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A key tool in our earlier work on ends of manifolds high-dimensionalmanifolds was an ability to embed cobordisms provided by the Quillen PlusConstruction into those ends. Here we develop a `spherical modification' trickwhich provides a constructive approach to obtaining such embeddings. Moreimportantly, this approach allows for more general embedding results. In thispaper we develop generalizations of the plus construction and show how thecorresponding cobordisms can be embedded in manifolds satisfying appropriatefundamental group properties. Results obtained here play an important role inour ongoing study of noncompact manifolds.

Spherical alterations of handles: embedding the manifold plus construction