A Comparison of Models for the Fulton-Macpherson Operads

In this thesis we explore the structure of the Fulton-Macpherson operads F_N by providing two new models for them. It is shown in work of Salvatore that these operads are cofibrant by claiming the existence of an isomorphism of operads from WF_N to F_N. Here, W is a functor which, for a large class of topological operads, produces cofibrant replacements. It would be satisfying to be
able to write down explicitly what these isomorphisms are. Our new models are an attempt to move towards this.

The building blocks of the first model appeared in the Ph.D. thesis of Daniel Singh but they were not assembled into an operad here. This model has a more algebraic feel than others in the literature which gives it technical advantages. We use this to demonstrate many of the well-known properties of the Fulton-Macpherson operads. In particular, we are able to write down explicit isomorphisms between F_1 and the Stasheff operad which we have not seen previously in the literature. This model is isomorphic to other models of the Fulton-Macpherson operads.

The second model is a realisation of an operad in posets. This poset operad is built from combinatorial objects called chains of preorders. These objects encode maps from a finite set A to some Euclidean space R^N. In particular, we can impose restrictions to encode injective maps of this type. This model is equivalent up to homotopy to the Fulton-Macpherson operads in a way which we define. It is also homotopy equivalent to the Smith operads, another example of topological operads defined combinatorially. The main advantage of this
model is that it has an obvious spine which may pave the way to writing down the desired isomorphisms from WF_N to F_N.