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