Category of fractions and acyclic spaces
Let \(X\) be an acyclic space, i.e., a CW-complex whose reduced integral homology vanishes. An acyclic decomposition of \(X\) is a tower of fibrations NEWLINE\[NEWLINE \cdots \rightarrow X^n \rightarrow X^{n-1} \rightarrow \cdots \rightarrow X^{1} \rightarrow X^{0} = (\text{point}) NEWLINE\]NEWLINE together with a weak homotopy equivalence NEWLINE\[NEWLINE X \simeq \lim_{\leftarrow} X^n , NEWLINE\]NEWLINE where the \(n\)-stage \(X^n\) is acyclic, and it is \(j\)-simple for all \(j > n\), and the fibre of \(X^n \rightarrow X^{n-1}\) is \((n-1)\)-connected [\textit{E. Dror}, Topology 11, 339--348 (1972; Zbl 0244.55006)].NEWLINENEWLINEIn the present paper, the authors show that the acyclic tower can be obtained through a general categorical completion process using the generalized Adams completion [\textit{A. Deleanu, A. Frei} and \textit{P. J. Hilton}, Cah. Topol. Géom. Différ. 15, 61--82 (1974; Zbl 0291.18003)] which includes the stable Adams completion [\textit{J. F. Adams}, Stable homotopy and generalised homology. Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press.(1974; Zbl 0309.55016)], the \(P\)-localization and \(p\)-profinite completion [\textit{D. Sullivan}, Geometric topology. Part I. Mimeographed notes, Massachusetts Inst. Tech., Cambridge, (1970), cf. Zbl 1078.55001], and the \(R\)-completion [\textit{A. K. Bousfield} and \textit{D. M. Kan}, Homotopy limits, completions and localizations. Lecture Notes in Mathematics. 304. Berlin-Heidelberg-New York: Springer-Verlag. (1972; Zbl 0259.55004)].NEWLINENEWLINEThey prove that the \(n\)-stage \(X^n\) can be obtained as the generalized Adams completion of an acyclic space with respect to the set \(S_n\) of all \((n+1)\)-equivalences in the pointed homotopy category of pointed CW-complexes inducing isomorphisms in the reduced integral homology.
scientific article; zbMATH DE number 6187694