The Topology of CW Complexes

Lundell, Albert T.; Weingram, Stephen

doi:10.1007/978-1-4684-6254-8

Cited by 188 publications

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“…The topological invariance of simplices under simplicial retractions (see Lemma 3.1.1), which is crucial for the proof of the triangulability of simplicial sets (see Corollary 4.6.12), is due to Barratt (1956); an alternative, but not simpler, approach can be found in Lundell & Weingram (1969). The intersection property (see Theorem 3.1.5) was conjectured by A.N.…”

Section: Notes To Chaptermentioning

confidence: 99%

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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: Notes To Chaptermentioning

confidence: 99%

“…A subtle answer to this question relying on algebraic X-theory is given in Wall {1965). The proof of the fact that the closure of any cell of a regulär CW-complex is a subcomplex (see Theorem 1.4.10) is inspired by the presentation in Lundell & Weingram (1969).…”

Section: Notes To Chaptermentioning

confidence: 99%

Cellular Structures in Topology

1990

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“…The same technique would work though if we were able to restrict ranges of the map on individual cells of the CW complex to aspherical subspaces of the codomain. This idea leads to the notion of a carrier and to the aspherical carrier theorem [12,II §9]. We generalize this notion to arbitrary spaces using a cover to replace the cell structure in the domain.…”

Section: Carrier Theoremsmentioning

confidence: 99%

Carrier and nerve theorems in the extension theory

Nagórko

2006

Proc. Amer. Math. Soc.

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Abstract. We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk and A. Weil, which state that the nerve of a regular cover is homotopy equivalent to the underlying space.Next we prove a nerve theorem for a class of spaces with uniformly bounded extension dimension. In particular we prove that the canonical map from a separable metric n-dimensional space into the nerve of its weakly regular open cover induces isomorphisms on homotopy groups of dimensions less than n.

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“…Every are, by definition, the homology groups H (K,G) and cohomology groups H n (K,G) of the simplicial set K with coefficients in G. We have mentioned these groups already in 2.16 and have observed there that they are naturally isomorphic to the ordinary homology and cohomology groups ,H (IKI,G) andH n (lKl,G) of the space |K|, which we had defined in Chapiter VI (cf. [LW,p. 192ff ] for a perhaps more elementary proof than our proof in §2).…”

Section: Caution If a Subspace Z Is Open In X Then It May Happen Thamentioning

confidence: 99%

Weakly Semialgebraic Spaces

Knebusch¹

1989

Lecture Notes in Mathematics

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The Topology of CW Complexes

Cited by 188 publications

References 0 publications

Cellular Structures in Topology

Cellular Structures in Topology

Carrier and nerve theorems in the extension theory

Weakly Semialgebraic Spaces

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