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The homology of real subspace arrangements

Rains, Eric M. (2010) The homology of real subspace arrangements. Journal of Topology, 3 (4). pp. 786-818. ISSN 1753-8416. https://resolver.caltech.edu/CaltechAUTHORS:20101221-082008592

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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).


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1112/jtopol/jtq027DOIArticle
http://jtopol.oxfordjournals.org/content/3/4/786PublisherArticle
http://arxiv.org/abs/math/0610743arXivDiscussion Paper
Additional Information:© 2010 London Mathematical Society. Received 10 December 2009. Journal of Topology Advance Access published October 14, 2010. The author was supported in part by NSF Grant No. DMS-0401387. The author would like to thank his coauthors P. Etingof, A. Henriques, and J. Kamnitzer on [7] for introducing him to these questions, and especially Henriques for discussions relating to blow-ups in Z[1/2]-cohomology. In addition, the author would like to thank S. Devadoss and especially S. Yuzvinsky for motivating discussions, as well as several referees for useful comments.
Funders:
Funding AgencyGrant Number
NSFDMS-0401387
Issue or Number:4
Classification Code:2000 Mathematics Subject Classification: 14N20 (primary), 14F25 (secondary).
Record Number:CaltechAUTHORS:20101221-082008592
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20101221-082008592
Official Citation:Rains, E. M. (2010). "The homology of real subspace arrangements." Journal of Topology 3(4): 786-818.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:21465
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Jan 2011 18:31
Last Modified:03 Oct 2019 02:24

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