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Monodromy of fiber-type arrangements and orbit configuration spaces

Monodromy of fiber-type arrangements and orbit configuration spaces Abstract. We prove similar theorems concerning the structure of bundles involving complements of ®ber-type hyperplane arrangements and orbit con®guration spaces. These results facilitate analysis of the fundamental groups of these spaces, which may be viewed as generalizations of the Artin pure braid group. In particular, we resolve two disparate conjectures. We show that the Whitehead group of the fundamental group of the complement of a ®ber-type arrangement is trivial, as conjectured by Aravinda, Farrell, and Rouchon [AFR]. For the orbit con®guration space corresponding to the natural action of a ®nite cyclic group on the punctured plane, we determine the structure of the Lie algebra associated to the lower central series of the fundamental group. Our results show that this Lie algebra is isomorphic to the module of primitives in the homology of the loop space of a related orbit con®guration space, as con jectured by Xicotencatl [Xi]. 2000 Mathematics Subject Classi®cation: 20F36, 52C35; 19B99, 20F40. Introduction Let M be a manifold without boundary of dimension at least two. The con®guration space of n ordered points in M is the subspace of the product space M n de®ned by F MY n fx1 Y F F F Y xn http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Monodromy of fiber-type arrangements and orbit configuration spaces

Cohen, Daniel C.
Forum Mathematicum , Volume 13 (4) – May 17, 2001
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References (25)

Publisher
de Gruyter
Copyright
Copyright © 2001 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2001.020
Publisher site
See Article on Publisher Site

Abstract

Abstract. We prove similar theorems concerning the structure of bundles involving complements of ®ber-type hyperplane arrangements and orbit con®guration spaces. These results facilitate analysis of the fundamental groups of these spaces, which may be viewed as generalizations of the Artin pure braid group. In particular, we resolve two disparate conjectures. We show that the Whitehead group of the fundamental group of the complement of a ®ber-type arrangement is trivial, as conjectured by Aravinda, Farrell, and Rouchon [AFR]. For the orbit con®guration space corresponding to the natural action of a ®nite cyclic group on the punctured plane, we determine the structure of the Lie algebra associated to the lower central series of the fundamental group. Our results show that this Lie algebra is isomorphic to the module of primitives in the homology of the loop space of a related orbit con®guration space, as con jectured by Xicotencatl [Xi]. 2000 Mathematics Subject Classi®cation: 20F36, 52C35; 19B99, 20F40. Introduction Let M be a manifold without boundary of dimension at least two. The con®guration space of n ordered points in M is the subspace of the product space M n de®ned by F MY n fx1 Y F F F Y xn

Journal

Forum Mathematicum – de Gruyter

Published: May 17, 2001

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