Rational homotopy nilpotency of self-equivalences
Let \(\Gamma\) denote an associative \(H\)-group. Its homotopy nilpotency \(\text{Hnil} (\Gamma)\) is defined as the maximal integer \(n\) such that the commutator map \(c_n:\Gamma ^n\to \Gamma\) is essential. This paper deals with homotopy nilpotency modulo torsion for \(\Gamma\) a loop space, resp. a space of homotopy self-equivalences of a finite CW-complex \(X\). For a CW-complex \(X\) with rational universal covering, it is shown that \(\text{Hnil} (\Omega X)\) agrees with the Whitehead length of \(X\) defined in terms of iterated Whitehead products; this generalizes work of \textit{M. Arkowitz} and \textit{C. R. Curjel} [Q. J. Math., Oxf., II. Ser. 15, 316-326 (1964; Zbl 0124.16204)]. For several spaces \(\Gamma \) of self-equivalences of a finite 1-connected CW-complex \(X\) (component of the identity, inducing identity on homotopy groups etc.), the author studies their rational homotopy by means of the Sullivan minimal model of \(X\). The homotopy nilpotency modulo torsion of \(\Gamma\) is shown to agree with the Lie algebra nilpotency of the cohomology of a corresponding complex of derivations on the minimal model of \(X\). Amongst several examples it turns out that the homotopy nilpotency of a wedge of odd spheres is infinite.
scientific article; zbMATH DE number 1022198
scientific article