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 to 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).

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