arXiv subjects

Robert Myers

Publications and source records attributed to Robert Myers.

26 records · Page 2Linked to original sources

arXiv1997

Compactifying sufficiently regular covering spaces of compact 3-manifolds

In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, $P^2$-irreducible 3-manifold corresponding to a non-trivial, finitely-generated subgroup of its fundamental group is infinite, then either the covering space is almost compact or the subgroup is infinite cyclic and has normalizer a non-finitely-generated subgroup of the rational numbers. In the first case additional information is obtained which is then used to relate Thurston's hyperbolization and virtual bundle conjectures to some algebraic conjectures about certain 3-manifold groups.

math.GT↗

arXiv1996

R^2-irreducible universal covering spaces of P^2-irreducible open 3-manifolds

An irreducible open 3-manifold $W$ is {\bf R}$^2$-irreducible if every proper plane in $W$ splits off a halfspace. In this paper it is shown that if such a $W$ is the universal cover of a connected, {\bf P}$^2$-irreducible open 3-manifold $M$ with finitely generated fundamental group, then either $W$ is homeomorphic to {\bf R}$^3$ or the group is a free product of infinite cyclic groups and infinite closed surface groups. Given any such finitely generated group uncountably many $M$ are constructed with that fundamental group such that their universal covers are {\bf R}$^2$-irreducible, are not homeomorphic to {\bf R}$^3$, and are pairwise non-homeomorphic. These results are related to the conjecture that closed, orientable, irreducible, aspherical 3-manifolds are covered by {\bf R}$^3$.

math.GT↗

arXiv1996

Contractible open 3-manifolds with free covering translation groups

This paper concerns the class of contractible open 3-manifolds which are ``locally finite strong end sums'' of eventually end-irreducible Whitehead manifolds. It is shown that whenever a 3-manifold in this class is a covering space of another 3-manifold the group of covering translations must be a free group. It follows that such a 3-manifold cannot cover a closed 3-manifold. For each countable free group a specific uncountable family of irreducible open 3-manifolds is constructed whose fundamental groups are isomorphic to the given group and whose universal covering spaces are in this class and are pairwise non-homeomorphic.

math.GT↗

arXiv1996

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

Suppose $M$ is a closed, connected, orientable, \irr\ \3m\ such that $G=π_1(M)$ 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}.)

math.GT↗

arXiv1996

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 $M_i$.

math.GT↗

arXiv1996

Attaching boundary planes to irreducible open 3-manifolds

Given any connected, open 3-manifold $U$ having finitely many ends, a non-compact 3-manifold $M$ is constructed having the following properties: the interior of $M$ is homeomorphic to $U$; the boundary of $M$ is the disjoint union of finitely many planes; $M$ is not almost compact; $M$ is eventually end-irreducible; there are no proper, incompressible embeddings of $S^1 \times \bold R$ in $M$; every compact subset of $M$ is contained in a larger compact subset whose complement is anannular; there is a compact subset of $M$ whose complement is $\bold P^2$-irreducible. If $U$ is irreducible it also has the following two properties: every proper, non-trivial plane in $M$ is boundary-parallel; every proper surface in $M$ each component of which has non-empty boundary and is non-compact and simply connected lies in a collar on $\partial M$. This construction can be chosen so that $M$ admits no homeomorphisms which take one boundary plane to another or reverse orientation. For the given $U$ there are uncountably many non-homeomorphic such $M$. Two auxiliary results may be of independent interest. First, general conditions are given under which infinitely many ``trivial'' compact components of the intersection of two proper, non-compact surfaces in an irreducible 3-manifold can be removed by an ambient isotopy. Second, $n$ component tangles in a 3-ball are constructed such that every non-empty union of components of the tangle has hyperbolic exterior.

math.GT↗

arXiv1995

The Value of Singularities

We point out that spacetime singularities play a useful role in gravitational theories by eliminating unphysical solutions. In particular, we argue that any modification of general relativity which is completely nonsingular cannot have a stable ground state. This argument applies both to classical extensions of general relativity, and to candidate quantum theories of gravity.

gr-qc↗

arXiv1992

What unitary matrix models are not?

We report results of two investigations of the double-scaling equations for the unitary matrix models. First, the fixed area partition functions have all positive coefficients only for the first four critical points. This implies that the critical points at $k\ge5$ describe non-unitary continuum theories. Secondly, we examine a conjectured connection to branched polymers, but find that the scaling solutions of the unitary models do not agree with those of a particular model describing branched polymers.

hep-th↗