Robert Myers
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 $W$.
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Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along their boundary intervals. Every such strip has a foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals, whence we get a foliation $Δ$ on all of $Z$. Many types of foliations on surfaces with leaves homeomorphic to the real line have such "striped" structure. That fact was discovered by W. Kaplan (1940-41) for foliations on the plane $\mathbb{R}^2$ by level-set of pseudo-harmonic functions $\mathbb{…
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 a…
This paper uses Brin and Thickstun's theory of end reductions of non-compact 3-manifolds to study groups of covering translations of irreducible contractible open 3-manifolds W which are not homeomorphic to R^3. We associate to W an object S(W) called the simplicial complex of minimal R^2-irreducible end reductions of W. Whenever W covers another 3-manifold the group of covering translations is isomorphic to a fixed point free group of automorphisms of S(W). We apply this result to give uncountably many examples of R^2-irreducible such W which cover 3-manifolds with infinite cyclic fundamental…
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}.)
By a construction of Berstein and Edmonds every proper branched cover f between manifolds is a factor of a branched covering orbit map from a locally connected and locally compact Hausdorff space called the monodromy space of f to the target manifold. For proper branched covers between 2-manifolds the monodromy space is known to be a manifold. We show that this does not generalize to dimension 3 by constructing a self-map of the 3-sphere for which the monodromy space is not a locally contractible space.
We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic $4$-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well known invariants of homology spheres that obstruct acyclic bounding 4-manifolds, and then use a method based on equivariant Yang-Mills moduli spaces to rule out extensions of the actions.
In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under the uniform topology. Further more, combining with similarity transformation, this enables us to induce a global homotopy property of groups of uniform homeomorphisms of metric spaces with Euclidean ends. In particular, we show that the identity component of the group of uniform h…
This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical which is a purely homotopy theoretical condition implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K- and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are to…
We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and applications of the Farrell-Jones Conjecture for algebraic $K$-and $L$-theory for a group ring $RG$ and coefficients in an additive category. These conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of …
Topological actions of wreath products
本文给出了花冠积在正则覆盖上的拓扑作用的基本群显式描述，并应用于计算可定向紧曲面上典型光滑函数在微分同胚群作用下的轨道同伦类型。
We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively curved, locally symmetric manifold. Another application is the classification of all contractible Riemannian manifolds covering both compact and (noncompact, complete) finite volume manifold. There are also applications to the Hopf Conjecture, a new proof of Kazhdan's Conjecture (Frankel's Theorem) in complex geometry, etc. Ideas in the proof come from Lie the…
In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is not stably Z, then M does not cover any manifold (except itself). In the non-manifold case, Wright's method showed that when a one-ended, simply connected, locally compact ANR X with pro-monomorphic fundamental group at infinity admits an action of Z by covering transformations then the fundamental group at infinity of X is (up to pro-isomorphism) an inverse …
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