Splitting homomorphisms and the Geometrization Conjecture

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture, and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.Comment: 11 pages, Some typos are correcte

End sums of irreducible open 3-manifolds

An end sum is a non-compact analogue of a connected sum. Suppose we are given two connected, oriented $n$-manifolds $M_1$ and $M_2$. Recall that to form their connected sum one chooses an $n$-ball in each $M_i$, removes its interior, and then glues together the two $S^{n-1}$ boundary components thus created by an orientation reversing homeomorphism. Now suppose that $M_1$ and $M_2$ are also open, i.e. non-compact with empty boundary. To form an end sum of $M_1$ and $M_2$ one chooses a halfspace $H_i$ (a manifold \homeo\ to ${\bold R}^{n-1} \times [0, \infty)$) embedded in $M_i$, removes its interior, and then glues together the two resulting ${\bold R}^{n-1}$ boundary components by an orientation reversing homeomorphism. In order for this space $M$ to be an $n$-manifold one requires that each $H_i$ be {\bf end-proper} in $M_i$ in the sense that its intersection with each compact subset of $M_i$ is compact. Note that one can regard $H_i$ as a regular neighborhood of an end-proper ray (a 1-manifold \homeo\ to $[0,\infty)$) \ga_i in $M_i$

Contractible open 3-manifolds which non-trivially cover only non-compact 3-manifolds

Suppose $M$ is a closed, connected, orientable, \irr\ \3m\ such that $G=\pi_1(M)$ is infinite. One consequence of Thurston's geometrization conjecture is that the universal covering space $\widetilde{M}$ of $M$ must be \homeo\ to \RRR. This has been verified directly under several different additional assumptions on $G$. (See, for example, \cite{2}, \cite{3}, \cite{6}, \cite{19}.

On covering translations and homeotopy groups of contractible open n-manifolds

This paper gives a new proof of a result of Geoghegan and Mihalik which states that whenever a contractible open $n$-manifold $W$ which is not homeomorphic to $\mathbf{R}^n$ is a covering space of an $n$-manifold $M$ and either $n \geq 4$ or $n=3$ and $W$ is irreducible, then the group of covering translations injects into the homeotopy group of $W$.Comment: 4 pages, LaTeX, amsart styl