Extension problem of diffeomorphisms of a 3-torus over some 4-manifolds
A K3 surface is a union of two copies of a half K3 surface N. It is proved that the diffeomorphism type of \(N\cup N\) does not depend on the choice of attaching orientation reversing diffeomorphism of the boundary 3-torus. A Poincaré homology 3-sphere H(1,1,1) is cobordant to a 3- torus \(T^ 3\) by a cobordism W(1,1,1) given by attaching three 2 handles to \(T^ 3\times [0,1]\) at three standard generators on \(T^ 3\) with framing numbers 1, 1 and 1. The paper actually proves that any orientation preserving diffeomorphism f of \(T^ 3\) extends to a diffeomorphism F of the cobordism W(1,1,1) so that \(F| H(1,1,1)=id\). Since \(N\supset W(1,1,1)\) so that \(\partial W(1,1,1)\supset\partial N\), the first statement follows directly. A similar result for W(0,0,0) is also proved.
extending diffeomorphism from a boundary component to a cobordism
scientific article; zbMATH DE number 3874228