I'm currently reading Tillmann's paper 'Higher genus surface operad detects infinite loop spaces' but am having trouble understanding the construction of said operad. Specifically, there are two points that I'd like clarification for:
The author says that the objects in the category $\mathcal{E}_{g,n,1}$ are of the form $(F,\sigma)$ where $F$ is a 'surface of topological type $F_{g,n+1}$ constructed from $D,P$ and $T$ by gluing the marked boundary of one of the surfaces to one of the free boundaries of another using the given parametrization'. I'm confused about the phrase "using the given parametrization"; is the parametrization considered part of the surface $F$?
To be more specific, could we equivalently define the objects in this category as triples $(F,S,\sigma)$ where $F$ is a fixed surface with genus $g$ and $n+1$ boundary components and $S$ is a set of (isotopy classes of) simple closed curves in $F$ such that the complement of their images is a disjoint union of copies of $D,P$ and $T$ such that re-gluing along these curves follows the rules outlined in the paper?
The author then tells us to fix isotopies $\phi_1:\gamma(P;\_,P)\to\gamma(P;P,\_)$ and $\phi_2:\gamma(P;\_,D)\to\gamma(P;D,\_)$. I'm guessing that here $\gamma(P;\_,P)$ means the space given by gluing the fixed boundary of $P$ to the second free boundary of $P$ (and similarly for $D$). But then the author seems to be saying to chose an isotopy between spaces, which does not make sense to me. An isotopy is a relation between maps of spaces not between spaces themselves. How am I meant to interpret e.g. an isotopy $\phi:S^2\setminus\lbrace x_1,x_2,x_3,x_4\rbrace\to S^2\setminus\lbrace y_1,y_2,y_3,y_4\rbrace$?
Thanks for any help.