Cellular Structures in Topology

Abstract

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.

Hurewicz fibrations, almost submetries and critical points of smooth maps

“…It will be clear from the proof of (d) that E ′ is II-countable provided that E has the homotopy type of a II-countable, CW-complex Y with a countable 1-skeleton Y 1 . Indeed, the injection i : [Hat,Proposition 1.26] or [FriPic,Corollary 2.4.7].…”

“…Since the covering projection P is a Hurewicz fibration, [Spa,Theorem II.2.3], and each fibre P −1 (e) is a discrete space, hence a CW-complex, the conclusion follows by applying [FriPic,Theorem 5.4.2].…”

“…Then, by [FriPic,Proposition 5.4.1], F ′ has the homotopy type of a CW-complex. Moreover, by (e) of Proposition 2.1, E ′ is separable and metrizable.…”

“…By Proposition 2.6 (a) and Proposition 2.5 (c.1), Ω y ′ 0 Y ′ is homotopy equivalent to the locally contractible space F ′ 0 = (π ′ ) −1 (y ′ 0 ), with P Y (y ′ 0 ) = y 0 . Moreover, F ′ 0 is paracompact because it is a closed subspace of the CW-complex X ′ , which is paracompact; [Miy], [FriPic,Proposition 1.3.5]. Finally, since the covering dimension of the CW-complex X ′ is exactly its CW-dimension, [FriPic,Proposition 1.5.14], and since F ′ 0 is a closed subspace of X ′ we have dim F ′ 0 ≤ dim X ′ < +∞, [Eng].…”

“…Moreover, F ′ 0 is paracompact because it is a closed subspace of the CW-complex X ′ , which is paracompact; [Miy], [FriPic,Proposition 1.3.5]. Finally, since the covering dimension of the CW-complex X ′ is exactly its CW-dimension, [FriPic,Proposition 1.5.14], and since F ′ 0 is a closed subspace of X ′ we have dim F ′ 0 ≤ dim X ′ < +∞, [Eng]. To conclude, we now use Proposition 2.7 as in the proof of Theorem 1.1.…”