  A key tool in our earlier work on ends of manifolds high-dimensional
manifolds was an ability to embed cobordisms provided by the Quillen Plus
Construction into those ends. Here we develop a `spherical modification' trick
which provides a constructive approach to obtaining such embeddings. More
importantly, this approach allows for more general embedding results. In this
paper we develop generalizations of the plus construction and show how the
corresponding cobordisms can be embedded in manifolds satisfying appropriate
fundamental group properties. Results obtained here play an important role in
our ongoing study of noncompact manifolds.
