Formality of the complements of subspace arrangements with geometric lattices

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. Bibliography: 10 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 235–247.     

On subspace arrangements of type

Let denote the subspace arrangement formed by all linear subspaces in given by equations of the form

₁x_i1=₂x_i2==_kx_ik,

where 1i₁<<i_kn and (₁,…,_k){+1,−1}^k.Some important topological properties of such a subspace arrangement depend on the topology of its intersection lattice. In a previous work on a larger class of subspace arrangements by Björner and Sagan (J. Algebraic Combin. 5 (1996) 291–314) the topology of the intersection lattice turned out to be a particularly interesting and difficult case.We prove in this paper that Pure(Π_n,k^±) is shellable, hence that Π_n,k^± is shellable for k>n/2. Moreover, we prove that unless i≡n−2 (mod k−2) or i≡n−3 (mod k−2), and that is free abelian for i≡n−2 (mod k−2). In the special case of Π_2k,k^± we determine homology completely. Our tools are generalized lexicographic shellability, as introduced in Kozlov (Ann. Combin. 1 (1997) 67–90), and a spectral sequence method for the computation of poset homology first used in Hanlon (Trans. Amer. Math. Soc. 325 (1991) 1–37).We state implications of our results on the cohomology of the complements of the considered arrangements.     

The homotopy type of the complement of a coordinate subspace arrangement

The homotopy type of the complement of a complex coordinate subspace arrangement is studied by utilising some connections 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.     

Hyperplane arrangements with a lattice of regions

A hyperplane arrangement is a finite set of hyperplanes through the origin in a finite-dimensional real vector space. Such an arrangement divides the vector space into a finite set of regions. Every such region determines a partial order on the set of all regions in which these are ordered according to their combinatorial distance from the fixed base region.We show that the base region is simplicial whenever the poset of regions is a lattice and that conversely this condition is sufficient for the lattice property for three-dimensional arrangements, but not in higher dimensions. For simplicial arrangements, the poset of regions is always a lattice.In the case of supersolvable arrangements (arrangements for which the lattice of intersections of hyperplanes is supersolvable), the poset of regions is a lattice if the base region is suitably chosen. We describe the geometric structure of such arrangements and derive an expression for the rank-generating function similar to a known one for Coxeter arrangements. For arrangements with a lattice of regions we give a geometric interpretation of the lattice property in terms of a closure operator defined on the set of hyperplanes.The results generalize to oriented matroids. We show that the adjacency graph (and poset of regions) of an arrangement determines the associated oriented matroid and hence in particular the lattice of intersections.The work of Anders Björner was supported in part by a grant from the NSF. Paul Edelman's work was supported in part by NSF Grants DMS-8612446 and DMS-8700995. The work of Günter Ziegler was done while he held a Norman Levinson Graduate Fellowship at MIT.     

Hilbert series of subspace arrangements

The vanishing ideal I of a subspace arrangement V₁∪V₂∪?∪V_m⊆V is an intersection I₁∩I₂∩?∩I_m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I₁I₂?I_m without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing.     

Lie algebras whose lattice of ideals is closely related with a subspace lattice

Relationships between the structure of a Lie algebra and that of its lattice of ideals is studied for those Lie algebras whose ideal lattice is very close to that of an almost-abelian Lie algebra. It is shown here that if the base field is algebraically closed, finite or the real one, for any n ≥3 the only solvable Lie algebra whose lattice of ideals is isomorphic to that of the (n+l)-dimensional almost-abelian Lie algebra is itself.     

The etale homotopy theory of a geometric fibration

A geometric fibration, f:XY, is a smooth map of schemes which locally on Y admits a smooth, relative compactification. The etale homotopy type of the geometric fibre when completed away from the residue characteristics of Y, , is shown to be weakly homotopy equivalent to the completion of the Hurewicz fibre of the etale homotopy type of f,F(f _et ^r )^{^}. This implies a homotopy sequence for f. A key topological fact is verified in the appendix: for any pointed Hurewicz fibre triple FEB, the action of _l(F) on the homotopy type of various covering spaces of F extends to an action of _l(E).Partially supported by the N.S.F., I.H.E.S., and University of Warwick.     

Rational homotopy of Leibniz algebras

We construct a non-commutative rational homotopy theory by replacing the pair (Lie algebras, commutative algebras) by the pair (Leibniz algebras, Leibniz-dual algebras). Both pairs are Koszul dual in the sense of operads (Ginzburg–Kapranov). We prove the existence of minimal models and the Hurewicz theorem in this framework. We define Leibniz spheres and prove that their homotopy is periodic. Received: 19 September 1997 / Revised version: 23 February 1998     

On cohomology algebras of complex subspace arrangements

The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations.

     

On the algebra associated with a geometric lattice

Let L be a geometric lattice. Following P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. math. 56 (1980), 167–189, we associate with L a graded commutative algebra A(L). In this paper we introduce a new invariant ψ of the algebra A(L) which suffices to distinguish algebras for which all other known invariants coincide. This result is applied to the study of arrangements of complex hyperplanes, with L being the intersection lattice. In this case A(L) is isomorphic to the cohomology algebra of the associated hyperplane complement. The goal is to find examples of arrangements with non-isomorphic lattices but homotopy equivalent complements. The invariant introduced here effectively narrows the list of candidates. Nevertheless, we exhibit combinatorially inequivalent arrangements for which all known invariants, including ψ, coincide.     

Concerning the shape of a geometric lattice

A well-known conjecture states that the Whitney numbers of the second kind of a geometric lattice (simple matroid) are logarithmically concave. We show this conjecture to be equivalent to proving an upper bound on the number of new copoints in the free erection of the associated simple matroid M. A bound on the number of these new copoints is given in terms of the copoints and colines of M. Also, the points-lines-planes conjecture is shown to be equivalent to a problem concerning the number of subgraphs of a certain bipartite graph whose vertices are the points and lines of a geometric lattice.     

Motivic structures on higher homotopy groups of hyperplane arrangements

In this note, we show the existence of motivic structures on certain objects arising from the higher (rational) homotopy groups of non-nilpotent spaces. Examples of such spaces include several families of hyperplane arrangements. In particular, we construct an object in Nori’s category of motives whose realization is a certain completion of \(\pi _{n}({\mathbb P}^{n} {\setminus } \{L_{1}, \ldots , L_{n+2}\})\) where the \(L_{i}\) are hyperplanes in general position. Similar results are shown to hold in Vovoedsky’s setting of mixed motives.     

Rational homotopy groups of generalised symmetric spaces

Abstract. We obtain explicit formulas for the rational homotopy groups of generalised symmetric spaces, i.e., the homogeneous spaces for which the isotropy subgroup appears as the fixed point group of some finite order automorphism of the group. In particular, this gives explicit formulas for the rational homotopy groups of all classical compact symmetric spaces. Received: 18 March 2002 / Published online: 14 February 2003 The author is supported by the {\it DFG Graduiertenkolleg “Mathematik im Bereich ihrer Wechselwirkung mit der Physik”} and is a member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential Programme.     

Rational tetrahedra with edges in geometric progression

This paper discusses tetrahedra with rational edges forming a geometric progression, focussing on whether they can have rational volume or rational face areas. We examine the 30 possible configurations of such tetrahedra and show that no face of any of these has rational area. We show that 28 of these configurations cannot have rational volume, and in the remaining two cases there are at most six possible examples, and none have been found.