Document Zbl 0886.55011 - zbMATH Open
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Homotopical localizations of spaces. (English) Zbl 0886.55011
The author studies periodic phenomena in unstable homotopy theory using localizations of spaces (space means simplicial set here). He works in the pointed homotopy category \(H_{O_*}\) of CW-complexes. First he gives the notion of an \(f\)-localization of spaces which generalizes both the \(E_*\) localization \(X\to X_E\) for a homology theory \(E_*\) and the \(A\)-nullification \(X\to P_AX\) for a space \(A\) obtained using a suitable \(E_*\)-equivalence \(f\) and the trivial map \(A\to^*\).
The main result is a general fibration theorem: for a map \(f\) of connected spaces he proves that the localization functors \(L_{\Sigma f}\), \(L_f\Omega\) preserve homotopy fiber sequences up to error terms whose \(p\)-completions have at most three nontrivial homotopy groups for each prime \(p\). He also determines the nullifications of nilpotent “generalized polyGEMS”. As consequences of these results he shows that \(K(n)_*\)-localizations and other \(E_*\) localizations “almost” preserve homotopy fiber sequences of \(H\)-spaces. Applications in \(K\)-theory are also given.