Approximate fibrations and a movability condition for maps

Coram, Donald; Duvall, P. F.

doi:10.2140/pjm.1977.72.41

Cited by 57 publications

(27 citation statements)

References 8 publications

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“…It follows from the movability argument [3,Corollary 3.4] that p is an approximate fibration over C .…”

Section: Proof Of 2) Since Deg R G = 1 We Have An Epimorphism (Rmentioning

confidence: 99%

Connected sums of manifolds which induce approximate fibrations

Kim

2000

Proc. Amer. Math. Soc.

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

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“…It follows from the movability argument [3,Corollary 3.4] that p is an approximate fibration over C .…”

Section: Proof Of 2) Since Deg R G = 1 We Have An Epimorphism (Rmentioning

confidence: 99%

Connected sums of manifolds which induce approximate fibrations

Kim

2000

Proc. Amer. Math. Soc.

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“…If a is an open cover of 7, then a proper map /: X -» 7 is said to be an a-fibration if for all maps F: Z X [0,1] -> 7 and g: Z -» X for which fg = F0, there is a map G: Z X [0,1] -» X such that G0 = g and fG is a-close to F If e > 0, then we also use e to denote the open cover of 7 by balls of diameter e. Thus, we speak of e-fibrations. A map /: X -* Y between ANRs is an approximate fibration provided it is an a-fibration for every open cover a of 7 This notion was introduced in [14]. If /: IxA"^ 7 X A" is an f.p.…”

Section: Ttnvb(rm X F) -mentioning

confidence: 99%

Bounded Homotopy Equivalences of Hilbert Cube Manifolds

Hughes¹

1985

Transactions of the American Mathematical Society

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Abstract. Let M and F be Hubert cube manifolds with F compact. The purpose of this paper is to study homotopy equivalences/: M -» R"' X F which have bounded control in the Redirection. Roughly, these homotopy equivalences form a semi-simplicial complex #^4(Rm X F), the controlled Whitehead space. Using results about approximate fibrations, #"^(Rm x F ) is related to the semi-simplicial complex of bounded concordances on R"' X F. Then the homotopy groups of #^(Rm X F) are computed in terms of the lower algebraic AT-theoretic functors K_i.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Special manifolds and shape fibrator properties

Vasilevska

2006

Topology and its Applications

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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.: +(

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Approximate fibrations and a movability condition for maps

Cited by 57 publications

References 8 publications

Connected sums of manifolds which induce approximate fibrations

Connected sums of manifolds which induce approximate fibrations

Bounded Homotopy Equivalences of Hilbert Cube Manifolds

Special manifolds and shape fibrator properties

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