On the tensor algebra of a non-Abelian group and applications (Q1199151)

The authors use the monoidal structure on the category of crossed complexes [\textit{R. Brown} and \textit{P. J. Higgins}, J. Pure Appl. Algebra 47, 1-33 (1987; Zbl 0621.55009)] to introduce the notion of the tensor algebra \(J(e)\) of a crossed complex \(e\). Then its relations with the James construction [\textit{I. M. James}, Ann. Math., II. Ser. 62, 170-197 (1955; Zbl 0064.415)] is examined. For \(e\) a group, \(G\) the derived algebra \(I_ * G\) is constructed which is isomorphic to the algebra \(C(JG)\) of the chains on \(JG\). Some applications to the homotopy of the space \(\Sigma BG\) are presented.