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Simplicial volume and fillings of hyperbolic manifolds
"... Abstract. Let M be a hyperbolic n–manifold whose cusps have torus crosssections. In [FM10], the authors constructed a variety of nonpositively and negatively curved spaces as “2π–fillings ” of M by replacing the cusps of M with compact “partial cones ” of their boundaries. These 2π–fillings are clos ..."
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Abstract. Let M be a hyperbolic n–manifold whose cusps have torus crosssections. In [FM10], the authors constructed a variety of nonpositively and negatively curved spaces as “2π–fillings ” of M by replacing the cusps of M with compact “partial cones ” of their boundaries. These 2π–fillings are closed pseudomanifolds, and so have a fundamental class. We show that the simplicial volume of any such 2π–filling is positive, and bounded above by Vol(M) vn, where vn is the volume of a regular ideal hyperbolic n–simplex. This result generalizes the fact that hyperbolic Dehn filling of a 3–manifold does not increase hyperbolic volume. In particular, we obtain information about the simplicial volumes of some 4–dimensional homology spheres described by Ratcliffe and Tschantz, answering a question of Belegradek and establishing the existence of 4–dimensional homology spheres with positive simplicial volume. 1.
Moduli spaces of nonnegative sectional curvature and nonunique souls, arXiv:0912.4869 (math.DG
, 2009
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Extensions, Automorphisms, and Definability
 CONTEMPORARY MATHEMATICS
"... This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to ..."
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This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here
Geom Dedicata DOI 10.1007/s1071100994021 ORIGINAL PAPER Taming 3manifolds using scalar curvature
, 2009
"... In this paper we address the issue of uniformly positive scalar curvature on noncompact 3manifolds. In particular we show that the Whitehead manifold lacks such a metric, and in fact that R 3 is the only contractible noncompact 3manifold with a metric of uniformly positive scalar curvature. We als ..."
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In this paper we address the issue of uniformly positive scalar curvature on noncompact 3manifolds. In particular we show that the Whitehead manifold lacks such a metric, and in fact that R 3 is the only contractible noncompact 3manifold with a metric of uniformly positive scalar curvature. We also describe contractible noncompact manifolds of higher dimension exhibiting this curvature phenomenon. Lastly we characterize all connected oriented 3manifolds with finitely generated fundamental group allowing such a metric.