Research

Since the time I received my Ph.D. (in 1975) I have worked on research in geometric topology, especially the theory of topological embeddings in codimension two and the geometric topology of 4-manifolds.

In my dissertation I proved several finite-dimensional "complement theorems." These are highly geometric forms of duality theorems for compact subsets of Rⁿ: under certain reasonable hypotheses two compact subsets of Rⁿ have homeomorphic complements if and only if they have the same shape. I have continued to take an interest in questions about the relationship between embedding theory and shape theory and have continued to publish papers on the subject.

Soon after I graduated, my major interests shifted to the problem of approximating topological embeddings in codimension two. While at the Unversity of Texas I proved that every topological embedding of the 2-disk into 4-space can be approximated by locally flat PL embeddings. Since that time I have had a continuing interest in both the theory of topological embeddings in codimension two, especially the theory of topological embeddings of surfaces in 4-manifolds, and in the theory of topological 4-manifolds. Most of my subsequent research efforts have been concentrated in those two areas. A list of the resulting publications is available on my home page. My research was supported by NSF research grants in 1978-1985, 1987-1991, and 1996-1999.

Here is an interesting example of an embedding of the 2-sphere in 4-space. The diagram below shows level pictures of a smooth embedding of R² into R⁴. The levels at the top are straight lines. As the lines move down, they grow more and more hooks which are linked to each other as shown in the picture. The middle level has infinitely many hooks. The diagram shows only the upper half of the embedded plane; the lower half is symmetric.

One-point-compactification yields a 2-sphere S in the 4-sphere S⁴. This 2-sphere is topologically embedded with one wild point. The interesting feature of both examples (the smooth embedding of R² and the wild embedding of S²) is the fact that the fundamental group of the complement is infinite cyclic, but the second homotopy group of the complement is nontrivial. In fact, the second homotopy group of the complement is not even finitely generated as a module over Z[p₁(S⁴- S)]. The loops x₁, x₂, x₃, ... represent a generating set for p₂(S⁴- S). Each loop can be capped off with a disk above and below by simply foating the loop up or down to a level where it is no longer linked around the level line. The upper and lower disks together form a 2-sphere which represents an element of p₂(S⁴- S). The relationship between these generators is that x_i =(t-1)x_i+1, where t is the generator of p₁(S⁴- S).

[A postscript version of the diagram is also available.]