Eva Maria Feichtner, Sergey Yuzvinsky
We show that, for an arrangement of subspaces in a complex vector space with geometric intersection lattice, the complement of the arrangement is formal. We prove that the Morgan rational model for such an arrangement complement is formal as a differential graded algebra.
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We prove a "purity implies formality" statement in the context of the rational homotopy theory of smooth complex algebraic varieties, and apply it to complements of hypersurface arrangements. In particular, we prove that the complement of a toric arrangement is formal. This is analogous to the classical formality theorem for complements of hyperplane arrangements, due to Brieskorn, and generalizes a theorem of De Concini and Procesi.
We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont's rational formality result to formality over $\mathbb Z$. The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the ar…
We explore the relation between cohomology jump loci in a finite Galois cover, formality properties and algebraic monodromy action. We show that the jump loci of the base and total space are essentially the same, provided the base space is 1-formal and the monodromy action in degree 1 is trivial. We use reduced multinet structures on line arrangements to construct components of the first characteristic variety of the Milnor fiber in degree 1, and to prove that the monodromy action is non-trivial in degree 1. For an arbitrary line arrangement, we prove that the triviality of the monodromy in de…
Let $G$ be a matroid on ground set \A. The Orlik-Solomon algebra $A(G)$ is the quotient of the exterior algebra \E on \A by the ideal \I generated by circuit boundaries. The quadratic closure $\bar{A}(G)$ of $A(G)$ is the quotient of \E by the ideal generated by the degree-two component of \I. We introduce the notion of \nbb set in $G$, determined by a linear order on \A, and show that the corresponding monomials are linearly independent in the quadratic closure $\bar{A}(G)$. As a consequence, $A(G)$ is a quadratic algebra only if $G$ is line-closed. An example of S.~Yuzvinsky proves the conve…
We describe the constructible derived category of sheaves on the $n$-sphere, stratified in a point and its complement, as a dg module category of a formal dg algebra. We prove formality by exploring two different methods: As a combinatorial approach, we reformulate the problem in terms of representations of quivers and prove formality for the 2-sphere, for coefficients in a principal ideal domain. We give a suitable generalization of this formality result for the 2-sphere stratified in several points and their complement. As a geometric approach, we give a description of the underlying dg alge…
To any graph and smooth algebraic curve $C$ one may associate a "hypercurve" arrangement and one can study the rational homotopy theory of the complement $X$. In the rational case ($C=\mathbb{C}$), there is considerable literature on the rational homotopy theory of $X$, and the trigonometric case ($C = \mathbb{C}^\times$) is similar in flavor. The case of when $C$ is a smooth projective curve of positive genus is more complicated due to the lack of formality of the complement. When the graph is chordal, we use quadratic-linear duality to compute the Malcev Lie algebra and the minimal model of …
Hyperplane arrangements with non-formal Milnor fibers
本文提出基于multinet结构的组合条件，用于判定超平面排列的Milnor纤维非形式，并构造了具有该性质的无限族单项排列。
On the connection between coordinate and diagonal arrangement complements
研究对角排列补集D(K)的性质，证明坐标排列补集U(K)是D(K)的双悬垂，实数域上则为单悬垂。
We consider the arrangements of subtori in a flat d - dimensional torus T. Let us consider an arrangement on n subtori of codimension one, let f be the number of connected components of the complement in T to the union of subtori. We found the set of all possible numbers f for given n and d and arbitrary arrangements of subtori.
We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of the arrangement.
The rational (non-)formality of the non-3-equal manifolds
研究显示，当d≥2时，M^{(3)}_d(n)在n≤6时有理形式，而n>6时非有理形式，与经典配置空间不同。
In combinatorial problems it is sometimes possible to define a $G$-equivariant mapping from a space $X$ of configurations of a system to a Euclidean space $\mathbb{R}^m$ for which a coincidence of the image of this mapping with an arrangement $\mathcal{A}$ of linear subspaces insures a desired set of linear conditions on a configuration. Borsuk-Ulam type theorems give conditions under which no $G$-equivariant mapping of $X$ to the complement of the arrangement exist. In this paper, precise conditions are presented which lead to such theorems through a spectral sequence argument. We introduce a…
The complement of the codimension 2 complex coordinate subspace arrangement is shown to be homotopy equivalent to a wedge of spheres.
