Sections of Serre fibrations with 2-manifold fibers

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney's theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.     

Proper fibrations with Hilbert cube manifold fibers

The main result of this paper asserts that any proper fibration over a finite-dimensional ANR with Hilbert cube manifold fibers is a locally trivial bundle.     

Vassiliev invariants and the Poincaré conjecture

This article examines the relationship between 3-manifold topology and knot invariants of finite type. We prove that in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants. If, on the other hand, Vassiliev invariants distinguish knots in each homotopy sphere, then the Poincaré conjecture is true (i.e. every homotopy 3-sphere is homeomorphic to the standard 3-sphere).     

Special manifolds and shape fibrator properties

Our main interest in this paper is further investigation of the concept of (PL) fibrators (introduced by Daverman [R.J. Daverman, PL maps with manifold fibers, J. London Math. Soc. (2) 45 (1992) 180-192]), in a slightly different PL setting. Namely, we are interested in manifolds that can detect approximate fibrations in the new setting. The main results state that every orientable, special (a new class of manifolds that we introduce) PL n-manifold with non-trivial first homology group is a fibrator in the new category, if it is a codimension-2 fibrator (Theorem 8.2) or has a non-cyclic fundamental group (Theorem 8.4). We show that all closed, orientable surface S with χ(S)<0 are fibrators in the new category.     

Basic intersection cohomology of conical fibrations

We introduce notions of singular fibration and singular Seifert fibration. These notions naturally generalize that o locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations determined by such fibrations, we prove the de Rham theorem for basic intersection cohomology recently introduced by the present authors. One of the main examples of such a structure is the natural projection to the space of fibers of a singular Riemannian foliation determined by a Lie group action on a compact smooth manifold.     

The converse of the Solid Torus Theorem

Given positive integers p and q, a (p,q)-solid torus is a manifold diffeomorphic to Dp+1×Sq while a (p,q)-torus in a closed manifold M is the image of a differentiably embedding Sp×Sq→M. We prove that if n=p+q+1 with p=q=1 or p≠q, then M is homeomorphic to Sn whenever every (p,q)-torus bounds a (p,q)-solid torus. We also prove for p=q that every closed n-manifold for which every (p,p)-torus bounds an irreducible manifold is irreducible. Consequently, every closed 3-manifold for which every torus bounds an irreducible manifold is irreducible.     

Examples of cohomology manifolds which are not homologically locally connected

Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions n?3.     

The hyperspace of hereditarily decomposable subcontinua of a cube is the Hurewicz set

The hyperspaces of hereditarily decomposable continua and of decomposable subcontinua without pseudoarcs in the cube of dimension greater than 2 are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. Moreover, in such a cube, all indecomposable subcontinua form a homotopy dense subset of the hyperspace of (nonempty) subcontinua.     

Basic intersection cohomology of conical fibrations

We introduce notions of singular fibration and singular Seifert fibration. These notions naturally generalize that o locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations determined by such fibrations, we prove the de Rham theorem for basic intersection cohomology recently introduced by the present authors. One of the main examples of such a structure is the natural projection to the space of fibers of a singular Riemannian foliation determined by a Lie group action on a compact smooth manifold.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 235–257.Original Russian Text Copyright © 2005 by M. Saralegi-Aranguren, R. Wolak.This revised version was published online in April 2005 with a corrected issue number.     

On compactification of schemes

For a Serre fibration with a fibre of theK(π, n)'s product type, obstructions to the section problem in each degree are defined by means of the Hirsch complex of fibration. This allows us to give the homotopy classification of sections (maps) as well as other applications. In particular, forG-bundles, these obstructions are related to theA _∞-module structure on the homology of the fibre and, consequently, some results in the fixed point theory are obtained.     

Approximate fibrations and bundle maps on Hilbert cube manifolds

This paper is concerned with parameterized families of approximate fibrations from a compact Hilbert cube manifold M to a compact polyhedron B. The main result shows how to straighten out certain of these families to be nearly like a product. As an application of this technique, it is shown that an approximate fibration p:M → B can be approximated arbitrarily closely by bundle maps if and only if p is homotopic via approximate fibrations to a bundle map. Another result is that the space of bundle maps from M to B is locally n-connected for each n ? 0.     

