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78
On Yetter’s invariant and an extension of the DijkgraafWitten invariant to categorical groups
 Theory Appl. Categ
"... We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the ..."
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We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined
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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Cited by 14 (11 self)
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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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Cited by 12 (2 self)
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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...
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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Cited by 12 (4 self)
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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
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
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.
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.
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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Cited by 8 (3 self)
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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 Classifying Space of a Topological 2Group
, 2008
"... Categorifying the concept of topological group, one obtains the notion of a ‘topological 2group’. This in turn allows a theory of ‘principal 2bundles’ generalizing the usual theory of principal bundles. It is wellknown that under mild conditions on a topological group G and a space M, principal G ..."
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Cited by 7 (1 self)
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Categorifying the concept of topological group, one obtains the notion of a ‘topological 2group’. This in turn allows a theory of ‘principal 2bundles’ generalizing the usual theory of principal bundles. It is wellknown that under mild conditions on a topological group G and a space M, principal Gbundles over M are classified by either the Čech cohomology ˇ H 1 (M, G) or the set of homotopy classes [M, BG], where BG is the classifying space of G. Here we review work by Bartels, Jurčo, Baas–Bökstedt–Kro, and others generalizing this result to topological 2groups and even topological 2categories. We explain various viewpoints on topological 2groups and the Čech cohomology ˇ H 1 (M, G) with coefficients in a topological 2group G, also known as ‘nonabelian cohomology’. Then we give an elementary proof that under mild conditions on M and G there is a bijection ˇH 1 (M, G) ∼ = [M, BG] where BG  is the classifying space of the geometric realization of the nerve of G. Applying this result to the ‘string 2group ’ String(G) of a simplyconnected compact simple Lie group G, it follows that principal String(G)2bundles have rational characteristic classes coming from elements of H ∗ (BG, Q)/〈c〉, where c is any generator of H 4 (BG, Q).