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Journal of the London Mathematical Society 1997 56(3):645-656; doi:10.1112/S0024610797005607
© 1997 by London Mathematical Society
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© The London Mathematical Society

On Towers Approximating Homological Localizations

Carles Casacuberta and José L. Rodríguez

Departament de Matemàtiques, Universitat Autònoma de Barcelona E-08193 Bellaterra, Spain. E-mail: casac{at}mat.uab.es
Departament de Matemàtiques, Universitat Autònoma de Barcelona E-08193 Bellaterra, Spain. E-mail: jlrodri{at}mat.uab.es

Received 27 March 1995. Revision received 3 October 1995.

Our object of study is the natural tower which, for any given map f:A->B and each space X, starts with the localization of X with respect to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory E*, yielding, for example, an analogue of Quillen's plus-construction for E*. We discuss in detail the case of ordinary homology with coefficients in Z/p or Z[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f:A->pt), which allows us, among other things, to generalize Neisendorfer's observation that p-completion of simply-connected spaces coincides with nullification with respect to a Moore space M(Z[1/p], 1).


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