On Steenrod 𝕃-homology, generalized manifolds, and surgery

Hegenbarth, Friedrich; Repovš, Dušan

doi:10.1017/s0013091520000012

Cited by 4 publications

(4 citation statements)

References 22 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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“…We outline the plan how we shall prove Theorem 1.2. In § 2, we shall recall the construction of the map t : N (X n ) → H st n (X n ; L + ) from Hegenbarth and Repovš [9]. In § 3, we shall prove that the map t : N (X n ) → H st n (X n ; L + ) is the composition of maps in the following commutative diagram…”

Section: Introductionmentioning

confidence: 99%

Generalized manifolds, normal invariants, and 𝕃-homology

Hegenbarth¹,

Repovš²

2021

Proceedings of the Edinburgh Mathematical Society

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Let $X^{n}$ be an oriented closed generalized $n$ -manifold, $n\ge 5$ . In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597–607), we have constructed a map $t:\mathcal {N}(X^{n}) \to H^{st}_{n} ( X^{n}; \mathbb{L}^{+})$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$ -manifold. Here, $\mathcal {N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,\,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$ . An important non-trivial question arose whether the map $t$ is bijective (note that this holds in the case when $X^{n}$ is a topological $n$ -manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.

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On Steenrod 𝕃-homology, generalized manifolds, and surgery

Cited by 4 publications

References 22 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

Generalized manifolds, normal invariants, and 𝕃-homology

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