Homology of gauge groups associated with special unitary groups

We study the homology of gauge groups associated with principal SU(n) bundles over the four-sphere. After computing the mod p homology of based gauge groups of SU(n) by combined use of the Serre spectral sequence and the Eilenberg-Moore spectral sequence, we compute the modp homology of gauge groups of SU(n) using the Serre spectral sequence for the evaluation fibration.     

Entropy conjecture for continuous maps of nilmanifolds

In 1974 Michael Shub asked the following question [29]: When is the topological entropy of a continuous mapping of a compact manifold into itself is estimated from below by the logarithm of the spectral radius of the linear mapping induced in the cohomologies with real coefficients? This estimate has been called the Entropy Conjecture (EC). In 1977 the second author and Micha? Misiurewicz proved [23] that EC holds for all continuous mappings of tori. Here we prove EC for all continuous mappings of compact nilmanifolds. Also generalizations for maps of some solvmanifolds and another proof via Lefschetz and Nielsen numbers, under the assumption the map is not homotopic to a fixed points free map, are provided.     

Brush spaces and the fixed point property

We introduce the notions of a brush space and a weak brush space. Each of these spaces has a compact connected core with attached connected fibers and may be either compact or non-compact. Many spaces, both in the Hausdorff non-metrizable setting and in the metric setting, have realizations as (weak) brush spaces. We show that these spaces have the fixed point property if and only if subspaces with core and finitely many fibers have the fixed point property. This result generalizes the fixed point result for generalized Alexandroff/Urysohn Squares in Hagopian and Marsh (2010) [4]. We also look at some familiar examples, with and without the fixed point property, from Bing (1969) [1], Connell (1959) [3], Knill (1967) [7] and note the brush space structures related to these examples.     

Locally G-homogeneous Busemann G-spaces

We present short proofs of all known topological properties of general Busemann G-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally G-homogeneous Busemann G-spaces are homeomorphic and strongly topologically homogeneous. This is a key result in the context of the classical Busemann conjecture concerning the characterization of topological manifolds, which asserts that every n-dimensional Busemann G-space is a topological n-manifold. We also prove that every Busemann G-space which is uniformly locally G-homogeneous on an orbal subset must be finite-dimensional.     

On the knot complement problem for non-hyperbolic knots

This paper explicitly provides two exhaustive and infinite families of pairs (M,k), where M is a lens space and k is a non-hyperbolic knot in M, which produces a manifold homeomorphic to M, by a non-trivial Dehn surgery. Then, we observe the uniqueness of such knot in such lens space, the uniqueness of the slope, and that there is no preserving homeomorphism between the initial and the final M's. We obtain further that Seifert fibered knots, except for the axes, and satellite knots are determined by their complements in lens spaces. An easy application shows that non-hyperbolic knots are determined by their complement in atoroidal and irreducible Seifert fibered 3-manifolds.     

Three-dimensional manifolds having metrics with the same geodesics

We prove that if two Riemannian metrics have the same geodesics on a closed three-dimensional manifold which is homeomorphic neither to a lens space nor to a Seifert manifold with zero Euler number, then the metrics are proportional.     

Families of Lagrangian fibrations on hyperkähler manifolds

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known (due to Huybrechts) that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact, simple hyperkähler manifold with _b2?7$b_2?7$ admits only finitely many deformation types of holomorphic Lagrangian fibrations. We also prove that all known hyperkähler manifolds are never Kobayashi hyperbolic.     

A solution to de Groot’s absolute cone conjecture

A compactum X is an ‘absolute cone’ if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for n≤3 and false for n≥5. For n=4, the absolute cone conjecture is true if and only if the 3-dimensional Poincaré Conjecture is true.     

Strongly countable dimensional compacta form the Hurewicz set

The hyperspaces of strongly countable dimensional compacta of positive dimension and of strongly countable dimensional continua of dimension greater than 1 in the Hilbert cube are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. These facts hold true, in particular, for covering dimension dim and cohomological dimension dimG, where G is any Abelian group.