Controlled homotopy equivalences and structure sets of manifolds

Hegenbarth, Friedrich; Repovš, Dušan

doi:10.1090/s0002-9939-2014-12131-9

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…This paper is a continuation of our systematic study of the characterization problem for generalized n-manifolds, n ≥ 5, (see Cavicchioli et al [5,6] and Hegenbarth and Repov² [23,24,25,26,27,28]). This is a very important class of spaces which in the algebraic sense strongly resemble topological manifolds, whereas in the geometric sense they can fail to be locally Euclidean at any point (see e.g., Cannon [4], Edwards [11], and Repov² [42,43,44]).…”

Section: Introductionmentioning

confidence: 92%

Limits of Manifolds in the Gromov–Hausdorff Metric Space

Hegenbarth

Repovš

2022

Mediterr. J. Math.

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We apply the Gromov-Hausdor metric d G for characterization of certain generalized manifolds. Previously, we have proved that with respect to the metric d G , generalized n-manifolds are limits of spaces which are obtained by gluing two topological n-manifolds by a controlled homotopy equivalence (the so-called 2-patch spaces). In the present paper, we consider the so-called manifold-like generalized n-manifolds X n , introduced in 1966 by Marde²i¢ and Segal, which are characterized by the existence of δmappings f δ of X n onto closed manifolds M n δ , for arbitrary small δ > 0, i.e. there exist onto maps f δ : X n → M n δ such that for everyhas diameter less than δ. We prove that with respect to the metric d G , manifold-like generalized n-manifolds X n are limits of topological n-manifolds M n i . Moreover, if topological n-manifolds M n i satisfy a certain local contractibility condition M(ϱ, n), we prove that generalized n-manifold X n is resolvable.H * (X n , X n \ {x}; Z) ∼ = H * (R n , R n \ {0}; Z), for every x ∈ X.

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Section: Introductionmentioning

confidence: 92%

Limits of Manifolds in the Gromov–Hausdorff Metric Space

Hegenbarth

Repovš

2022

Mediterr. J. Math.

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“…This paper is a continuation of our systematic study of the characterization problem for generalized n-manifolds, n ≥ 5, (see Cavicchioli et al [5,6] and Hegenbarth and Repovš [23,24,25,26,27,28]). This is a very important class of spaces which in the algebraic sense strongly resemble topological manifolds, whereas in the geometric sense they can fail to be locally Euclidean at any point (see e.g., Cannon [4], Edwards [11], and Repovš [42,43,44]).…”

Section: Introductionmentioning

confidence: 93%

Limits of manifolds in the Gromov-Hausdorff metric space

Hegenbarth,

Repovš

2023

Preprint

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We apply the Gromov-Hausdorff metric d G for characterization of certain generalized manifolds. Previously, we have proved that with respect to the metric d G , generalized n-manifolds are limits of spaces which are obtained by gluing two topological n-manifolds by a controlled homotopy equivalence (the so-called 2-patch spaces). In the present paper, we consider the so-called manifold-like generalized n-manifolds X n , introduced in 1966 by Mardešić and Segal, which are characterized by the existence of δmappings f δ of X n onto closed manifolds M n δ , for arbitrary small δ > 0, i.e. there exist onto maps f δ : X n → M n δ such that for every u ∈ M n δ , f −1 δ (u) has diameter less than δ. We prove that with respect to the metric d G , manifold-like generalized n-manifolds X n are limits of topological n-manifolds M n i . Moreover, if topological n-manifolds M n i satisfy a certain local contractibility condition M(̺, n), we prove that generalized n-manifold X n is resolvable.H * (X n , X n \ {x}; Z) ∼ = H * (R n , R n \ {0}; Z), for every x ∈ X.

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“…can be split into adic surgery problems which define [f, b] (see Hegenbarth and Repovš [9] and Ranicki [25]). This is due to transversality with respect to a dual cell structure on X n .…”

Section: Theorem 13 (Ferry [5] Milnormentioning

confidence: 99%

On Steenrod 𝕃-homology, generalized manifolds, and surgery

Hegenbarth¹,

Repovš²

2020

Proceedings of the Edinburgh Mathematical Society

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The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a generalized n-manifold Xn, in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the nth Steenrod homology group $H^{st}_{n} (X^{n}, \mathbb {L}^+)$, where 𝕃+ is the connected covering spectrum of the periodic surgery spectrum 𝕃, avoiding the use of the geometric splitting procedure, the use of which is standard in surgery on topological manifolds.

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Controlled homotopy equivalences and structure sets of manifolds

Cited by 5 publications

References 12 publications

Limits of Manifolds in the Gromov–Hausdorff Metric Space

Limits of Manifolds in the Gromov–Hausdorff Metric Space

Limits of manifolds in the Gromov-Hausdorff metric space

On Steenrod 𝕃-homology, generalized manifolds, and surgery

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