Desuspending and delooping cellular inequalities (Q1360896)

This paper concerns an approach to homotopy theory which focusses on how spaces may be built from a single atom using certain basic operations. Specifically, for a space \(A\), let \(C(A)\) denote the class generated by \(A\) using the operations of taking homotopy push-outs and wedges. Then \(X\in C(A)\) is denoted by \(X>>A\) (i.e. \(X\) is \(A\)-cellular) and is referred to as a cellular inequality. Another cellular inequality, \(X>A\) (i.e. \(X\) is \(A\)-acyclic) may also be defined denoting the fact that \(X\) is built from \(A\) by the operations above as well as by extensions by fibrations. Cellular inequalities were introduced by \textit{E. Dror Farjoun} [ Cellular spaces, null spaces and homotopy localization, Lect. Notes Math. 1622 (1995; Zbl 0842.55001)] and have found much application. The paper at hand considers the problem of determining when cellular inequalities may be desuspended or delooped. In particular, if \(\Sigma X>>\Sigma A\), when is it true that \(X>>A\) as well? The author gives several equivalent conditions to desuspending a cellular inequality, among which is the interesting condition that \(\Sigma X>> \Sigma A\) may be desuspended if and only if there exists a connected space \(B\) with \(C(A)=C(\Omega B)\). The \(\Sigma X>\Sigma A\) case and the dual delooping situation are also considered with analogous results. Although the paper centers on the desuspension-delooping problem, it is packed with fundamental techniques and results in this area as well as connections with classical homotopical topics such as Ganea's fibre-cofibre construction, James's construction and a (generalized) Freudenthal suspension theorem.