We introduce new general techniques for computing the geometric index of a link L in the interior of a solid torus T. These techniques simplify and unify previous ad hoc methods used to compute the geometric index in specific examples and allow a simple computation of geometric index for new examples where the index was not previously known. The geometric index measures the minimum number of times any meridional disc of T must intersect L. It is related to the algebraic index in the sense that adding up signed intersections of an interior simple closed curve C in T with a meridional disc gives +/- the algebraic index of C in T. One key idea is introducing the notion of geometric index for solid chambers of the form B^2 x [0,1] in T. We prove that if a solid torus can be divided into solid chambers by meridional discs in a specific (and often easy to obtain) way, then the geometric index can be easily computed.

COBISS.SI-ID: 18167129

We show that for every sequence (n_i), where each n_i is either an integer greater than 1 or is \infty, there exists a simply connected open 3-manifold M with a countable dense set of ends {e_i} so that, for every i, the genus of end e_i is equal to n_i. In addition, the genus of the ends not in the dense set is shown to be less than or equal to 2. These simply connected 3-manifolds are constructed as the complements of certain Cantor sets in S^3. The methods used require careful analysis of the genera of ends and new techniques for dealing with infinite genus.

COBISS.SI-ID: 18016345

Gabai showed that the Whitehead manifold is the union of two submanifolds each of which is homeomorphic to R^3 and whose intersection is again homeomorphic to R^3. Using a family of generalizations of the Whitehead Link, we show that there are uncountably many contractible 3-manifolds with this double 3-space property. Using a separate family of generalizations of the Whitehead Link and using an extension of interlacing theory, we also show that there are uncountably many contractible 3-manifolds that fail to have this property.

COBISS.SI-ID: 18165593

We show that for any set of primes P there exists a space M_P which is a homology and cohomology 3-manifold with coefficients in Z_p for p \in P and is not a homology or cohomology 3-manifold with coefficients in Z_q for q \notin P. In addition, M_P is not a homology or cohomology 3-manifold with coefficients in Z or Q.

COBISS.SI-ID: 17248089

A solenoid is an inverse limit of circles. When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with nonhomeomorphic complements.

COBISS.SI-ID: 17540697