The Novikov conjecture and low-dimensional topology
This paper is a contribution to the question of whether in dimension 4 ''cutting and pasting'' can produce new manifolds. This means the process of cutting a manifold along a separating codimension-one submanifold N and glueing the pieces together by a homeomorphism (or diffeomorphism) of N homotopic to the identity. The case is studied in which the initial and terminal manifolds are \(M_ 1=S^ 1\times L_ 1\), resp. \(M_ 2=S^ 1\times L_ 2\), \(L_ 1\) and \(L_ 2\) being nonhomeomorphic 3-dim. lens spaces. It is shown by 3-manifold theory that, under the additional hypothesis that N doesn't contain fake 3-discs, \(M_ 2\) cannot be obtained from \(M_ 1\) by cutting and pasting. This additional hypothesis later is removed by the use of L-theory and known cases of the Novikov conjecture. The author conjectures that in dimension 4 no new manifolds can be produced by cutting and pasting.
scientific article; zbMATH DE number 3850098