The homology of real subspace arrangements

Abstract

Associated to any subspace arrangement is a "De Concini-Procesi model", a certain smooth compactification of its complement, which in the case of the braid arrangement produces the Deligne-Mumford compactification of the moduli space of genus 0 curves with marked points. In the present work, we calculate the integral homology of real De Concini-Procesi models, extending earlier work of Etingof, Henriques, Kamnitzer and the author on the (2-adic) integral cohomology of the real locus of the moduli space. To be precise, we show that the integral homology of a real De Concini-Procesi model is isomorphic modulo its 2-torsion with a sum of cohomology groups of subposets of the intersection lattice of the arrangement. As part of the proof, we construct a large family of natural maps between De Concini-Procesi models (generalizing the operad structure of moduli space), and determine the induced action on poset cohomology. In particular, this determines the ring structure of the cohomology of De Concini-Procesi models (modulo 2-torsion).