theory that requires the elimination property only among modular pairs of circuits, and we present a cryptomorphic phrasing thereof in terms of Crapo's axioms for flats. This new point of view leads to a corresponding strengthening of the circuit axioms for oriented matroids.

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arXiv2016

Colorings and flows on CW complexes, Tutte quasi-polynomials and arithmetic matroids

In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials encompasses invariants defined for CW complexes by Beck-Breuer-Godkin-Martin and Duval-Klivans-Martin. Furthermore, we answer a question by Bajo-Burdick-Chmutov, concerning the modified Tutte-Krushkal-Renhardy polynomials defined by these authors: to this end, we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function.

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arXiv2015

Bergman Complexes of Lattice Path Matroids

We give an explicit description of the poset of cells of Bergman complexes of Lattice Path Matroids and establish a criterion for its simpliciality, in terms of the shape of the bounding paths.

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arXiv2015

The integer cohomology algebra of toric arrangements

We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the results to full generality. As a byproduct we obtain: -a "combinatorial" version of Brieskorn's lemma in terms of Salvetti complexes of complexified arrangements, -a uniqueness result for realizations of arithmetic matroids with at least one basis of multiplicity 1.

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arXiv2015

Cryptomorphisms for abstract rigidity matroids

This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of circuits, cocircuits, bases, hyperplanes). Moreover, the study of hyperplanes in abstract rigidity matroids leads us to state (and support with significant evidence) a conjecture about characterizing the class of abstract rigidity matroids by means of certain "prescribed substructures". We then prove a recursive version of this conjecture.

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arXiv2013

An equivariant discrete model for complexified arrangement complements

We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M is the oriented matroid of an arrangement A of linear hyperplanes in real space, the Z_4 action corresponds to the diagonal action of C* on the complement M of the complexification of A: |Q| is equivariantly homotopy-equivalent to M under the identification of Z_4 with {+1, i, -1, -i}, and |Q(M, e)| is homotopy-equivalent to the complement of the decone of A relative to the hyperplane corresponding to e. All constructions and arguments are carried out at the level of the underlying posets. If a group G acts on the set of topes of M preserving adjacency, then G acts simplicially on |Q|. We also show that the class of fundamental groups of such complexes is strictly larger than the class of fundamental groups of complements of complex hyperplane arrangements. Specifically, the group of the non-Pappus arrange- ment is not isomorphic to any realizable arrangement group.

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arXiv2013

Minimality of toric arrangements

We prove that the complement of a toric arrangement has the homotopy type of a minimal CW complex. As a corollary we obtain that the integer cohomology of these spaces is torsion free. We use Discrete Morse Theory, providing a sequence of cellular collapses that leads to a minimal complex.

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arXiv2012

Combinatorial polar orderings and recursively orderable arrangements

Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal complex is particularly easy, called {\em recursively orderable} arrangements, can therefore be combinatorially defined. We initiate the study of this class, giving a complete characterization in dimension 2 and proving that every supersolvable complexified arrangement is recursively orderable.

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arXiv2012

Foundations for a theory of complex matroids

We explore a combinatorial theory of linear dependency in complex space, "complex matroids", with foundations analogous to those for oriented matroids. We give multiple equivalent axiomatizations of complex matroids, showing that this theory captures properties of linear dependency, orthogonality, and determinants over C in much the same way that oriented matroids capture the same properties over R. In addition, our complex matroids come with a canonical circle action analogous to the action of C* on a complex vector space. Our phirotopes (analogues of determinants) are the same as those studied previously by Below, Krummeck, and Richter-Gebert and by Delucchi. We further show that complex matroids cannot have vector axioms analogous to those for oriented matroids.

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arXiv2011

Face vectors of subdivided simplicial complexes

Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then prove an interesting symmetry of these roots about the real number -2. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric motivation to generalize the combinatorial statements to arbitrary subdivision methods: any subdivision method will exhibit similar limit behavior and symmetry. Our techniques allow us to compute explicit formulas for the values of the limit roots in the case of barycentric subdivision.

