Exponential separation in 4–manifolds

Abstract

We use a new geometric construction, grope splitting, to give a sharp bound for separation of surfaces in 4–manifolds. We also describe applications of this technique in link-homotopy theory, and to the problem of locating ${\pi }_1$–null surfaces in 4–manifolds. In our applications to link-homotopy, grope splitting serves as a geometric substitute for the Milnor group.