Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- List of symbols
- I Axioms for homotopy theory and examples of cofibration categories
- II Homotopy theory in a cofibration category
- III The homotopy spectral sequences in a cofibration category
- IV Extensions, coverings, and cohomology groups of a category
- V Maps between mapping cones
- VI Homotopy theory of CW-complexes
- VII Homotopy theory of complexes in a cofibration category
- VIII Homotopy theory of Postnikov towers and the Sullivan–de Rham equivalence of rational homotopy categories
- IX Homotopy theory of reduced complexes
- Bibliography
- Index
Introduction
Published online by Cambridge University Press: 22 March 2010
- Frontmatter
- Contents
- Preface
- Introduction
- List of symbols
- I Axioms for homotopy theory and examples of cofibration categories
- II Homotopy theory in a cofibration category
- III The homotopy spectral sequences in a cofibration category
- IV Extensions, coverings, and cohomology groups of a category
- V Maps between mapping cones
- VI Homotopy theory of CW-complexes
- VII Homotopy theory of complexes in a cofibration category
- VIII Homotopy theory of Postnikov towers and the Sullivan–de Rham equivalence of rational homotopy categories
- IX Homotopy theory of reduced complexes
- Bibliography
- Index
Summary
In his lecture at the international congress of mathematicians (1950) J.H.C. Whitehead outlined the idea of algebraic homotopy as follows:
In homotopy theory, spaces are classified in terms of homotopy classes of maps, rather than individual maps of one space in another. Thus, using the word category in the sense of S. Eilenberg and Saunders Mac Lane, a homotopy category of spaces is one in which the objects are topological spaces and the ‘mappings’ are not individual maps but homotopy classes of ordinary maps. The equivalences are the classes with two-sided inverses, and two spaces are of the same homotopy type if and only if they are related by such an equivalence. The ultimate object of algebraic homotopy is to construct a purely algebraic theory, which is equivalent to homotopy theory in the same sort of way that ‘analytic’ is equivalent to ‘pure’ projective geometry.
This goal of algebraic homotopy in particular includes the following basic homotopy classification problems:
Classify the homotopy types of polyhedra X, Y…, by algebraic data. Compute the set of homotopy classes of maps, [X, Y], in terms of the classifying data for X and Y. Moreover, compute the group of homotopy equivalences, Aut(X).
There is no restriction on the algebraic theory which might solve these problems, except the restriction of ‘effective calculability’. Indeed, algebraic homotopy is asking for a theory which, a priori, is not known and which is not uniquely determined by the problem.
- Type
- Chapter
- Information
- Algebraic Homotopy , pp. xiii - xvPublisher: Cambridge University PressPrint publication year: 1989