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arXiv2011

A Salvetti complex for Toric Arrangements and its fundamental group

We describe a combinatorial model for the complement of a complexified toric arrangement by using nerves of acyclic categories. This generalizes recent work of Moci and Settepanella on thick toric arrangements. Moreover, we compute its fundamental group by finding a (finite) presentation.

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arXiv2007

Shelling-type orderings of regular CW-complexes and acyclic matchings of the Salvetti complex

Motivated by the work of Salvetti and Settepanella we introduce certain total orderings of the faces of any shellable regular CW-complex (called `shelling-type orderings') that can be used to explicitly construct maximum acyclic matchings of the poset of cells of the given complex. Building on an application of this method to the classical zonotope shellings we describe a class of maximum acyclic matchings for the Salvetti complex of a linear complexified arrangement. To do this, we introduce and study a new combinatorial stratification of the Salvetti complex. For the obtained acyclic matchings we give an explicit description of the critical cells that depends only on the chosen linear extension of the poset of regions. It is always possible to choose the linear extension so that the critical cells can be explicitly constructed from the chambers of the arrangement via the bijection to no-broken-circuit sets defined by Jewell and Orlik. Our method can be generalized to arbitraty oriented matroids.

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arXiv2006

Combinatorial remarks on a classical theorem of Deligne

We examine Deligne's classical proof of the asphericity of simplicial arrangements from the viewpoint of the combinatorics of the poset of regions of the arrangement. This turns out to be very natural. In particular, we show that an arrangement is simplicial only if it satisfies Deligne's property on positive paths, thus answering a question posed by Luis Paris.

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arXiv2006

Nested set complexes of Dowling lattices and complexes of Dowling trees

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice Q(G) and of its subposet of the G-symmetric partitions Q_G which was recently introduced by Hultman together with the complex of G-symmetric phylogenetic trees T_G. Hultman shows that T_G and Q_G are homotopy equivalent and Cohen-Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov shows that in fact T_G is subdivided by the order complex of Q_G. We introduce the complex of Dowling trees T(G) and prove that it is subdivided by the order complex of Q(G) and contains T_G as a subcomplex. We show that T(G) is obtained from T_G by successive coning over certain subcomplexes. We explicitly and independently calculate how many homology spheres are added in passing from T_G to T(G).

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arXiv2006

Simplicial shellable spheres via combinatorial blowups

The construction of the Bier sphere Bier(K) for a simplicial complex K is due to Bier. Björner, Paffenholz, Sjöstrand and Ziegler generalize this construction to obtain a Bier poset Bier(P,I) from any bounded poset P and any proper ideal I of P. They show shellability of Bier(P,I) for the case where P is the boolean lattice, and obtain thereby 'many shellable spheres' in the sense of Kalai. We put the Bier construction into the general framework of the theory of nested set complexes of Feichtner and Kozlov. We obtain 'more shellable spheres' by proving the general statement that combinatorial blowups, hence stellar subdivisions, preserve shellability.

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arXiv2005

Subdivision of complexes of k-Trees

Consider the poset of partitions of {1,...(n-1)k+1} with block sizes congruent to 1 modulo k. We prove that its order complex is a subdivision of the complex of k-trees, thereby answering a question posed by Feichtner. The result is obtained by an ad-hoc generalization of concepts from the theory of nested set complexes to non-lattices.

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arXiv2004

Diagram models for the covers of the Salvetti complex

To every affine real arrangement of hyperplanes we associate a family of diagrams of spaces over the face poset of the arrangement. We show that any cover of the complement of the complexification of the arrangement is homotopy equivalent to the homotopy colimit of one of the diagrams. More precisely, we show that any cover of the Salvetti complex is isomorphic to the order complex of the poset limit of one of the diagrams. We thus obtain explicit simplicial models for covers of the Salvetti complex.

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