Codimension 2 nonfibrators with finite fundamental groups

Daverman, Robert J.

doi:10.1090/s0002-9939-99-05192-8

Cited by 8 publications

(1 citation statement)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following is the main question that we address in this paper: Which direct products of Hopfian manifolds are shape m simpl o-fibrators? The question of whether the collection of codimension-k PL (or shape m simpl ) fibrators is closed under taking Cartesian product remains unsolved, but seems not likely (because of the examples presented in [10]). Some partial answers to this question for codimension-k PL fibrators (as well as PL fibrators) have been given in [12,20,21,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

Coperfectly Hopfian groups and shape fibrator's properties

Vasilevska

2019

Topology and its Applications

View full text Add to dashboard Cite

This paper provides further investigation of the concept of shape m simplfibrators (previously introduced by the author). The main results identify shape m simpl -fibrators among direct products of Hopfian manifolds. First it is established that every closed orientable manifold homotopically determined by π1 with coperfectly Hopfian group (a new class of Hopfian groups that are introduced here) is a shape m simpl o-fibrator if it is a codimension-2 fibrator (Theorem 5.4). The main result (Theorem 6.2) states that the direct product of two closed orientable manifolds (of different dimension) homotopically determined by π1 and with coperfectly Hopfian fundamental groups (one normally incommensurable with the other one) is a shape m simpl o-fibrator, if it is a Hopfian manifold and a codimension-2 fibrator.

show abstract

Section: Introductionmentioning

confidence: 99%

Coperfectly Hopfian groups and shape fibrator's properties

Vasilevska

2019

Topology and its Applications

View full text Add to dashboard Cite

show abstract

Fibrator properties of manifolds

Daverman

2005

Topology and its Applications

View full text Add to dashboard Cite

Special manifolds and shape fibrator properties

Vasilevska

2006

Topology and its Applications

View full text Add to dashboard Cite

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. Manifold homotopically determined by π 1 0. Introduction In 1977, Coram and Duvall [3,4] introduced the concept of approximate fibrations as a generalization of both Hurewicz fibrations and cell-like maps. Approximate fibrations are proper mappings that satisfy an approximate version of the homotopy lifting property-the defining property for fibrations. Why study approximate fibrations? Coram and Duvall showed that approximate fibrations have shape theoretic properties analogous to the homotopy theoretic properties of Hurewicz fibrations. For each approximate fibration p : M → B, where M, B are locally compact ANRs, they showed that all fibers of p are fundamental absolute neighborhood retracts [1, p. 266], and moreover that if B is path connected, then any two fibers have the same shape. The most useful property is the existence of an exact sequence involving the homotopy groups of domain, target and shape-theoretical homotopy groups of any point inverse of p. When working with the PL approximate fibration these properties of an approximate fibration reduce to the usual properties of Hurewicz fibration because the fibers are ANRs, so the ith shape homotopy groups are isomorphic to ith homotopy groups. A question that arises is this: how can we detect approximate fibration so we can use these nice properties? Sometimes a proper map defined on an arbitrary manifold of a specific dimension can be recognized as an approximate fibration due to having point inverses all of a certain homotopy type (or shape). So the above question that has been addressed for more than two decades can be restated in a different form: * Tel.: +(

show abstract

Codimension 2 nonfibrators with finite fundamental groups

Cited by 8 publications

References 16 publications

Coperfectly Hopfian groups and shape fibrator's properties

Coperfectly Hopfian groups and shape fibrator's properties

Fibrator properties of manifolds

Special manifolds and shape fibrator properties

Contact Info

Product

Resources

About