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

  • 1 School of Mathematical Sciences, University of Adelaide, Australia, 5005.


The Kreck monoids l2q+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 l2q+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 l2q+1(υ, υ′) ⊂ l2q+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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