Abstract
In this paper new methods of classifying smooth, piecewise linear (P.L.) or topological submanifolds are developed as consequences of a classification theory for manifolds that are homology equivalent, over various systems of coefficients. These methods are particularly suitable for the placement problem for submanifolds of codimension two. The role of knot theory in this larger problem is studied systematically by the itroduction of the local knot gorup of an arbitrary manifold. Computations of this group are used to determine when sufficiently close embeddings in codimension two “differ” by a knot. A geometric periodicity is derived for the knot cobordism groups.
The method of this paper can also be applied to get classification results on submanifolds invariant under group actions and on submanifolds fixed by group actions. In particular, algebraic calculations of the equivariant knot cobordism groups are given in this paper and some geometric consequences are derived. Other applications of the methods of this paper include a general solution of the codimension two surgery problem, given below, and corresponding results on smoothings of Poincaré embeddings in codimension two.
The proofs of many of the results use computations of new algebraic $K$-theory functors. In a future paper, the present methods will be applied to the sutdy of P.L. embeddings and their singularities.
