Approximate Fibrations Research Articles

Directed approximate fibrations

Directed approximate fibrations

Erratum to: Dihedral manifold approximate fibrations over the circle

The original publication of the article has been published with an incorrect proof of Theorem 8.1 in [1]. Specifically, we know neither how to verify nor disprove the claim that the homeomorphism J1 : W → W satisfies the dihedral relation. Thus, we retract Theorem 8.1. Consequently, the assertion that (3) implies (4) in Theorem 1.1 is unproven. The remaining implications in the proof of Theorem 1.1 hold. Theorems 1.2, 1.3, and 1.4 are independent of Theorem 8.1 and remain valid. In particular, the material in Sects. 2–7 holds (as well as Lemma 8.2).

MANIFOLDS WITH TRIVIAL HOMOLOGY GROUPS IN SOME RANGE AS CODIMENSION-K FIBRATORS

Approximate fibrations provide a useful class of maps. Fibrators give instant detection of maps in this class, and PL fibrators do the same in the PL category. We show that rational homology spheres with some additional conditions are codimension-k PL fibrators and PL manifolds with trivial homology groups in some range can be codimension-k (k > 2) PL fibrators.

Dihedral manifold approximate fibrations over the circle

Consider the cyclic group C 2 of order two acting by complex-conjugation on the unit circle S 1. The main result is that a finitely dominated manifold W of dimension > 4 admits a cocompact, free, discontinuous action by the infinite dihedral group D ∞ if and only if W is the infinite cyclic cover of a free C 2-manifold M such that M admits a C 2-equivariant manifold approximate fibration to S 1. The novelty in this setting is the existence of codimension-one, invariant submanifolds of M and W. Along the way, we develop an equivariant sucking principle for orthogonal actions of finite groups on Euclidean space.

SOME MANIFOLDS WITH NONZERO EULER CHARACTERISTIC AS PL FIBRATORS

Approximate fibrations form a useful class of maps. By definition fibrators provide instant detection of maps in this class, and PL fibrators do the same in the PL category. We show that every closed s-hopfian t-aspherical manifold N with some algebraic conditions and X(N) 0 is a codimension-(2t + 2) PL fibrator.

Examples of exotic stratifications

We produce examples of manifold stratified pairs in which the lower strata do not have neighborhoods that are mapping cylinders of fiber bundles, or even block bundles. Moreover, the examples do not improve in this regard under stabilization by products with tori. The examples are locally conelike and the lower strata do have neighborhoods which are mapping cylinders of manifold approximate fibrations. They are constructed by combining the classification of manifold approximate fibrations with the authors' classification of neighborhood germs.

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.

PARTIALLY ASHPHERICAL MANIFOLDS WITH NONZERO EULER CHARACTERISTIC AS PL FIBRATORS

Approximate fibrations form a useful class of maps. By definition fibrators provide instant detection of maps in this class, and PL fibrators do the same in the PL category. We show that every closed s-hopfian t-aspherical manifold N with sparsely Abelian, hopfian fundamental group and X(N) <TEX>$\neq$</TEX> 0 is a codimension-(t + 1) PL fibrator.

Fibrator properties of manifolds

Approximate fibrations form a useful class of maps, in part, because they provide computable relationships involving the domain, image and homotopy fiber. Fibrator properties, which pertain to and depend upon the homotopy fiber, allow instant recognition of approximate fibrations. This is a survey of results about those fibrator properties.

PL fibrator properties of partially aspherical manifolds

Approximate fibrations form a useful class of maps. By definition fibrators provide instant detection of maps in this class, and PL fibrators do the same in the PL category. This paper formalizes a natural concept of partial asphericity and establishes fibrator properties of certain partially aspherical closed manifolds. One consequence is that any connected sum of aspherical PL manifolds with residually finite fundamental groups is a codimension-(2 n−2) PL fibrator.

THE PL FIBRATORS AMONG GEOMETRIC 4-MANIFOLDS

Fibrators are closed manifolds which afford instant recognition of approximate fibrations. In this note we determine which 4-manifolds with geometric structure are fibrators.

