arXiv subjects

Stefan Papadima

Publications and source records attributed to Stefan Papadima.

44 records · Page 3Linked to original sources

arXiv2006

When does the associated graded Lie algebra of an arrangement group decompose?

Let \A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra \H. Suppose \H_3 is a free abelian group of minimum possible rank, given the values the Möbius function μ: Ł_2\to \Z takes on the rank 2 flats of \A. Then the associated graded Lie algebra of G decomposes (in degrees 2 and higher) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by ϕ_r(G)=\sum_{X\in Ł_2} ϕ_r(F_{μ(X)}), for r\ge 2. We illustrate this new Lower Central Series formula with several families of examples.

math.CO↗

arXiv2005

Algebraic invariants for right-angled Artin groups

A finite simplicial graph Γdetermines a right-angled Artin group G_Γ, with generators corresponding to the vertices of Γ, and with a relation vw=wv for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of G_Γ, directly from the graph Γ.

math.GR↗

arXiv2004

Homotopy Lie algebras, lower central series and the Koszul property

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie algebra associated to the lower central series of pi_1(X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H^*(X,Q). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of pi_1(X) and the completion of [Omega S^{2k+1}, Omega Y]. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph.

math.AT↗

arXiv2004

Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements

We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. In particular we show that any projective hypersurface has affine parts which are bouquets of spheres. The main tools are the polar curves and the affine Lefschetz theory developped by H. Hamm, D.T. Lê 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 the minimality of hyperplane arrangements (see Randell math.AT/0011101 for another proof of this result). This is then used to compute some higher homotopy groups of hyperplane arrangements using the ideas from Papadima-Suciu, see math.AT/0002251. The second version contains applications of the above ideas to the polar Cremona transformations and gives a positive answer to Dolgachev's Conjecture (see Michigan Math. J. 48 (2000), volume dedicated to W. Fulton). The third version corrects some errors and provides new applications.

math.AT↗

arXiv2003

Isometry-invariant geodesics and nonpositive derivations of the cohomology

We introduce a new class of zero-dimensional weighted complete intersections, by abstracting the essential features of rational cohomology algebras of equal rank homogeneous spaces of compact connected Lie groups. We prove that, on a 1-connected closed manifold M whose rational cohomology algebra belongs to this class, every isometry has a non-trivial invariant geodesic, for any metric on M. We use rational surgery to construct large classes of new examples for which the above result may be applied.

math.DG↗

arXiv2003

Multiplicative models for configuration spaces of algebraic varieties

W. Fulton and R. MacPherson described a Sullivan dg-algebra model for the space of n-configurations of labeled points in a smooth compact complex algebraic variety X. I. Kriz then gave a simpler model that depends only on the cohomology ring of X. We construct an even simpler and substantially smaller model for this configuration space. Using this smaller model, we also define another, new dg-algebra model, for the space of n-configurations in the variety X with a point removed. Following an idea of V.G. Drinfel'd, we endow the collection of these new "punctured" models with a simplicial bigraded differential algebra structure.

math.AT↗

arXiv2003

Chen Lie algebras

The Chen groups of a finitely-presented group G are the lower central series quotients of its maximal metabelian quotient, G/G''. The direct sum of the Chen groups is a graded Lie algebra, with bracket induced by the group commutator. If G is the fundamental group of a formal space, we give an analog of a basic result of D. Sullivan, by showing that the rational Chen Lie algebra of G is isomorphic to the rational holonomy Lie algebra of G modulo the second derived subalgebra. Following an idea of W.S. Massey, we point out a connection between the Alexander invariant of a group G defined by commutator-relators, and its integral holonomy Lie algebra. As an application, we determine the Chen Lie algebras of several classes of geometrically defined groups, including surface-like groups, fundamental groups of certain classical link complements, and fundamental groups of complements of complex hyperplane arrangements. For link groups, we sharpen Massey and Traldi's solution of the Murasugi conjecture. For arrangement groups, we prove that the rational Chen Lie algebra is combinatorially determined.

math.GR↗

arXiv2002

Higher homotopy groups of complements of complex hyperplane arrangements

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 properties of this module. For graphic arrangements associated to graphs with no 3-cycles, we obtain information on π_2, directly from the graph. The π_1-coinvariants of π_2 may distinguish the homotopy 2-types of arrangement complements with the same π_1, and the same Betti numbers in low degrees.

math.AT↗