Mathematical Sciences: Topological Embeddings in Piecewise Linear Manifolds

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Venema will investigate properties of manifolds, both topological manifolds and the more restricted class of piecewise linear (PL) manifolds, with special emphasis on topological imbeddings into PL 4-manifolds. (1) Suppose X is a compact subset of the PL 4-manifold M. Under what conditions does X have arbitrarily close neighborhoods which collapse to 1-dimensional spines? (2) Suppose F is a closed surface and W has the homotopy type of F. Does there exist a topological imbedding of F into W which is a homotopy equivalence? (3) Under what natural additional hypotheses can one prove that two compacta imbedded in a manifold have homeomorphic complements if and only if they have the same shape? (4) Is the approximation theorem valid for codimension two imbeddings of any manifolds other than k-cells? These questions and others point to basic deficiencies in our understanding of very natural geometric objects, namely topological and PL-manifolds, even in low dimensions. Such manifolds arise in physics as the solutions of ordinary and differential equations set in the 4-dimensional space-time in which we live.