Stably diffeomorphic manifolds and l2q+1(ℤ[π])

Diarmuid Crowley; Jörg Sixt

doi:10.1515/form.2011.016

Article

Stably diffeomorphic manifolds and l_2q+1(ℤ[π])

Diarmuid Crowley and Jörg Sixt

Published/Copyright: April 14, 2010

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From the journal Volume 23 Issue 3

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

Received: 2008-10-17

Revised: 2009-06-15

Published Online: 2010-04-14

Published in Print: 2011-May

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Articles in the same Issue

https://doi.org/10.1515/form.2011.016

Keywords for this article

Stable diffeomorphism; surgery obstruction monoids; cancellation; polycyclicby-finite groups