Approximate fibrations

Coram, Donald; Duval, P.F.

doi:10.1216/rmj-1977-7-2-275

Cited by 81 publications

(56 citation statements)

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“…is one for any vertex v. Since N is a hopfian manifold and π 1 (N ) has Property NCSA, R|p −1 (y) : p −1 (y) → p −1 (v) is a homotopy equivalence and N is a codimension-(s+1) PL fibrator, as required [1]. Theorem 2.9.…”

Section: Partially Acyclic Manifolds As Codimension-k Fibratorsmentioning

confidence: 87%

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

Im¹

2010

Communications of the Korean Mathematical Society

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Section: Partially Acyclic Manifolds As Codimension-k Fibratorsmentioning

confidence: 87%

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

Im¹

2010

Communications of the Korean Mathematical Society

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“…Throughout the remainder of this section we use O to denote the origin in R 2 . Since p is an approximate fibration over R 2 \{O}, the homotopy exact sequence for the approximate fibration [CD1,Corollary 3.5] shows that …”

Section: Conditions Yielding Approximate Fibrationsmentioning

confidence: 99%

“…The manifold N is called a fibrator if all such maps p are approximate fibrations. To appreciate the fibrator concept, it helps to regard approximate fibrations as beneficial; they are essentially as efficacious as the notable class of fibrations (see [CD1], [CD2] for evidence).…”

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confidence: 99%

Codimension 2 nonfibrators with finite fundamental groups

Daverman

1999

Proc. Amer. Math. Soc.

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Abstract. Fibrators are n-manifolds which automatically induce approximate fibrations, in the following sense: given any proper mapping p from an (n + k)-manifold onto a finite-dimensional metric space such that, up to shape, each point-preimage is a copy of the fibrator, p is necessarily an approximate fibration. This paper sets forth new examples, for the case k = 2, of nonfibrators whose fundamental groups are finite. Nonfibrators are surprisingly scarce. The most obvious manifold to consider, the 1-sphere, fails to be a fibrator, due to a partition of the Möbius band into circles, and, perhaps even more familiarly, in codimension 2 due to the singularities typically present in Seifert fibrations. Somewhat similarly, the n-sphere, n > 1, is a nonfibrator, but only in case the dimension of the supermanifold exceeds 2n. Among closed 2-manifolds merely the torus and Klein bottle fail to be fibrators in codimension 2 [D1]. No manifold N admitting a regular self-covering map N → N with a cyclic group of covering transformations can be a codimension 2 fibrator [D1,

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“…All of our results depend on the notion of an approximate fibration, which was originally defined by Coram and Duvall [2] in terms of lifting properties. A map p : E → B has the approximate homotopy lifting property for a space X if given an open cover ε of B and maps g : X → E and H : X × I → B such that pg = H 0 , then there exists a map G : X × I → E such that G 0 = g and pG and H are ε-close.…”

Section: Introductionmentioning

confidence: 99%

A Concordance Extension Theorem

Jones

1996

Trans. Amer. Math. Soc.

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Abstract. Let p : E → B be a manifold approximate fibration between closed manifolds, where dim(E) ≥ 4, and let M (p) be the mapping cylinder of p. In this paper it is shown that if g : B × I → B × I is any concordance on B, then there exists a concordance G :As an application, if N n and M n+j are closed manifolds where N is a locally flat submanifold of M and n ≥ 5 and j ≥ 1, then a concordance g :This uses the fact that under these hypotheses there exists a manifold approximate fibration p : E → N , where E is a closed (n + j − 1)-manifold, such that the mapping cylinder M(p) is homeomorphic to a closed neighborhood of N in M by a homeomorphism which is the identity on N .

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Approximate fibrations

Cited by 81 publications

References 0 publications

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

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

Codimension 2 nonfibrators with finite fundamental groups

A Concordance Extension Theorem

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