Pre-crossed modules and rack homology (Q6588445)

It has been observed that racks and Leibniz algebras come with a new kind of homology. These invariants were first defined intrinsically, and then it was realized that they have a natural interpretation in terms of the corresponding cubical and DG structures. This paper aims to point out that there is a similar homology for pre-crossed modules. Since both pre-crossed modules and augmented racks are one-dimensional shadows of associative products, the former being simplicial while the latter pre-cubical, it is natural that these new invariants are related to the rack homology.\N\NAn augmented rack\N\[\N\pi:X\rightarrow G\N\]\N\ gives rise to the pre-crossed module\N\[\N\overline{\pi}:F\left( X\right) \rightarrow G\N\]\Nwhere \(F\left( X\right) \)\ is the free group on \(X\)\ and \(\overline{\pi}\)\ is induced by \(\pi\). The principal result in this paper is Theorem 3.1 claiming that the pre-crossed homology of this pre-crossed module coincides with the rack homology \(X\overset{\pi}{\rightarrow}G\). Another case when one can identify the pre-crossed homology is that of a pre-crossed module \(X\overset{\pi}{\rightarrow}G\)\ with a trivial action, where it turns out to be the tensor algebra on the homology of the group \(X\) (Theorem 4.1).\N\NIn order to relate the rack homology to the pre-crossed homology the author uses the group-completion theorem of Quillen in the appendix to [\textit{E. M. Friedlander} and \textit{B. Mazur}, Filtrations on the homology of algebraic varieties. Providence, RI: American Mathematical Society (AMS) (1994; Zbl 0841.14019)]. As for the computation of the homology for the pre-crossed modules with the trivial action, it rests on the identification of the classifying space for the pre-crossed homology as a certain twisted version of the Milnor-Carlsson construction of a circle [\textit{G. Carlsson}, Topology 23, 85--89 (1984; Zbl 0532.55024)].