Fiber bundles with degenerations and their applications to computing fundamental groups
type: Item
instance of: scholarly article
title: Fiber bundles with degenerations and their applications to computing fundamental groups
zbMATH Open document ID: 0812.57004
DOI: 10.1007/BF01265678
published in: Geometriae Dedicata
publication date: 1994-01-21T00:00:00Z
review text: Let \(p : E \to B\) be a surjective morphism of connected smooth manifolds, with generic fiber \(F\). The author derives a short exact sequence relating the fundamental groups of \(F\), \(B\), and \(E\) under the assumptions that (i) \(p\) is a locally trivial bundle projection except over a union of codimension two submanifolds of \(B\), and (ii) \(p\) has a local section at each point of \(B\). In case \(p\) has a sufficiently generic global section the sequence looks like the tail of the long exact homotopy sequence, with a correction term determined by the monodromies over loops in \(B\) linking the singular locus of \(p\). These results are applied to the fundamental groups of complements of hypersurfaces, using generic projection, and to fundamental groups of hyperplane complements, using iterated linear projections with singular fibers. In particular, the author computes presentations of the generalized pure braid groups of type \(B_ n\) and \(D_ n\).
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 55R65
zbMATH DE Number: 486901
zbMATH Keywords: surjective morphism of connected smooth manifolds
zbMATH Keywords: generic fiber
zbMATH Keywords: fundamental groups
zbMATH Keywords: complements of hypersurfaces
zbMATH Keywords: hyperplane complements
zbMATH Keywords: presentations of the generalized pure braid groups
author: A. Leibman
reviewed by: Michael Falk
MaRDI profile type: Publication
cites work: The Minimal Model of the Complement of an Arrangement of Hyperplanes
cites work: The lower central series of a fiber-type arrangement
cites work: On the Fundamental Group of an Algebraic Curve
cites work: Q4145882
cites work: Q5513066
cites work: The \(D_ n\) generalized pure braid group
cites work: Q5623828
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description: scientific article; zbMATH DE number 486901