Cutting and pasting of pairs (Q579681)

This paper is concerned with showing certain relationships among groups associated with (controllable) cutting and pasting equivalences. More explicitly if \(M_ n\) denotes the set of diffeomorphism classes of n- dimensional closed \(C^{\infty}\)-manifolds then the cutting and pasting relation on \(M_ n\) is generated by the relation \(N_ 1\sim N_ 2\) if there are n-dimensional compact \(C^{\infty}\)-manifolds P, Q such that \(N_ i\) is diffeomorphic to the result of glueing P to Q along their boundaries by a diffeomorphism \(f_ i: \partial P\to \partial Q\), \(i=1,2\). Denote by \(SK_ n\) the Grothendieck group associated with the semigroup (under disjoint union) \(M_ n/\sim\) and by \(SK_{m,n}\) the group obtained by the corresponding construction in which N is replaced by a pair of manifolds (M,N), where N is an n-dimensional submanifold of the m-dimensional manifold M. It is shown that \(SK_{m,n}\approx SK_ m\oplus SK_ n\) and consequences are considered. A modification (controllable cutting and pasting) of the above construction results in groups \(SKK_ m\) and \(SKK_{m,n}\) and a splitting \(SKK_{m,n}=SKK_ m\oplus G\), where G is an appropriate group associated to B(O,(m-n)), the classifying space of (m-n)-dimensional vector bundles. Consequences are considered including relationships involving the unoriented cobordism groups.