Manifold Approximate Fibrations Are Approximately Bundles

Hughes, C. Bruce

doi:10.1515/form.1991.3.309

Cited by 12 publications

(36 citation statements)

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“…These generalize unstratified results of Chapman [4], Hughes [10], and Hughes-TaylorWilliams [13]. Stratified sucking gives a condition for a map to be close to a manifold stratified approximate fibration, whereas stratified straightening can be thought of as a uniqueness result which gives a condition for two manifold stratified approximate fibrations to be controlled homeomorphic.…”

Section: The Main Toolssupporting

confidence: 74%

“…Next we generalize the definition of an approximate fibration (as given in [13]) to the stratified setting. Let X = X m ⊇ · · · ⊇ X 0 and Y = Y n ⊇ · · · ⊇ Y 0 be filtered spaces and let p : X → Y be a map (p is not assumed to be stratum preserving).…”

Section: Stratified Spaces and Stratified Approximate Fibrationsmentioning

confidence: 99%

“…For undefined terms related to the controlled category see [13]. We remark that there are also relative and ∆ k -parameter versions of stratified sucking.…”

Section: The Main Toolsmentioning

confidence: 99%

“…A neighborhood germ of Y is an equivalence class of stratified neighborhoods of Y . When Y has just one stratum (i.e., Y is a manifold), one can use the following result (which generalizes [13], [14]) to give a classifying space description of the manifold stratified approximate fibrations which occur in the theorem above. For notation, let B be a connected i-manifold and let q : V → R i be a manifold stratified approximate fibration where all components of strata of V have dimension greater than 4.…”

Section: Theorem 53 (First Topological Isotopy)mentioning

confidence: 99%

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Hughes

1996

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. A theory of tubular neighborhoods for strata in manifold stratified spaces is developed. In these topologically stratified spaces, manifold stratified approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include a multiparameter isotopy extension theorem, neighborhood germ classification and a topological version of Thom's First Isotopy Theorem.

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Section: The Main Toolssupporting

confidence: 74%

Section: Stratified Spaces and Stratified Approximate Fibrationsmentioning

confidence: 99%

“…For undefined terms related to the controlled category see [13]. We remark that there are also relative and ∆ k -parameter versions of stratified sucking.…”

Section: The Main Toolsmentioning

confidence: 99%

Section: Theorem 53 (First Topological Isotopy)mentioning

confidence: 99%

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Hughes

1996

Electron. Res. Announc. Amer. Math. Soc.

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“…(Hughes, Taylor and Williams [5] obtained a topological regular neighborhood theorem for arbitrary locally flat submanifolds in a manifold of dimension ≥ 5, in which the neighborhood is the mapping cylinder of a manifold approximate fibration).…”

Section: ξ : Q → B T Op (Q)mentioning

confidence: 99%

Homology manifold bordism

Johnston

Ranicki

2000

Trans. Amer. Math. Soc.

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Abstract. The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact AN R homology manifolds of dimension ≥ 6 is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.First, we establish homology manifold transversality for submanifolds of dimension ≥ 7: if f : M → P is a map from an m-dimensional homology manifold M to a space P , and Q ⊂ P is a subspace with a topological q-block bundle neighborhood, and m−q ≥ 7, then f is homology manifold s-cobordant to a map which is transverse to Q,Second, we obtain a codimension q splitting obstruction s Q (f ) ∈ LS m−q (Φ) in the Wall LS-group for a simple homotopy equivalence f : M → P from an m-dimensional homology manifold M to an m-dimensional Poincaré space P with a codimension q Poincaré subspace Q ⊂ P with a topological normal bundle, such that s Q (f ) = 0 if (and for m − q ≥ 7 only if) f splits at Q up to homology manifold s-cobordism.Third, we obtain the multiplicative structure of the homology manifold bordism groups Ω H *

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Approximately fibering a manifold over an aspherical one

2017

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Abstract. The paper is devoted to the problem when a map from some closed connected manifold to an aspherical closed manifold approximately fibers, i.e., is homotopic to Manifold Approximate Fibration. We define obstructions in algebraic K-theory. Their vanishing is necessary and under certain conditions sufficient. Basic ingredients are Quinn's Thin h-Cobordism Theorem and End Theorem, and knowledge about the Farrell-Jones Conjectures in algebraic Kand L-theory and the MAF-Rigidity Conjecture by Hughes-Taylor-Williams.

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Manifold Approximate Fibrations Are Approximately Bundles

Cited by 12 publications

References 0 publications

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Homology manifold bordism

Approximately fibering a manifold over an aspherical one

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