# Higher homotopy structures in the spineless cactus operad

### Dr I. Galvez Carillo (Universitat Politecnica de Catalunya)

Monday 11th March, 2013 15:00-16:00 Mathematics 203

#### Abstract

This is joint work in progress with L. Lombardi and A. Tonks. The cactus operad was introduced by Voronov motivated by the work of Chas and Sullivan. We will consider one of its versions, the spineless cactus operad, equivalent to the operad of the little disks. Our aim in this project is to produce an explicit map from the cofibrant operad of Gerstenhaber infinity algebras to this cactus operad. So far we have constructed an explicit $A_{\infty}$ action on this operad, using an intermediate map from $As^ {(2)}_\infty$ the cofibrant replacement of the $As^{(2)}$ operad described by the virtual mathematician G. W. Zinbiel. The question of further higher homotopy structures will be considered as well.