I want to understand what means the homotopy of the little discs operad. I'm starting to research in this area and I have some questions.
1) I don't understand why little discs operad is a topological space, what is the topology of the configuration space of n discs? I know each element of the configuration space is a topological space (like a disc with holes) but the morfisms that is used there goes between the configutarions spaces, not between the discs... I thought this is a Set operad and not a Top operad but I'm not sure.
2) If We have a topological spaces I know we can create the homotopy group. Then I can forget about which type of operad really is and take the homotopy group of each element in a configuration space. But I have a problem when I try to understand this like an Operad morfism because if we have two element $f$ in the homotopy group of a disc with one hole $D_1$ and $g$ in disc with two holes $D_2$ and $\gamma$ the operator in the structure of little disc operad, what is $\gamma(f,g)$ in the homotopy group of the disc with two holes $\gamma(D_1,D_2)$?. $\gamma$ is not a topological morfism between $D_1\times D_2\rightarrow \gamma(D_1, D_2)$ then how I can create a element from $f$ and $g$ in the homotopy group of $\gamma(D_1,D_2)$.
I'm sure that I am misleading something. Any information would be great!
For (1) : The configuration space of $n$ little disks can be considered as an open set in a vector space of dimension $3n$, where coordinates are given by the centers of the disks ($2n$ parameters) and the radius ($n$ more parameters). The condition that disks are disjoints and contained in the interior of the unit disk is clearly an open condition.
For (2): what you are asking is not very clear to me. If you have one loop in the spaces of $n$ little disks and one other loop in the space of $m$ little disks, you can compose them (by fixing a parameter on each loop to perform the composition), but you will get a torus embedded in the space of $m+n-1$ little disks.