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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 theory ..."
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Cited by 52 (13 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
Correction to: “Complete normality and metrization theory of manifolds
 Top. Appl
"... Abstract. 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 on ..."
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Cited by 6 (2 self)
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Abstract. 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. The modern settheoretic topology of manifolds can be said to begin with the 1975 construction by Mary Ellen Rudin of a nonmetrizable perfectly normal manifold using the settheoretic axiom ♦. This solved a problem that had been posed by Wilder at the end of his 1949 textbook [11] and thereby made it possible to consistently extend the wealth of algebraic topology techniques used by Wilder beyond the context of metrizable manifolds, at least consistently with the usual axioms of ZFC. Shortly thereafter, with the help of Phillip Zenor [7], Rudin was able to
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."
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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly welldeveloped theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental
RNC WORKSHOP BOUNDARY VALUES OF MAPPINGS OF FINITE DISTORTION
"... Abstract. We find a sufficient condition for a weakly differentiable homeomorphism in Euclidean space to have a homeomorphic extension to the boundary of its domain of definition. In a certain sense, this condition is best possible. 1. ..."
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Abstract. We find a sufficient condition for a weakly differentiable homeomorphism in Euclidean space to have a homeomorphic extension to the boundary of its domain of definition. In a certain sense, this condition is best possible. 1.
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.
QUASICONFORMALITY, QUASISYMMETRY, AND REMOVABILITY IN LOEWNER SPACES
 VOL. 101, NO. 3 DUKE MATHEMATICAL JOURNAL
, 2000
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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
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.
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