Embeddings of open manifolds

Let be the simplicial group of homeomorphisms of . The following theorems are proved.

Theorem A. Let be a topological manifold of dim 5 with a finite number of tame ends , . Let be the simplicial group of end preserving homeomorphisms of . Let be a periodic neighborhood of each end in , and let be manifold approximate fibrations. Then there exists a map such that the homotopy fiber of is equivalent to , the simplicial group of homeomorphisms of which have compact support.

Theorem B. Let be a compact topological manifold of dim 5, with connected boundary , and denote the interior of by . Let be the restriction map and let be the homotopy fiber of over . Then is isomorphic to for , where is the concordance space of .

Theorem C. Let be a manifold approximate fibration with dim 5. Then there exist maps and for , such that , where is a compact and connected manifold and is the infinite cyclic cover of .

     

Coverings of 3-manifolds by open balls and two open solid tori

Hempel and McMillan showed that a closed 3-manifold that can be covered by three open balls is a connected sum of S³- and S²-bundles over S¹. In this paper we obtain a classification of all closed 3-manifolds that can be covered by two open balls and one open solid torus or by one open ball and two open solid tori.     

A compactification of open varieties

In this paper we prove a general method to compactify certain open varieties by adding normal crossing divisors. This is done by showing that blowing up along an arrangement of subvarieties can be carried out. Important examples such as Ulyanov's configuration spaces and complements of arrangements of linear subspaces in projective spaces, etc., are covered. Intersection ring and (nonrecursive) Hodge polynomials are computed. Furthermore, some general structures arising from the blowup process are also described and studied.

     

Cohomology of open torus manifolds

The notion of an open torus manifold is introduced. A compact open torus manifold is a torus manifold introduced earlier. It is shown that the equivariant cohomology ring of an open torus manifold M is the face ring of a simplicial poset when every face of the orbit space Q is acyclic. This result extends an earlier result by Masuda and Panov, and the proof here is more direct. Reisner’s theorem is then applied to our setting, and a necessary and sufficient condition is given for the equivariant cohomology ring of M to be Cohen-Macaulay in terms of the orbit space Q.     

On differentiability of open mappings

The well-known Men’shov and Gehring-Lehto theorems on the differentiability of topological mappings of plane domains are generalized to the case of continuous open mappings of many-dimensional domains. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1587–1600, December, 1998. This work was financially supported by the INTAS Foundation (grant No. 94-1474).     

Integrals of open two-dimensional lattices

We present an explicit formula for integrals of the open two-dimensional Toda lattice of type A_n. This formula is applicable for various reductions of this lattice. As an illustration, we find integrals of the G₂ Toda lattice. We also reveal a connection between the open An Toda and Shabat-Yamilov lattices.     

Deformations of open stein manifolds

We consider the deformation of the complex structure on an open Stein manifold. We show that a tame, compactly supported deformation of a Stein manifold is trivial. The remainder of our results are for deformations of the standard complex structure on Cⁿ. A deformation of Cⁿ which tends to a constant deformation faster that r^-3 is trivial. Harmonic deformation tensors (w.r.t to the standard Euclidean metric) which are regular at infinity are constant.     

A selection of open problems

This is a collection of open problems which touch on Neil Hindman's mathematics and were collected in conjunction with the Conference on Ramsey Theory and Topological Algebra in his honor.     

Pseudo-rotations of the open annulus

In this paper, we study pseudo-rotations of the open annulus, i.e. conservative homeomorphisms of the open annulus whose rotation set is reduced to a single irrational number (the angle of the pseudo-rotation). We prove in particular that, for every pseudo-rotation h of angle ρ, the rigid rotation of angle ρ is in the closure of the conjugacy class of h. We also prove that pseudo-rotations are not persistent in C^r topology for any r ≥ 0.     

Effectively open real functions

A function f is continuous iff the pre-image f^-1[V] of any open set V is open again. Dual to this topological property, f is called open iff the image f[U] of any open set U is open again. Several classical open mapping theorems in analysis provide a variety of sufficient conditions for openness.By the main theorem of recursive analysis, computable real functions are necessarily continuous. In fact they admit a well-known characterization in terms of the mapping Vf^-1[V] being effective: given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f^-1[V]. Analogously, effective openness requires the mapping Uf[U] on open real subsets to be effective.The present work combines real analysis with algebraic topology and Tarski's quantifier elimination to effectivize classical open mapping theorems and to establish several rich classes of real functions as effectively open.     

Chip-firing via open covers

We investigate chip-firing with respect to open covers of discrete graphs and metric graphs. For the case of metric graphs we show that given an open cover and a sink q, stabilization of a divisor D is unique and that there is a distinguished configuration equivalent to D, which we call the critical configuration. Also, we show that given a double cover of the metric graph by stars, which is the continuous analogue of the sandpile model, the critical configurations are in bijection with reduced divisors. Passing to the discrete case, we interpret open covers of a graph as simplicial complexes on the vertex and observe that chip-firing with respect to a simplicial complex is equivalent to the model introduced by Paoletti [G. Paoletti. July 11 2007: Master in Physics at University of Milan, defending thesis “Abelian sandpile models and sampling of trees and forests”; supervisor: Prof. S. Caracciolo. http://pcteserver.mi.infn.it/caraccio/index.html]. We generalize this setup for directed graphs using weighted simplicial complexes on the vertex set and show that the fundamental results extend. In the undirected case we present a generalization of the Cori-Le Borgne algorithm for chip-firing models via open covers, giving an explicit bijection between the critical configurations and the spanning trees of a graph.(http://www.elsevier.com/locate/endm)