Dennis J. Garity

Rigid cantor sets in ³ with simply connected complement

We prove that there exist uncountably many inequivalent rigid wild Cantor sets in R 3 with simply connected complement. Previous constructions of wild Cantor sets in R 3 with simply connected complement, in particular the Bing-Whitehead Cantor sets, had strong homogeneity properties. This suggested it might not be possible to construct such sets that were rigid. The examples in this paper are constructed using a generalization of a construction of Skora together with a careful analysis of the local genus of points in the Cantor sets.

Property C and closed maps

Abstract A standard theorem from dimension theory states that a closed (m+1) to 1 map defined on a finite dimensional space can raise dimension by at most m. Dimension raising maps on countable dimensional spaces and on weakly infinite dimensional spaces have been investigated by A.V. Arhangelskii, A.I. Vainstein and E.G. Sklyarenko. A typical theorem is that a closed map on such spaces raises dimension only if some point has an uncountable number of preimages. A class of infinite dimensional spaces closely related to the two types mentioned above is the class of C spaces. R. Pols example in 1980 and work of F.D. Ancel have generated renewed interest in C spaces. We prove results about dimension raising closed maps defined on C spaces that are analogous to the results mentioned above.

Distinguishing Bing-Whitehead Cantor sets

Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger than three. These Cantor sets in

Closed curves and geodesics with two self-intersections on the Punctured torus

S^{3}

Contractible 3-manifolds and the double 3-space property

are constructed by using Bing or Whitehead links as stages in defining sequences. Ancel and Starbird, and separately Wright characterized the number of Bing links needed in such constructions so as to produce Cantor sets. However it was unknown whether varying the number of Bing and Whitehead links in the construction would produce non equivalent Cantor sets. Using a generalization of geometric index, and a careful analysis of three dimensional intersection patterns, we prove that Bing-Whitehead Cantor sets are equivalently embedded in

Simply connected open 3-manifolds with rigid genus one ends

S^3

Inequivalent Cantor sets in ³ whose complements have the same fundamental group

if and only if their defining sequences differ by some finite number of Whitehead constructions. As a consequence, there are uncountably many non equivalent such Cantor sets in

Topology and Chaos

S^{3}

Simply Connected 3-Manifolds with a Dense Set of Ends of Specified Genus

constructed with genus one tori and with simply connected complement.

New Techniques for Computing Geometric Index

We classify the free homotopy classes of closed curves with minimal self intersection number two on a once punctured torus,T, up to homeomorphism. Of these, there are six primitive classes and two imprimitive. The classification leads to the topological result that, up to homeomorphism, there is a unique curve in each class realizing the minimum self intersection number. The classification yields a complete classification of geodesics on hyperbolicT which have self intersection number two. We also derive new results on the Markoff spectrum of diophantine approximation; in particular, exactly three of the imprimitive classes correspond to families of Markoff values below Halls ray.