Highly connected embeddings in codimension two

Abstract:

In this paper we study semilocal knots over $f$ into $\xi$, that is, embeddings of a manifold $N$ into $E(\xi )$, the total space of a $2$-disk bundle over a manifold $M$, such that the restriction of the bundle projection $p:E(\xi ) \to M$ to the submanifold $N$ is homotopic to a normal map of degree one, $f:N \to N$. We develop a new homology surgery theory which does not require homology equivalences on boundaries and, in terms of these obstruction groups, we obtain a classification (up to cobordism) of semilocal knots over $f$ into $\xi$. In the simply connected case, the following geometric consequence follows from our classification. Every semilocal knot of a simply connected manifold $M\# K$ in a bundle over $M$ is cobordant to the connected sum of the zero section of this bundle with a semilocal knot of the highly connected manifold $K$ into the trivial bundle over a sphere.