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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Neeb, Karl-Hermann / Noguchi, Junjiro / Shahidi, Freydoon / Sogge, Christopher D. / Wienhard, Anna

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Volume 23, Issue 3 (Jan 2011)

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Stably diffeomorphic manifolds and l 2q+1(ℤ[π])

Diarmuid Crowley
  • School of Mathematical Sciences, University of Adelaide, Australia, 5005.
  • Email:
/ Jörg Sixt
  • School of Mathematical Sciences, University of Adelaide, Australia, 5005.
  • Email:
Published Online: 2010-04-14 | DOI: https://doi.org/10.1515/form.2011.016

Abstract

The Kreck monoids l 2q+1(ℤ[π]) detect s-cobordisms amongst certain bordisms between stably diffeomorphic 2q-dimensional manifolds and generalise the Wall surgery obstruction groups, . In this paper we identify l 2q+1(ℤ[π]) as the edge set of a directed graph with vertices a set of equivalence classes of quadratic forms on finitely generated free ℤ[π] modules. Our main theorem computes the set of edges l 2q+1(υ, υ′) ⊂ l 2q+1(ℤ[π]) between the classes of the forms υ and υ′ via an exact sequence

Here sbIso(υ, υ′) denotes the set of “stable boundary isomorphisms” between the algebraic boundaries of υ and υ′. As a consequence we deduce new classification results for stably diffeomorphic manifolds.

Keywords.: Stable diffeomorphism; surgery obstruction monoids; cancellation; polycyclicby-finite groups

About the article

Received: 2008-10-17

Revised: 2009-06-15

Published Online: 2010-04-14

Published in Print: 2011-05-01



Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/form.2011.016. Export Citation

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