Finiteness Conditions for CW-Complexes

Wall, C. T. C.

doi:10.2307/1970382

Cited by 473 publications

(292 citation statements)

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“…The classical approach to the fundamental group of simplicial complexes by means of the 4 edge path group', due to Poincare (1895), was, with time, carried over to CW-complexes; for special CW-complexes, one could consult Schubert (1964,1968), and, for the general case, Massey (1984). The ideas about increasing the Connectivity of maps were first published in Wall (1965), where credit is given to unpublished work of J. Milnor.…”

Section: Increasing the Connectivity Of Mapsmentioning

confidence: 99%

Cellular Structures in Topology

1990

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This book describes the construction and the properties of CW-complexes. These spaces are important because firstly they are the correct framework for homotopy theory, and secondly most spaces that arise in pure mathematics are of this type. The authors discuss the foundations and also developments, for example, the theory of finite CW-complexes, CW-complexes in relation to the theory of fibrations, and Milnor's work on spaces of the type of CW-complexes. They establish very clearly the relationship between CW-complexes and the theory of simplicial complexes, which is developed in great detail. Exercises are provided throughout the book; some are straightforward, others extend the text in a non-trivial way. For the latter; further reference is given for their solution. Each chapter ends with a section sketching the historical development. An appendix gives basic results from topology, homology and homotopy theory. These features will aid graduate students, who can use the work as a course text. As a contemporary reference work it will be essential reading for the more specialized workers in algebraic topology and homotopy theory.

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Section: Increasing the Connectivity Of Mapsmentioning

confidence: 99%

Cellular Structures in Topology

1990

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“…Controlled Wall theory. The purpose of this section is to establish controlled versions of some aspects of Wall's finiteness obstruction [10]. As in §4 we will have to deal with the problem of multiple basepoints, so we will be using the controlled matrices which were defined there.…”

Section: Addendummentioning

confidence: 99%

“…Assume further that (L, J) is «-connected and the inclusion / =» L is a domination rel K. One easily shows that the inclusion-induced homomorphism of Z7r,(AT)-modules, tt"(J, K) -» tt"(L, K), is split, so tt"(J, K) **> irn+x(L, J) © tt"(L, K). It follows from the results of [10] that if L is homotopy equivalent to a finite complex, then tt"(L, K) is stably free. This quickly implies that there are free f.g. Z7r,(7i)-modules Fx, F2 and an isomorphism Trn(J, K)(BFX®F2~ TTn(J, K)@FX®F2 which takes tt"(J, K) © Fx onto Tr"(L, K) © F2 and which takes F2 onto %+x(L, J) © Fx.…”

Section: Addendummentioning

confidence: 99%

Controlled boundary and ℎ-cobordism theorems

Chapman¹

1983

Trans. Amer. Math. Soc.

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Abstract. In this paper two theorems are established which are consequences of some earlier approximation results of the author. The first theorem is a controlled boundary theorem for finite-dimensional manifolds. By this we mean an ordinary boundary theorem plus small e-control in a given parameter space. The second theorem is a controlled ft-cobordism theorem for finite-dimensional manifolds with small e-control in a given parameter space. These results generalize the End Theorem and the Thin /¡-Corbordism of Quinn.1. Introduction. The purpose of this paper is to establish two theorems which are consequences of the approximation results of [2]. The first theorem is a controlled boundary theorem for finite-dimensional manifolds. By this we mean an ordinary boundary theorem, as treated in Siebenmann [8], plus small e-control in a given parameter space. The second theorem is a controlled /¡-cobordism theorem for finite-dimensional manifolds with similar small e-control in a given parameter space. These results generalize the End Theorem and the Thin /z-Cobordism Theorem of Quinn [7] in the sense that they do not require the parameter space to be locally simply connected and they do not require that the map to the parameter space be locally constant on irx. This is accomplished by replacing the stable algebra (i.e. geometric module) approach of Quinn [7, §8] by the more topological approximation results of [2].

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“…If G is finitely presented, this is equivalent by Wall [5,6] to the existence of an Eilenberg-Mac Lane complex K(G, 1) of finite type (i.e., with finitely many cells in every dimension). Up to now, all known torsion-free groups of type FPoo have had finite cohomological dimension; in fact, they have admitted a finite K(G, l)-complex.…”

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confidence: 99%