Manifolds with intermediate tautness
The main achievement of this paper is to show that there are k-taut submanifolds of Euclidean space which are not taut. \(M\subset E^ n\) is called k-taut if for every closed round ball (or complement of a ball) B the induced homomorphism \(H_ i(M\cap B)\to H_ i(M)\) is injective for \(i=0,1,...,k.\). Some examples are ellipsoids with two distinct principal radii, other examples are topologically products of type \(S^ p\times S^ q\). In particular if M is taut then \(M\times S^ 2\) is shown to admit a 0-taut embedding which is not 1-taut. There are also examples with higher codimension.
scientific article; zbMATH DE number 4192570