Assembly maps and realization of splitting obstructions

Abstract

In 1987 Kharshiladze introduced the concept of type for an element in a Wall group, and proved that the elements of first and second type cannot be realized by normal maps of closed manifolds. This approach is sufficiently easy for computing the assembly maps and sometimes very effective. Here we give a geometrical interpretation of this approach by using the Browder–Quinn surgery obstruction groups for filtered manifolds. To understand the obtained relations we give algebraic definitions of the element types which are based on the algebraic surgery theory of Ranicki. Our approach describes a level of indeterminate for the algebraic passing to surgery on a codimension k submanifold of a given Browder–Livesay filtration. Then we study the realization of splitting obstructions by simple homotopy equivalences of closed manifolds, and compute the assembly maps for some classes of groups.