A generalization of a construction of Whitney

He well-known construction of H. Whitney (also called “Whitney trick”) appeared in the paper [1], where it was used for eliminating self-intersections of a closedsmooth manifold immersed in R2n, n ) 3. The same construction played an important role in Smale’s proof of the h-cobordism theorem [2, 3]. We remind thatWhitney construction allows, in short, to eliminate a pair of transversal intersection points of two oriented submanifolds embedded in a manifold of the summary dimension by an appropriate isotopy of one of the submanifolds, on the condition that the signs of intersection points are opposite (and assuming some other conditions that are discussed in this paper). The significance of this construction for differential topology is that, consecutively removing pairs of intersection points with opposite signs, we finally obtain a pair of embeddings with minimal possible number of intersections, equal up to sign to the intersection index of the corresponding homology classes; in particular, in the case where the intersection index is zero, we get a disjoin pair of embeddings.The goal of this paper is to prove a generalization of Whitney construction to the case where the intersection of two submanifolds is positive dimensional, in other words the dimension of ambient manifold is less than the summary dimension of submanifolds. As it turns out, in this case (assuming the same extra conditions as in Whitney case) one can use appropriate isotopy of one of the embeddings to replace some two components of the intersection by their connected sum, so that in the end the intersection becomes connected (or empty). Regretfully, this is maximum of what one can expect: in this case we cannot define an invariant for the pair of embeddings, similar to intersection index for the case of Whitney, that could be used to control the topological type of “minimal” intersection.The above generalization turned out necessary to the author while studying the isotopy classes of embedings S3 → S3 × S3 which, in turn, required consideringintersections of some embeddings I × S3 → I × S3 × S3 (i. e. isotopies). Thisgeneralization seems rather straightforward; however, the author needed precisestatement, which he was not able to find in literature.