Motivated by string topology and the arc operad, we introduce the notion of
quasi-operads and consider four (quasi)-operads which are different varieties of the
operad of cacti. These are cacti without local zeros (or spines) and cacti proper as
well as both varieties with fixed constant size one of the constituting loops. Using
the recognition principle of Fiedorowicz, we prove that spineless cacti are
equivalent as operads to the little discs operad. It turns out that in terms of
spineless cacti Cohen’s Gerstenhaber structure and Fiedorowicz’ braided
operad structure are given by the same explicit chains. We also prove that
spineless cacti and cacti are homotopy equivalent to their normalized versions as
quasi-operads by showing that both types of cacti are semi-direct products of the
quasi-operad of their normalized versions with a re-scaling operad based on
.
Furthermore, we introduce the notion of bi-crossed products of quasi-operads and
show that the cacti proper are a bi-crossed product of the operad of cacti
without spines and the operad based on the monoid given by the circle group
.
We also prove that this particular bi-crossed operad product is homotopy
equivalent to the semi-direct product of the spineless cacti with the group
. This
implies that cacti are equivalent to the framed little discs operad. These
results lead to new CW models for the little discs and the framed little discs
operad.
