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Bulletin of the London Mathematical Society Advance Access originally published online on December 15, 2006
Bulletin of the London Mathematical Society 2007 39(1):46-52; doi:10.1112/blms/bdl007
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© 2006 London Mathematical Society

An algebraic loop theorem and the decomposition of PD3-pairs

John Crisp

Institut de Mathématiques de Bourgogne (UMR 5584 du CNRS)
Université de Bourgogne
Avenue Alain Savary
B.P. 47 870
21078 Dijon
France

Received 9 May 2005. Revision received 11 October 2005.

Let (Y, X) denote a three-dimensional Poincaré pair (PD3-pair). By the work of Eckmann, Müller and Linnell we may suppose, up to a homotopy equivalence, that the boundary X is a closed 2-manifold. We show that if a component of X fails to be {pi}1-injective in Y, then there is an essential simple loop in X which is nullhomotopic in Y. It follows that there is a finite process of attaching 2-disks along essential simple loops on X, and filling spherical components of X, which transforms (Y, X) into a PD3-pair (Y', X') with aspherical incompressible boundary X' and such that {pi}1(Y) = {pi}1(Y'). The PD3-pair (Y', X') then admits a canonical decomposition as a connected sum of a finite number of aspherical PD3-pairs with incompressible boundary, together with a PD3-pair having virtually free (possibly finite) fundamental group and boundary a (possibly empty) disjoint union of projective planes.


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