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77
The Proportionality Principle of Simplicial Volume
, 2004
"... Manifolds, the basic objects of geometry, form the playground for both topological and smooth structures. Many challenges in modern mathematics are concerned with the nature of this interaction between algebraic topology, differential topology and differential geometry. For example all incarnations ..."
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Cited by 15 (0 self)
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Manifolds, the basic objects of geometry, form the playground for both topological and smooth structures. Many challenges in modern mathematics are concerned with the nature of this interaction between algebraic topology, differential topology and differential geometry. For example all incarnations
A NOTE ON A THEOREM OF MUNKRES
, 2004
"... Abstract. We prove that given a C ∞ Riemannian manifold with boundary, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result holds extends to C¹ manifolds, and that in dimensions 2,3 and 4 it also holds for PL manifolds. We employ the main result ..."
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Abstract. We prove that given a C ∞ Riemannian manifold with boundary, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result holds extends to C¹ manifolds, and that in dimensions 2,3 and 4 it also holds for PL manifolds. We employ the main result to prove that given any orientable C ∞ Riemannian manifold with boundary admits quasimeromorphic mappings onto ̂ R n. In addition some generalizations are given.
On open 3manifolds proper homotopy equivalent to geometrically simply connected polyhedra
 Topology Appl
, 1998
"... ..."
About Local Refinement of Tetrahedral Grids based on Bisection
 In 5th International Meshing Roundtable
, 1996
"... . We present a novel approach to the development of a threedimensional simplicial refinement strategies based on ideas that have been shown to be effective for twodimensional refinement. This procedure can be applied with other refinement algorithms with minimal adjustments and can be extended rec ..."
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. We present a novel approach to the development of a threedimensional simplicial refinement strategies based on ideas that have been shown to be effective for twodimensional refinement. This procedure can be applied with other refinement algorithms with minimal adjustments and can be extended recursively to even higherdimensional spaces. keywords. mesh refinement, 3D bisection, tetrahedra, adaptivity 1 Introduction As is well known, there are two main steps in local adaptive refinement: the refinement of a subset of elements, and the achievement of the conformity of the mesh. The elements that offer the simplest choice in any dimension are the simplices: triangles in two dimensions, tetrahedra in three dimensions and their analog in even higher dimensions. There are also some algorithms based on bisection or longestside midpoint insertion [1, 2, 3, 4, 5, 6]. 1.1 Notation and definitions Definition 1 (nsimplex) Suppose X0 ; X1 ; : : : ; Xn are n + 1 points such that the set of ve...
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
Surgery on closed 4manifolds with free fundamental group
 Math. Proc. Cambridge Philos. Soc
"... The 4dimensional topological surgery conjecture has been established for a class of groups, including the groups of subexponential growth (see [6], [13] for recent developments), however the general case remains open. The full surgery conjecture is known to be equivalent to the question for a class ..."
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The 4dimensional topological surgery conjecture has been established for a class of groups, including the groups of subexponential growth (see [6], [13] for recent developments), however the general case remains open. The full surgery conjecture is known to be equivalent to the question for a class of canonical problems with free fundamental group [5, Chapter 12]. The proof of the conjecture for “good ” groups relies on the disk embedding theorem (see [5]), which is not presently known to hold for arbitrary groups. However, in certain cases it may be shown that surgery works even when the disk embedding theorem is not available for a given fundamental group (such results still use the diskembedding theorem in the simplyconnected setting, proved in [3].) For example, this may be done when the surgery kernel is represented by π1null spheres [4], or more generally by a π1null submanifold satisfying a certain condition on Dwyer’s filtration on second homology [7]. Here we state another instance when the surgery conjecture holds for free groups. The following results are stated in the topological category. Theorem 1. Let X be a 4dimensional Poincaré complex with free fundamental group, and assume the intersection form on X is extended from the integers. Let f: M − → X be a degree 1 normal map, where M is a closed 4manifold. Then the vanishing of the Wall obstruction implies that f is normally bordant to a homotopy equivalence f ′ : M ′ − → X. In the canonical surgery problems, X has free fundamental group and trivial π2, however what makes them harder to analyze is the interplay between the homotopy type of X, and the topology of the boundary. Our result sidesteps this by considering closed manifolds. We also prove a related splitting result: Theorem 2. Let M be a closed orientable 4manifold with free fundamental group, and suppose the intersection form on M is extended from the integers. Then M is scobordant to a connected sum of ♯ n S 1 × S 3 with a simplyconnected 4manifold. Note that if the surgery conjecture fails for free groups, then for both theorems above there is, in general, no extension to 4manifolds with boundary. The assumption on the intersection pairing in theorem 2 is necessary, since there are forms
The MorseWitten complex via dynamical systems
 Expo. Math
"... to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed ..."
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to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in [We93] and is based on tools from hyperbolic dynamical systems. For instance, we apply the GrobmanHartman theorem and the λlemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.
Galois Theory of Second Order Covering Maps of Simplicial Sets
 J. Pure Appl. Algebra
, 1995
"... this paper is to develop such a theory for simplicial sets, as a special case of Galois theory in categories [6]. The second order notion of fundamental groupoid arising here as the Galois groupoid of a fibration is slightly different from the above notions but ..."
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this paper is to develop such a theory for simplicial sets, as a special case of Galois theory in categories [6]. The second order notion of fundamental groupoid arising here as the Galois groupoid of a fibration is slightly different from the above notions but
THE FUNDAMENTAL CROSSED MODULE OF THE COMPLEMENT Of A Knotted Surface
, 2009
"... We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, the ..."
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We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π2(M,M 1) →Gcan be rescaled to a homotopy invariant IG(M), depending only on the algebraic 2type of M. We describe an algorithm for calculating π2(M,M (1) ) as a crossed module over π1(M (1)), in the case when M is the complement of a knotted surface Σ in S 4 and M (1) is the handlebody of a handle decomposition of M made from its 0 and 1handles. Here, Σ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant IG yields a nontrivial invariant of knotted surfaces in S 4 with good properties with regard to explicit calculations.
Infinitedimensional representations of 2groups. Available as arXiv:0812.4969
"... A ‘2group ’ is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2groups have representations on ‘2vector spaces’, which are categories analogous to vector spaces. Unfortunately, Lie 2groups typically have few r ..."
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A ‘2group ’ is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2groups have representations on ‘2vector spaces’, which are categories analogous to vector spaces. Unfortunately, Lie 2groups typically have few representations on the finitedimensional 2vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinitedimensional 2vector spaces called ‘measurable categories ’ (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinitedimensional representations of certain Lie 2groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2intertwiners for any skeletal measurable 2group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and subintertwiners—features not seen in ordinary group representation theory. We classify irreducible and indecomposable representations and intertwiners. We also classify ‘irretractable ’ representations—another feature not seen in ordinary