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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ k$-Parabolic subspace arrangements


Authors: Hélène Barcelo, Christopher Severs and Jacob A. White
Journal: Trans. Amer. Math. Soc. 363 (2011), 6063-6083
MSC (2010): Primary 52C35, 05E99
Published electronically: June 17, 2011
MathSciNet review: 2817419
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Abstract: In this paper, we study $ k$-parabolic arrangements, a generalization of $ k$-equal arrangements for finite real reflection groups. When $ k=2$, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed that the fundamental group of the complement, over $ \mathbb{C}$, of the type $ W$ Coxeter arrangement is isomorphic to the pure Artin group of type $ W$. Khovanov (1996) gave an algebraic description for the fundamental group of the complement, over $ \mathbb{R}$, of the $ 3$-equal arrangement. We generalize Khovanov's result to obtain an algebraic description of the fundamental groups of the complements of $ 3$-parabolic arrangements for arbitrary finite reflection groups. Our description is a real analogue to Brieskorn's description.


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Additional Information

Hélène Barcelo
Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287 – and – Mathematical Sciences Research Institute, Berkeley, California 94720
Email: hbarcelo@msri.org

Christopher Severs
Affiliation: Mathematical Sciences Research Institute, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Reykjavík University, Menntavegur 1, IS 101, Reykjavík, Iceland
Email: csevers@msri.org

Jacob A. White
Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287
Address at time of publication: Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720
Email: jacob.a.white@asu.edu, jawhite@msri.org

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05336-5
PII: S 0002-9947(2011)05336-5
Keywords: Subspace arrangement, Eilenberg-MacLane space, discrete homotopy theory, right-angled Coxeter group
Received by editor(s): September 3, 2009
Received by editor(s) in revised form: February 24, 2010
Published electronically: June 17, 2011
Additional Notes: The second author was partially supported by NSF grant DMS-0441170, administered by the Mathematical Sciences Research Institute, while the author was in residence during the Complementary Program, Fall 2009–Spring 2010. We thank the institute for its support.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.