Higher Homotopy Groups of Complements of Complex Hyperplane Arrangements

✍ Scribed by Stefan Papadima; Alexander I. Suciu

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 241 KB
• Volume: 165
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2001.2023

✦ Synopsis

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 Zp 1 -module presentation of p p , the first non-vanishing higher homotopy group. We also give a combinatorial formula for the p 1 -coinvariants of p p . For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of p 2 and study some of the properties of this module. For graphic arrangements associated to graphs with no 3-cycles, the algorithm for computing p 2 is purely combinatorial. The Fitting varieties associated to p 2 may distinguish the homotopy 2-types of arrangement complements with the same p 1 , and the same Betti numbers in low degrees.