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Topology of homology manifolds
 Ann. of Math
, 1996
"... The study of the localglobal geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satis ed by manifolds [25, 57]. Bing's work on decomposition space th ..."
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Cited by 53 (14 self)
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The study of the localglobal geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satis ed by manifolds [25, 57]. Bing's work on decomposition space theory opened new perspectives. He constructed important examples of 3dimensional homology manifolds with nonmanifold points, which led to the study of other structural properties of these spaces, and also established his shrinking criterion that can be used to determine when homology manifolds obtained as decomposition spaces of manifolds are manifolds [4]. In the 1970s, the fundamental work of Cannon and Edwards on the double suspension problem led Cannon to propose a conjecture on the nature of manifolds, and generated a program that culminated with the EdwardsQuinn characterization of higherdimensional topological manifolds as ENR homology manifolds satisfying a weak general position property known as the disjoint disks property [17, 26,23]. Starting with the work of Quinn [45, 47], a new viewpoint has emerged. Recent advances [10] use techniques of controlled topology to produce a wealth of previously
Complete normality and metrization theory of manifolds
 TOP. APPL
"... A manifold is a connected Hausdorff space in which every point has a neighborhood homeomorphic to Euclidean nspace (n is unique). A space is collectionwise Hausdorff (cwH) if every closed discrete subspace D can be expanded to a disjoint collection of open sets each of which meets D in one point. T ..."
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Cited by 5 (2 self)
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A manifold is a connected Hausdorff space in which every point has a neighborhood homeomorphic to Euclidean nspace (n is unique). A space is collectionwise Hausdorff (cwH) if every closed discrete subspace D can be expanded to a disjoint collection of open sets each of which meets D in one point. There are exactly two examples of 1dimensional nonmetrizable hereditarily normal, hereditarily cwH manifolds: the long line and the long ray. The main new result is that if it is consistent that there is a supercompact cardinal, it is consistent that every hereditarily normal, hereditarily cwH manifold of dimension greater than 1 is metrizable.
A theory of binary digital pictures
 Computer Vision, Graphics and Image Processing 32
, 1985
"... We study 2 and 3dimensional digital geometry in the context of almost arbitrary adjacency relations. (Previous authors have based their work on particular adjacency relations.) We define a binary digital picture to be a pair whose components are a set of latticepoints and an adjacency relation on ..."
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Cited by 3 (0 self)
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We study 2 and 3dimensional digital geometry in the context of almost arbitrary adjacency relations. (Previous authors have based their work on particular adjacency relations.) We define a binary digital picture to be a pair whose components are a set of latticepoints and an adjacency relation on the whole lattice. We show how a wide class of digital pictures have natural "continuous analogs. " This enables us to use methods of continuous topology in studying digital pictures. We are able to prove general results on the connectivity of digital borders, which generalize results that have appeared in the literature. In the 3dimensional case we consider the possibility of using a uniform relation on the whole lattice. (In the past authors have used different types of adjacency for "object " and "background.") 9 1985 Academic Press, Inc. PREREQUISITES Familiarity with the content of [15] is probably essential for understanding some of the remarks in the introduction; the rest of the paper is more or less selfcontained, but familiarity with [15] might still be helpful. The graphtheoretic terminology we use is defined in the first chapter of [3]. A httle elementary topology is assumed in our discussion of continuous analogsthe relevant concepts are covered in the third and fourth chapters of [1].
The HilbertSmith Conjecture by
, 2001
"... In 1900, Hilbert proposed twentythree problems [8]. For an excellent discussion concerning these problems, see the Proceedings of Symposia In Pure Mathematics concerning “Mathematical Developments Arising From Hilbert Problems ” [3]. The abstract by C.T. Yang [22] gives a review of Hilbert’s Fifth ..."
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In 1900, Hilbert proposed twentythree problems [8]. For an excellent discussion concerning these problems, see the Proceedings of Symposia In Pure Mathematics concerning “Mathematical Developments Arising From Hilbert Problems ” [3]. The abstract by C.T. Yang [22] gives a review of Hilbert’s Fifth Problem “How is Lie’s concept of continuous
unknown title
"... arXiv version: fonts, pagination and layout may vary from GT published version Universal circles for quasigeodesic flows ..."
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arXiv version: fonts, pagination and layout may vary from GT published version Universal circles for quasigeodesic flows
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"... arXiv version: fonts, pagination and layout may vary from GT published version Universal circles for quasigeodesic flows ..."
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arXiv version: fonts, pagination and layout may vary from GT published version Universal circles for quasigeodesic flows
A Proof Of The HilbertSmith Conjecture by
, 2001
"... The HilbertSmith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (“Finitetoone mappi ..."
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The HilbertSmith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (“Finitetoone mappings of manifolds”, Trans. of Math. Sk. 65 (107), 1964.) His work is generalized to the orbit map of an effective action of a padic group on compact connected nmanifolds with the aid of some new ideas. There is no attempt to use Smith Theory even though there may be similarities.
QUASICONFORMALITY, QUASISYMMETRY, AND REMOVABILITY IN LOEWNER SPACES
 VOL. 101, NO. 3 DUKE MATHEMATICAL JOURNAL
, 2000
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Size functions for shape recognition in the presence of occlusions
, 2009
"... In Computer Vision the ability to recognize objects in the presence of occlusions is a necessary requirement for any shape representation method. In this paper we investigate how the size function of an object shape changes when a portion of the object is occluded by another object. More precisely, ..."
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In Computer Vision the ability to recognize objects in the presence of occlusions is a necessary requirement for any shape representation method. In this paper we investigate how the size function of an object shape changes when a portion of the object is occluded by another object. More precisely, considering a set X = A ∪ B and a measuring function ϕ on X, we establish a condition so that ℓ (X,ϕ) = ℓ (A,ϕA) + ℓ (B,ϕB) − ℓ (A∩B,ϕA∩B). The main tool we use is the MayerVietoris sequence of Čech homology groups. This result allows us to prove that size functions are able to detect partial matching between shapes by showing a common subset of cornerpoints.