Groups, cacti and framed little discs (Q2838077)

This paper presents a new comparison between the operad of framed little disks and the cacti operad which is particularly suited to comparisons between algebras over these operads.NEWLINENEWLINEAn algebra over the framed little disks operad is equivalent to a double loop space, which is in turn equivalent to the double loop space of the classifying space of some topological group. On the other hand, an algebra over the cacti operad is equivalent to the loop space of a topological group. The equivalence between these operads was established by Kaufmann, using an intermediate operad equivalent to both.NEWLINENEWLINEIn this paper, again an intermediate operad is used, but it can be described explicitly in such a way so as to blend features of the framed little disks and cacti operads. It is formed as a particular example of a construction of the mapping operad between two operads equipped with realization systems. The theory of realization systems is developed in the first part of the paper with the two operads of interest presented as key examples.NEWLINENEWLINEAnother advantage to this approach is that the comparison between the operads can be extended to a comparison between algebras over those operads. This feature is particularly interesting in the motivating example, since it is already known that the algebras are abstractly equivalent.