Connected sums of manifolds which induce approximate fibrations

Kim, Yongkuk

doi:10.1090/s0002-9939-00-05385-5

Proc. Amer. Math. Soc.

2000

DOI: 10.1090/s0002-9939-00-05385-5

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Connected sums of manifolds which induce approximate fibrations

Yongkuk Kim

Abstract: 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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Cited by 4 publications

References 17 publications

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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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Special manifolds and shape fibrator properties

Vasilevska

2006

Topology and its Applications

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

Im¹,

Kim²

2000

Proc. Amer. Math. Soc.

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Abstract. Fibrators help detect approximate fibrations. A closed, connected n-manifold N is called a codimension-2 fibrator if each map p : M → B defined on an (n + 2)-manifold M such that all fibre p −1 (b), b ∈ 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.

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Manifolds With Trivial Homology Groups in Some Range as Codimension-K Fibrators

Im¹

2010

Communications of the Korean Mathematical Society

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Connected sums of manifolds which induce approximate fibrations

Cited by 4 publications

References 17 publications

Special manifolds and shape fibrator properties

Special manifolds and shape fibrator properties

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

Manifolds With Trivial Homology Groups in Some Range as Codimension-K Fibrators

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