2000/02/29 by Ştefan Papadima, Stefan Papadima, Alexander I. Suciu · 3 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #math.AG #math.AT #math.CO #msc:14M12 #msc:32S22 #msc:52C35 #msc:55Q52
paper · pdf · doi:10.1006/aima.2001.2023
published as Advances in Math. 165 (2002), 71-100 · 24 pages, 3 figures
openalex publication_date 2002/01/01 · arxiv created 2002/09/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Zπ1-module presentation of πp, the first non-vanishing higher homotopy group. We also give a combinatorial formula for the π1-coinvariants of πp. For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of π2, and study some of the properties of this module. For graphic arrangements associated to graphs with no 3-cycles, we obtain information on π2, directly from the graph. The π1-coinvariants of π2 may distinguish the homotopy 2-types of arrangement complements with the same π1, and the same Betti numbers in low degrees.