Relative simplicial approximation
01 Jan 1964-Vol. 60, Iss: 1, pp 39-43
TL;DR: The relative simplicial approximation theorem as mentioned in this paper allows us to leave a simplicial map f unchanged on any subcomplex on which f happens to be already simplicial, on which the subcomplex is known to be simplicial.
Abstract: The absolute simplicial approximation theorem, which dates back to Alexander (l), states that there is a simplicial approximation g to any given continuous map f between two finite simplicial complexes (see for instance (2), p. 37 or (3), p. 86). The relative theorem given here permits us to leave f unchanged on any subcomplex, on which f happens to be already simplicial.
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TL;DR: In this article, the authors give a homotopy theoretic criterion for an imbedding of the n-sphere s into a higher dimension msphere to be "equivalent" to the standard imbding.
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TL;DR: In this article, it was shown that the splitting homomorphism induced by (U, V) is possible to obtain all such configurations in algebraic configurations, and that all such configuration occurs in precisely this way.
Abstract: where (w*, u*)(/) = («*(/), f *(/)). The homomorphism (w*, v*) is called the splitting homomorphism of ^(ft s0) induced by (U, V). In studying the results and questions presented by Stallings in [9], a rather obvious question comes to mind. With the three-manifold M and the Heegaard splitting (U, V) of M is related the algebraic configuration called the splitting homomorphism induced by (U, V), is it possible that all such algebraic configurations occur in precisely this way? That is, suppose S is a closed, orientable surface of genus «>0. Let ft and ft denote free groups of rank n and suppose r¡1, r¡2 are homomorphisms of tt(S, s0) onto ft and ft, respectively. If
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Book•
21 Apr 2013
TL;DR: The stable parametrized h-cobordism theorem as mentioned in this paper is based on simplicial sets and simplicial simplicial maps of simplicial set is a generalization of simple maps.
Abstract: Introduction 1 1.The stable parametrized h-cobordism theorem 7 1.1. The manifold part 7 1.2. The non-manifold part 13 1.3. Algebraic K-theory of spaces 15 1.4. Relation to other literature 20 2.On simple maps 29 2.1. Simple maps of simplicial sets 29 2.2. Normal subdivision of simplicial sets 34 2.3. Geometric realization and subdivision 42 2.4. The reduced mapping cylinder 56 2.5. Making simplicial sets non-singular 68 2.6. The approximate lifting property 74 2.7. Subdivision of simplicial sets over DELTAq 83 3.The non-manifold part 99 3.1. Categories of simple maps 99 3.2. Filling horns 108 3.3. Some homotopy fiber sequences 119 3.4. Polyhedral realization 126 3.5. Turning Serre fibrations into bundles 131 3.6. Quillen's Theorems A and B 134 4.The manifold part 139 4.1. Spaces of PL manifolds 139 4.2. Spaces of thickenings 150 4.3. Straightening the thickenings 155 Bibliography 175 Symbols 179 Index 181
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TL;DR: In this paper, it was shown that if 0 m ≤ n, then every closed, (m − l)-connected n-manifold can be imbedded in R 2n −m+1.
Abstract: This chapter discusses the piecewise linear imbeddings in R q of compact, n -dimensional, combinatorial manifolds that are (m – 1)-connected, where 0 m ≤ n . The condition m > 0 means that such a manifold is connected. If a closed, that is, compact, unbounded, n -manifold M is (m – l)-connected and 2 m > n , then it follows from the Poincare duality that M has the homotopy type of a w -sphere. Therefore, if it turns out that every such manifold is a combinatorial n -sphere, or even if it can be piecewise linearly imbedded in R n+1 , then the theorem proved in the chapter is valid for 0 m ≤ n . The chapter presents the proof of the theorem that states that if 0 m ≤ n , then every closed, (m – l)-connected n-manifold can be imbedded in R 2n–m+1 .
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TL;DR: In this paper, it was shown that there is no point-for-point continuous reciprocal correspondence between two manifolds possessing different numbers (P and R ), even if the correspondence were not required to be analytic.
Abstract: whose rôles are analogous to that played by connectivity in the theory of surfaces. The numbers P¿ and 77¿ are each greater by unity than the maximum number of ¿-dimensional cycles (cf. § 4 below) which may be traced in the manifold, but in calculating the numbers P,, certain conventions about sense are taken into account and the attention is confined to the sensed cycles. Now if a manifold be subdivided into a complex, or generalized polyhedron, then one plus the maximum number of independent ¿-dimensional cycles of the polyhedron (i. e., cycles made up of cells of the polyhedron) is also equal to Ri, or to Pi if the conventions about sense be adopted. This theorem, which is of considerable use in calculating the values of the numbers (P) and (R), has been proved by Poincaré on the assumption that the manifold, all the cycles of the manifold, and all the cells of the complex may be regarded as made up of a finite number of analytic pieces. But such an assumption opens the way to a theoretical objection in that the numbers (P) and (R) when calculated from the analytic cycles alone might conceivably fail to be topological invariants. To remove this objection, it would have to be shown that there never could exist a point-for-point continuous reciprocal correspondence between two manifolds possessing different numbers (P) and ( R ), even if the correspondence were not required to be analytic. In the following discussion, we shall take into account not only non-analytic cycles but also cycles possessing singularities of however complicated a nature.
36 citations