We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of intersections. We also associate a topological space: the complement of the union of tangent bundles of these subspheres in the tangent bundle of the ambient sphere. We call this space the tangent bundle complement. As in the case of hyperplane arrangements the aim of this new notion is to understand the interaction between the combinatorics of the intersections an…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of calculating the obstructions for the existence of equivariant maps. A variety of techniques are introduced and discussed with the emphasis on concrete and explicit calculations. This eventually leads (Theorems 18 and 19) to an almost exhaustive analysis of when such maps do or do not exist in the particular case of interest.
The homotopy type of the complement of a complex coordinate subspace arrangement is studied by fathoming out the connection between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy equivalent to a wedge of spheres is described. One consequence is an application in commutative algebra: certain local rings are proved to be Golod, that is, all Massey products in their homology vanish.
Topological line arrangements and their topological invariants
研究拓扑线排列补集拓扑，用同调方法证其补集上同调环同构于Orlik - Solomon代数，还研究补集同伦型，发现辛线排列补集有特定同伦型，且拓扑线排列组合类型的补集有多种非同伦等价情况。
We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, and we give an explicit Zπ_1-module presentation of π_p, the first non-vanishing higher homotopy group. We also give a combinatorial formula for the π_1-coinvariants of π_p. For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of π_2, and study some of the pro…
In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on the poset of layers. The analysis of discriminantal toric arrangements permits us to isolate certain conditions under which two toric arrangements have diffeomorphic complements. We also give combinatorial conditions determining whether the cohomology algebra is generated in degree one.
We study bicolored configurations of points in the Euclidean $n$-space that are constrained to remain either inside or outside a fixed Euclidean $m$-subspace, with $n - m \ge 2$. We define a higher-codimensional variant of the Swiss-Cheese operad, called the complementarily constrained disks operad $\mathsf{CD}_{mn}$, associated to such configurations. The operad $\mathsf{CD}_{mn}$ is weakly equivalent to the operad of locally constant factorization algebras on the stratified space $\{\mathbb{R}^{m} \subset \mathbb{R}^{n}\}$. We prove that this operad is formal over $\mathbb{R}$.
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 line…
Monodromy of supersolvable toric arrangements
研究可解化环面的拓扑特性，揭示其与经典配置空间 bundles 的关系，并确定严格可解化环面补集的不变量，如上同调环和基本群的下中央系列李代数。
We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. The proof uses a new invariant of the fundamental group of the complement to a line arrangement of a given combinatorial type with respect to isomorphisms inducing the canonical isomorphism of the first homology groups.
We consider smooth, complex quasi-projective varieties $U$ which admit a compactification with a boundary which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.
It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined, i.e., they depend uniquely on the underlying matroid. To do this, we associate to every such matroid a topological space, that we call the reduced realization …
To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study arrangements of pseudohyperplanes; they correspond to non-realizable oriented matroids. These arrangements arise as a consequence of the Folkman-Lawrence topological representation theorem. We propose a generalization of the complexification process in this context. In particular we construct a space naturally associated with these pseudo-arrangements which…
Maximal twisted Betti numbers of complex hyperplane arrangement complements
本文证明本质复超平面排列补空间上局部系统的贝蒂数仅在局部系统为常值时最大，正面回答了吉永与第一作者近期提出的问题。
Let A be a geometric arrangement such that codim(x) > 1 for every x in A. We prove that, if the complement space M(A) is rationally hyperbolic, then there exists an injective from a free Lie algebra L(u,v) to the homotopy Lie algebra of M(A).
This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik--Solomon rings, Salvetti complex, matroids, Spanier--Whitehead duality, twisted homology groups, monodromy theory and multidimensional hypergeometric functions. The emphasis on the most geometrical explanation is done; applications and analogies in the differential topology are outlined.
We define a monoid structure on the set of $k$-equal arrangements and use this structure to define limits of braid arrangements. We compute the cohomology of the associated limits of rational models of the arrangements complex complements. We collect these complexes together into one complex by creating a new differential and product on their direct sum and show that the resulting complex is exact.
