Most texts on algebraic topology emphasize homological algebra, with topological considerations limited to a few propositions about the geometry of simplicial complexes. There is much to be gained however, by using the more sophisticated concept of cell (CW) complex. Even for simple computations, this concept ordinarily allows us to bypass much tedious algebra and often gives geometric insight into the homology and homotopy theory of a space. For example, the easiest way to calculate and interpret the homology of Cpn, complex projective n-space, is by means of a cellular decomposition with only n+ 1 cells. Also, by a suitable construction we can "realize" the sin gular complex of a space as a CW complex and perhaps thus give a more geometric basis for some arguments involving singular homology theory for general spaces and a more concrete basis for singular ho motopy type. As a fInal example, if we start with the category of sim plicial complexes and maps, common topological constructions such as the formation of product spaces, identifIcation spaces, and adjunction spaces lead us often into the category of CW complexes. These topics, among others, are usually not treated thoroughly in a standard text, and the interested student must fInd them scattered through the literature. This book is a study of CW complexes. It is intended to supplement and be used concurrently with a standard text on algebraic topology.
Reihe
Auflage
Sprache
Verlagsort
Verlagsgruppe
Illustrationen
10
10 s/w Abbildungen
VIII, 216 p. 10 illus.
ISBN-13
978-1-4684-6254-8 (9781468462548)
DOI
10.1007/978-1-4684-6254-8
Schweitzer Klassifikation
0. Preliminaries.- I. Combinatorial Cell Complexes.- 1. Definitions.- 2. Examples.- 3. Carrier theory.- 4. Functions.- 5. Product complexes.- 6. Equivalence relations and quotients.- 7. Adjunction complexes.- II. CW Complexes.- 1. Definitions.- 2. Alternative descriptions of CW complexes.- 3. Remarks on the general topology of CW complexes.- 4. Paracompactness.- 5. Products, quotients, and adjunctions.- 6. Homotopy and local properties.- 7. The homotopy extension theorem.- 8. The cellular approximation theorem.- 9. Aspherical carrier theorem.- III. Regular and Semisimplicial CW Complexes.- 1. Regular and normal CW complexes.- 2. Regular CW complexes and invariance of domain.- 3. Semisimplicial complexes.- 4. The realization of semisimplicial complexes.- 5. Semisimplicial constructions.- 6. Simplicial subdivision of semisimplicial complexes.- 7. Barycentric subdivision of semisimplicial complexes.- 8. Regulated semisimplicial complexes.- 9. The functor *.- IV. Homotopy Type of CW Complexes.- 1. Homotopy equivalence and deformation retraction.- 2. Homotopy equivalence of adjunction spaces.- 3. Whitehead's theorems.- 4. Simplicial complexes with the metric topology.- 5. Equi-local convexity.- 6. Countable CW complexes.- 7. Finite CW complexes.- V. The Singular Homology of CW Complexes.- 1. Excision in the CW category.- 2. Cellular homology.- 3. Orientation, incidence, and degree.- 4. Regular CW complexes and proper maps.- 5. Quotient complexes.- 6. Product and adjunction complexes.- 7. Semisimplicial complexes.- 8. Realizing cellular maps.- Appendix I. Paracompact Spaces.- Appendix II. Extension Spaces and Neighborhood Retracts.