NEIGHBORHOODS OF STRATA IN MANIFOLD STRATIFIED SPACES Supported in part by NSF Grant DMS-9971367.

Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.

The Approximate Tubular Neighborhood Theorem

Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by mapping cylinder neighborhoods. Thus, this is the best possible topological tubular neighborhood theorem in the stratified setting.

2-groups and approximate fibrations

This paper strives for instant detection of approximate fibrations. A wide class of homotopy types sustains this effort, in the sense that those maps between manifolds for which all point preimages have the homotopy type of a fixed object in this class necessarily are approximate fibrations. Often, as exemplified here, the fundamental group or merely the first homology group carries the determining feature. The main result promises that the closed manifold N is a codimension-2 fibrator, meaning that it has the desired effect on proper maps from an ( n+2)-manifold M onto a metric space B whose fibers all have the homotopy type of N, if π 1( N) is an Abelian 2-group. Under more restrictive conditions, the same conclusion about N holds if either H 1( N) is a cyclic 2-group or if H 1( N) is an arbitrary finite cyclic group and no element of π 1( N) has finite order.

Products and adjunctions of manifold stratified spaces

Basic topological constructions of manifold stratified spaces and stratified approximate fibrations are studied. These include products of manifold stratified spaces, products and compositions of stratified approximate fibrations and Euclidean stabilization of stratified approximate fibrations. The main result shows that the adjunction of two manifold stratified spaces via a manifold stratified approximate fibration is a manifold stratified space.

Necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators

Fibrators help detect approximate fibrations. A closed, connected $n$-manifold $N$ is called a codimension-2 fibrator if each map $p:\ M \to B$ defined on an $(n+2)$-manifold $M$ such that all fibre $p^{-1}(b), b\in B$, are shape equivalent to $N$ is an approximate fibration. The most natural objects $N$ to study are s-Hopfian manifolds. In this note we give some necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators.

Neighborhoods in stratified spaces with two strata

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, a multiparameter isotopy extension theorem and an h-cobordism extension theorem.

Finite groups and approximate fibrations

A closed connected n-manifold N is called a codimension-2 fibrator (codimension-2 orientable fibrator, respectively) if every proper map p :M→B on an (orientable, respectively) (n+2)-manifold M each fiber of which is shape equivalent to N is an approximate fibration. The aim of this paper is to prove the following three statements: (i) If N is a hopfian manifold with |H 1(N)|⩽2 , then N is a codimension-2 orientable fibrator. (ii) If N is a closed manifold whose fundamental group is isomorphic to a finite product of Z 2 r 's for some r, then N is a codimension-2 fibrator. (iii) Let N be a hopfian n-manifold with H 1(N)≈ Z 2 . If the commutator subgroup [π 1(N),π 1(N)] of π 1(N) is a hyperhopfian group, then N is a codimension-2 fibrator. The method used in (i) and (ii) induces the following: If a codimension-2 PL fibrator N satisfies that both π 1(N) and π k−1(N) are finite and that π i(N)=0 for 2⩽i⩽k−2, then N is a codimension- k PL fibrator.

Connected sums of manifolds which induce approximate fibrations

Codimension-2 fibrators are n n -manifolds which automatically induce approximate fibration, in the following sense: given any proper mapping p p from an ( n + 2 ) (n+2) -manifold onto a 2 2 -manifold such that each point-preimage is a copy of the codimension-2 fibrator, p p is necessarily an approximate fibration. In this paper, we give some answers to the following question: given an n n -manifold N N which is a nontrivial connected sum, when is N N a codimension-2 fibrator?

Real projective spaces are nonfibrators

Fibrators are manifolds which, in context, automatically induce approximate fibrations. This paper sets forth a new method for constructing nonfibrators, by establishing that a closed connected manifold N fails to be a codimension k+1 fibrator provided there exists a homeomorphism h of N×S k onto itself such that proj·h:N×{ point}→N is not a homotopy equivalence. Consequently, real projective n-space fails to be a codimension n+1 fibrator, and certain manifolds covered by S 3 fail to be codimension 4 fibrators.