The Orlik--Solomon algebra of a locally geometric poset and cohomology of complex abelian arrangements
该研究构造了局部几何偏序集的Orlik–Solomon代数，利用其与复阿贝尔李群子群排列的关联，结合拓扑信息计算排列补集的有理上同调，尤其给出紧李群情形下的显式微分分次代数。
We show that if the fundamental groups of the complements of two line arrangements in the complex projective plane are isomorphic to the same direct sum of free groups, then the complements of the arrangements are homotopy equivalent. For any such arrangement, we construct another arrangement that is complexified-real, the intersection lattices of the arrangements are isomorphic, and the complements of the arrangements are diffeomorphic.
We show that the integral cohomology algebra of the complement of a toric arrangement is not determined by the poset of layers. Moreover, the rational cohomology algebra is not determined by the arithmetic matroid (however it is determined by the poset of layers).
Drawing parallels with hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\mathcal{A}$ of locally flat, codimension-1 submanifolds that intersect like hyperplanes. To such a collection we associate two combinatorial objects: the face category and the intersection poset. We also associate a topological space to the arrangement called the tangent bundle complement. It is the complement of union of tangent bundles of these submanifolds inside the tangent bundle of the ambient manifo…
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.
We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.
Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its quadratic closure, we express g_A as a semi-direct product of the well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows us to compute the homotopy Lie algebra associated to the cohomology ring of the complement of a complex hyperplane arrangement, provided some combinatorial assumptions are satisfied. As an application, we give examples…
Non-vanishing of homotopy groups of Manin--Schechtman arrangements
本文研究Manin-Schechtman排列的补集，证明其高维同调群非零，从而推翻其为K(π,1)空间的假设。
Let \A be a complex hyperplane arrangement, and let $X$ be a modular element of arbitrary rank in the intersection lattice of \A. We show that projection along $X$ restricts to a fiber bundle projection of the complement of \A to the complement of the localization $\A_X$ of \A at $X$. The fiber is the decone of a realization of the complete principal truncation of the underlying matroid of \A along the flat corresponding to $X$. This result gives a topological realization of results of Stanley, Brylawsky, and Terao on modular factorization. We show that (generalized) parallel connection of mat…
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.
A subspace arrangement defined by intersections of hyperplanes of the braid arrangement can be encoded by an edge colored hypergraph. It turns out that the characteristic polynomial of this type of subspace arrangement is given by a generalized chromatic polynomial of the associated edge colored hypergraph. The main result of this paper supplies a sufficient condition for the existence of non-trivial Massey products of the subspace arrangements complex complement. This is accomplished by studying a spectral sequence associated to the Lie coalgebras of Sinha and Walter.
Permutohedral complex and complements of diagonal subspace arrangements
本文研究了对角子空间排列补集的同调结构，并通过排列体与立方体的投影关系，揭示了余代数环乘法的构造方法。
We find an explicit combinatorial gradient vector field on the well known complex S (Salvetti complex) which models the complement to an arrangement of complexified hyperplanes. The argument uses a total ordering on the facets of the stratification of R^n associated to the arrangement, which is induced by a generic system of polar coordinates. We give a combinatorial description of the singular facets, finding also an algebraic complex which computes local homology. We also give a precise construction in the case of the braid arrangement.
Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this subspace arrangement are related to those of the complement of the original hyperplane arrangement. In particular, if the hyperplane arrangement is fiber-type, then, apart from grading, the Lie algebra obtained from the descending central series for the fundamental group of the complement of the hyperplane arrangement is isomorphic to the Lie algebra of primitive…
We compute the mixed Hodge structure on the cohomology ring of complements of complex coordinate subspace arrangements. The mixed Hodge structure can be described in terms of the special bigrading on the cohomology ring of complements of complex coordinate subspace arrangements. Originally this bigrading was introduced in the setting of toric topology by V.M. Buchstaber and T.E. Panov.
