One-dimensional sets and planar sets are aspherical

Cannon, J. W.; Conner, Gregory R.; Zastrow, Andreas

doi:10.1016/s0166-8641(01)00005-0

Cited by 31 publications

(30 citation statements)

References 1 publication

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“…Since planar sets are aspherical [Cannon et al 2002], their homotopy groups of dimension 2 or more are trivial. As a consequence, geodesic Voronoi cells have…”

Section: Geodesic Delaunay Triangulationsmentioning

confidence: 99%

Geodesic delaunay triangulations in bounded planar domains

Oudot

Guibas

Gao

et al. 2010

ACM Trans. Algorithms

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We introduce a new feature size for bounded domains in the plane endowed with an intrinsic metric. Given a point x in a domain X, the systolic feature size of X at x measures half the length of the shortest loop through x that is not null-homotopic in X. The resort to an intrinsic metric makes the systolic feature size rather insensitive to the local geometry of the domain, in contrast with its predecessors (local feature size, weak feature size, homology feature size). This reduces the number of samples required to capture the topology of X, provided that a reliable approximation to the intrinsic metric of X is available. Under sufficient sampling conditions involving the systolic feature size, we show that the geodesic Delaunay triangulation D X (L) of a finite sampling L is homotopy equivalent to X. Under similar conditions, D X (L) is sandwiched between the geodesic witness complex C W X (L) and a relaxed version C W X,ν (L). In the conference version of the paper, we took advantage of this fact and proved that the homology of D X (L) (and hence the one of X) can be retrieved by computing the persistent homology between C W X (L) and C W X,ν (L). Here, we investigate further and show that the homology of X can also be recovered from the persistent homology associated with inclusions of type, under some conditions on the parameters ν ≤ ν ′ . Similar results are obtained for Vietoris-Rips complexes in the intrinsic metric. The proofs draw some connections with recent advances on the front of homology inference from point cloud data, but also with several well-known concepts of Riemannian (and even metric) geometry. On the algorithmic front, we propose algorithms for estimating the systolic feature size of a bounded planar domain X, selecting a landmark set of sufficient density, and computing the homology of X using geodesic witness complexes or Rips complexes.

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“…Since planar sets are aspherical [Cannon et al 2002], their homotopy groups of dimension 2 or more are trivial. As a consequence, geodesic Voronoi cells have…”

Section: Geodesic Delaunay Triangulationsmentioning

confidence: 99%

Geodesic delaunay triangulations in bounded planar domains

Oudot

Guibas

Gao

et al. 2010

ACM Trans. Algorithms

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“…; e i mþ1 . By Zastrow's theorem on asphericity of planar subsets [5,16], this mapping can be extended to the ðm þ 1Þ-dimensional simplex ½e i 0 ; e i 1 ; . .…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…Since the fundamental group of the intersection of any two simply connected planar set is trivial (consider any Jordan curve J in the intersection -the bounded region determined by J belongs to both subsets), it follows that the intersection of every two elements of the family is simply connected. By Zastrow's theorem on asphericity of planar spaces ( [5,16]) and by the Hurewicz theorem (see e.g. [14, p. 397]), every simply connected planar space is acyclic.…”

Section: On Statements By Danzer-grü Nbaum-klee and Eckhoffmentioning

confidence: 99%

On Unions and Intersections of Simply Connected Planar Sets

Karimov

Repovš

Željko

2005

Mh Math

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Abstract. We construct several simple examples of planar compacta which show that without additional conditions, a theorem of Breen and a direct generalization of the Seifert-van Kampen theorem fail. We give answers to two conjectures of Bogatyi and a partial solution to his third conjecture. We give a counterexample to a statement in the classical survey paper by Danzer-Gr€ u unbaum-Klee, related to Molnár's result on intersections of simply connected planar sets.2000 Mathematics Subject Classification: 54F15, 55N10, 54D05

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“…In [4, §5] it was then rigorously proven for an analogously constructed space that this space is definitely not homotopy equivalent to any lower-dimensional space. The examples of [16] and of [4] had large (i.e. uncountable) fundamental groups.…”

Section: Introductionmentioning

confidence: 99%

On continua with homotopically fixed boundary

Malešič

Repovš

Rosicki

et al. 2007

Topology and its Applications

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The paper presents two subcontinua of R n , one Peano-continuum, and one cellular continuum with trivial fundamental group. Both of them have the remarkable property that neither the entire spaces nor (roughly speaking) any part of them is homotopy equivalent to a lower-dimensional space. This extends work of the last three authors and of Karimov from the planar case to the higher-dimensional case, but it also contains in the cellular case the first example with all these properties in dimension two.

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One-dimensional sets and planar sets are aspherical

Cited by 31 publications

References 1 publication

Geodesic delaunay triangulations in bounded planar domains

Geodesic delaunay triangulations in bounded planar domains

On Unions and Intersections of Simply Connected Planar Sets

On continua with homotopically fixed boundary

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