Some results in the study of noncompact 4-manifolds

For many years there existed a notable gap in manifold theory when it came to dimension 4. While recent breakthroughs by Freedman, Donaldson, and Quinn, have gone far towards changing this, many smaller gaps persist. In this dissertation we do our best to close a few of them.

A natural question to ask about a manifold with boundary is whether it has the simplest possible structure, i.e., whether or not M^m is homeomorphic to ∂M×[0,l). It was shown by Whitehead in 1937 that it is not sufficient to assume that ∂M ⊂ M is a homotopy equivalence. In 1969, Siebenmann's "Open Collar Theorem" showed that, with the additional assumption that M'^m have a "stable end" with the correct fundamental group, the desired conclusion holds whenever m≥5. Modulo the Poincare Conjecture, work by Husch, Price, Brown and Tucker extends this result to dimension 3. The main result of this dissertation is an extension of The Open Collar Theorem to dimension 4, provided ^π1(M^m) falls into a certain class of groups.

Another question which stood open only in dimension 4 involves embeddings of spheres. An embedded k-sphere ∑^k ≥ sⁿ is said to be "weakly flat" provided its complement is homeomorphic to the complement of the standard k-sphere in An especially interesting case occurs when k=n-2 . Results by Daverman (n=3) and Hollingsworth and Rushing (n≥5) characterize weakly flat (n-2)-spheres in Sⁿ as those which satisfy a certain embedding condition is (∑^n-2 is globally 1-alg), along with Sⁿ-∑^n-2being homotopy equivalent to a circle. The latter is needed to prevent knotting. We are able to extend this characterization to include 2-spheres in S⁴.

A final result presented here is a "stabilization" result for open 3-manifolds homotopy equivalent to closed surfaces. One would hope that such a 3-manifold would necessarily be a nice R¹-bundle over that surface. Unfortunately, possible problems at the ends of M³ prevent this from being a theorem. Our result guarantees that M×R¹ is exactly what one would hope it to be.