Let $\mathcal{A} $ be a complexified-real arrangement of lines in $\mathbb{C}^2.$ Let $H$ be any line in $ \mathcal{A} $. Then, form a new complexified-real arrangement $ \mathcal{B}_H = \mathcal{A} \cup \mathcal{C} $ where $ \mathcal{C} \cup \{H\} $ is a pencil of lines with multiplicity $ m\geq 3 $, the intersection point in the pencil is not a multiple point in $ \mathcal{A}, $ and every line in $ \mathcal{C} $ intersects every line in $ \mathcal{A}\setminus \{H\} $ in points of multiplicity two. In this article, we show that for $ H_1, H_2 \in \mathcal{A} $ we may have that $ \mathcal{B}_{…
Rational Homotopy Type of Complements of Submanifold Arrangements
本文通过构造cdga模型，研究了子流形排列补集的有理同调类型，并推广了多个关于排列补集的已知结果。
We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. The main tools are polar curves and the affine Lefschetz theory developped by H. Hamm and A. Némethi. In the special case of the hyperplane arrangements, we strengthen some results due to Orlik and Terao (see Math. Ann. 301(1995)) and obtain an independant proof for the minimality of hyperplane arrangements (see Randell math.AT/0011101 for another proof of this result).
It is shown that the diffeomorphism type of the complement to a real space line arrangement in any dimensional affine ambient space is determined only by the number of lines and the data on multiple points.
The homology groups of finite cyclic covering of line arrangement complement
本文研究复线排列补空间有限循环覆盖的第一整同调群，在类似 Cohen-Dimca-Orlik 的条件下证明其无挠，并将 Williams 关于复化线排列的结果推广到任意复线排列。
Cohomology of $\mathbb{Z}$-local systems on complex hyperplane arrangement complements
本文证明了复超平面排列补集上秩一 $\mathbb{Z}$-局部系的 Cohen-Dimca-Orlik 型定理，从而解决了 S. Sugawara 最近提出的猜想。
We prove that the complement of any affine 2-arrangement in R^d is minimal, that is, it is homotopy equivalent to a cell complex with as many i-cells as its i-th rational Betti number. For the proof, we provide a Lefschetz-type hyperplane theorem for complements of 2-arrangements, and introduce Alexander duality for combinatorial Morse functions. Our results greatly generalize previous work by Falk, Dimca--Papadima, Hattori, Randell, and Salvetti--Settepanella and others, and they demonstrate that in contrast to previous investigations, a purely combinatorial approach suffices to show minimali…
Centering toric arrangements of maximal rank
本文研究最大秩toric排列的补集流形拓扑结构，通过线性代数方法证明其与中心化toric排列同胚，验证其拓扑极小性。
Deletion-restriction is a fundamental tool in the theory of hyperplane arrangements. Various important results in this field have been proved using deletion-restriction. In this paper we use deletion-restriction to identify a class of toric arrangements for which the cohomology algebra of the complement is generated in degree $1$. We also show that for these arrangements the complement is formal in the sense of Sullivan.
The complement of an arrangement A of a finite number of affine hyperplanes in complex n-space has the structure of a poset of spaces indexed by the intersection poset, L(A). The space corresponding to G in L(A) is homotopy equivalent to the complement of the hyperplanes in the central arrangement A_G normal to G. This poset of spaces structure can be used to repair a spectral sequence argument in two earlier papers of Davis, Januszkiewicz, Leary and Okun for computing certain cohomology groups of arrangement complements. Similarly, toric hyperplane arrangements have the structure of a diagram…
We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement. In particular, we solve a conjecture of Drton and Klivans on the characteristic polynomial of finite reflection arrangements. The minimal complex is compatible with restrictions, and this allows us t…
We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the co…
In this paper, we give a fully detailed exposition of computing fundamental groups of complements of line arrangements using the Moishezon-Teicher technique for computing the braid monodromy of a curve and the Van-Kampen theorem which induces a presentation of the fundamental group of the complement from the braid monodromy of the curve. For example, we treated the cases where the arrangement has t multiple intersection points and the rest are simple intersection points. In this case, the fundamental group of the complement is a direct sum of infinite cyclic groups and t free groups. Hence, th…
Kwai Man Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups. He also conjectured that the converse is true as well. The main purpose of this paper is to prove this conjecture
The cohomology groups of finite cyclic covers of complexified real arrangement complements
本文研究复化实构形补空间有限循环覆盖的上同调群，利用Yoshinaga的室上链复形得到贝蒂数上界，引入类CDO组合条件证明此类覆盖整上同调无挠，部分推广了作者与Liu的相关结果。
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