Submanifolds of codimension two and homology equivalent manifolds

Cappell, Sylvain E.; Shaneson, Julius L.

Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 19-30.

Ce travail présente de nouvelles méthodes dans la théorie des plongements des variétés en codimension deux. On décrit des résultats sur la périodicité géométrique des groupes de cobordisme des nœuds. Les groupes des nœuds locaux d’une variété dans un espace fibré vectoriel de dimension deux sont introduits. Les calculs de ces groupes sont indiqués et appliqués aux plongements “$C^o$-près”. On énonce des théorèmes généraux sur l’existence des sous-variétés caractéristiques en codimension deux, ainsi que leurs applications aux nœuds équivariants. On donne aussi un théorème général d’existence pour les plongements P.L. non localement plats.

Ces méthodes emploient de nouveaux foncteurs dans la $K$-théorie hermitienne, que nous appelons ${\Gamma }_n$. Quelques-uns des résultats s’expriment en termes de ces foncteurs, qui satisfont d’ailleurs une “formule de Kunneth” pour ${\Gamma }_n(\mathbf{I}\times Z)$.

In this paper new methods of studying codimension two embeddings of manifolds are outlined. Results are stated on geometric periodicity of knot cobordism. The group of local knots of a manifold in a 2-plane bundle is introduced and computed, and applied to $C^o$-close embeddings. General codimension two splitting theorems are discussed, with applications to equivariant knots and knot cobordism. A general existence theorem for P.L. (non-locally flat) embeddings is also given.

The methods involve some new functors in Hermitian $K$-theory, denoted ${\Gamma }_n\left(\mathbf{I}\right)$. Some of the results are stated in terms of these functors and a “Kunneth formula” for ${\Gamma }_n(\mathbf{I}\times Z)$ is indicated.

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