The Boardman category of spectra, chain complexes and (co)-localizations. (English) Zbl 0921.55007

Bauer, Friedrich W.

Homology Homotopy Appl. 1, 95-116 (1999).

A chain functor associates a pair \((X,A)\) of topological spaces to chain complexes \(C(X,A)\) and \(C'(X,A)\). In addition, we require that the complexes \(C(A), C(X), C(X,A)\) and \(C'(X,A)\) share a certain relationship, which ensures that \((X,A)\mapsto H_*C(X,A)\) is a homology theory on the homotopy category of pairs of topological spaces. Singular homology is a special case (with \(C=C'\)). However, the more intricate structural data of “chain functor” are designed so that compactly supported homology theories on pairs of CW-spaces defined via chain functors correspond bijectively to such theories defined via spectra (in the sense of Boardman).
It follows that, whenever a chain complex construction on chain functors yields again a chain functor, then there is a corresponding construction on spectra. The present article is a survey of the author’s recent work in this direction. For example, the tensor product of chain functors is again a chain functor, and so there is a corresponding operation on spectra (which is not the smash product). Notably, this leads to a different view of (co-)localization in the homotopy category of spaces.

Reviewer: G.Peschke (Edmonton)

MSC:

55P42	Stable homotopy theory, spectra
55P60	Localization and completion in homotopy theory
55N20	Generalized (extraordinary) homology and cohomology theories in algebraic topology
55U15	Chain complexes in algebraic topology
55N07	Steenrod-Sitnikov homologies

Keywords:

chain functors; Boardman spectra; tensor products; localization; colocalization