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

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.


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