A Class of Decompositions of E n which are Factors of E n + 1

Bailey, John

doi:10.2307/1995389

Cited by 2 publications

(4 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In other words, R 4 admits a trivial partition into open arcs (being not a foliation) such that the quotient space B is not a 3-manifold. That example was improved by many authors, see e.g R. Rosen [38], J. Kim [22], J. Bailey [1], L. Rubin [39].…”

Section: One-dimensional Foliationsmentioning

confidence: 99%

One-dimensional foliations on topological manifolds

Maksymenko,

Polulyakh

2016

Preprint

View full text Add to dashboard Cite

Let X be an (n + 1)-dimensional manifold, ∆ be a one-dimensional foliation on X, and p : X → X/∆ be a quotient map. We will say that a leaf ω of ∆ is special whenever the space of leaves X/∆ is not Hausdorff at ω. We present necessary and sufficient conditions for the map p : X → X/∆ to be a locally trivial fibration under assumptions that all leaves of ∆ are non-compact and the family of all special leaves of ∆ is locally finite.

show abstract

Section: One-dimensional Foliationsmentioning

confidence: 99%

One-dimensional foliations on topological manifolds

Maksymenko,

Polulyakh

2016

Preprint

View full text Add to dashboard Cite

show abstract

“…As a convention we use C to denote the "middle thirds" Cantor set on [0, 1]. We let C=C x [0, 1 ] and call any set homeomorphic to C a Cantor set of arcs.The material in this paper was contained in a dissertation [2] written at the University of Tennessee under the direction of Ralph J. Bean to whom the author wishes to express his gratitude.To prove Theorem 1 we generalize the technique used by Bing in [4]. We define a decomposition B of Fn+1 by B={(g, w)eEnxE1…”

mentioning

confidence: 99%

“…The material in this paper was contained in a dissertation [2] written at the University of Tennessee under the direction of Ralph J. Bean to whom the author wishes to express his gratitude.…”

mentioning

confidence: 99%

See 1 more Smart Citation

A class of decompositions of 𝐸ⁿ which are factors of 𝐸ⁿ⁺¹

Bailey¹

1970

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

has given an example of a space which fails to be a 3-manifold at each of a Cantor set of points, but such that the cartesian product of the space and F1 is homeomorphic to F4 [4]. This result has led to the study of other nonmanifold factors of F4 ; a summary of results in this area may be found in Armentrout's paper in [5].Bing's example arises from a decomposition of F3 into points and arcs; the defining sequence for the arcs consists of collections of double tori resembling bones, hence the example is generally called the dogbone space. In this paper we generalize Bing's result with the following theorem. Standard notation is used with the following adaptations. We use 5(^4, e) to denote an e neighborhood of the set A, Cl denotes closure, and equality of topological spaces indicates only that the spaces are homeomorphic. As a convention we use C to denote the "middle thirds" Cantor set on [0, 1]. We let C=C x [0, 1 ] and call any set homeomorphic to C a Cantor set of arcs.The material in this paper was contained in a dissertation [2] written at the University of Tennessee under the direction of Ralph J. Bean to whom the author wishes to express his gratitude.To prove Theorem 1 we generalize the technique used by Bing in [4]. We define a decomposition B of Fn+1 by B={(g, w)eEnxE1: ge G, weE1}. The identity map is a homeomorphism of En¡GxE1 onto En+1¡B. A pseudo-isotopy / of En+1 x [0, 1] onto Fn+1 is then constructed in such a way that/(x, 0) is the identity map and/(x, 1) takes each element of B onto a distinct point of En + 1. This establishes that En+1IB=En + 1 [4], [8].

show abstract

A Class of Decompositions of E n which are Factors of E n + 1

Cited by 2 publications

References 0 publications

One-dimensional foliations on topological manifolds

One-dimensional foliations on topological manifolds

A class of decompositions of 𝐸ⁿ which are factors of 𝐸ⁿ⁺¹

Contact Info

Product

Resources

About