Diagonal complexes for punctured polygons

It is known that taken together, all collections of non-intersecting diagonals in a convex planar $n$-gon give rise to a (combinatorial type of a) convex $(n-3)$-dimensional polytope $\mathrm{As}_n$ called the Stasheff polytope, or associahedron. In the paper, we act in a similar way by taking a convex planar $n$-gon with $k$ labeled punctures. All collections of mutually nonintersecting and mutually non-homotopic topological diagonals yield a complex $\mathrm{As}_{n,k}$. We prove that it is a topological ball. We also show a natural cellular fibration $\mathrm{As}_{n,k}\to\mathrm{As}_{n,k-1}$. A special example is delivered by the case $k=1$. Here the vertices of the complex are labeled by all possible permutations together with all possible bracketings on $n$ distinct entries. This hints to a relationship with M. Kapranov's permutoassociahedron.