Assigning manifolds to graphs in a functorial way

Question

I am looking for ways to functorially assign manifolds (or more general topological spaces) to families of graphs. To be more precise, I am interested in functors from specific subcategories of the category of graphs (with isomorphisms or arbitrary homomorphisms as morphisms) into the category of manifolds (or more general topological spaces). One example would be simply viewing a graph as a topological space. But are there also examples where the spaces obtained in this way are e.g. $n$-manifolds for $n\geq 3$?

Answer 1

Here's one sort of artificial way: Begin with a fixed $n-1-$manifold $M^{n-1}$. Then, for each integer $k$, fix a manifold $M^n_k$, whose boundary is a disjoint union of $k$ copies of $M^{n-1}$. Then to each graph you can construct an $n-$manifold by associating to each vertex with valence $k$ a copy of $M^n_k$, and for each edge gluing along corresponding copies of $M^{n-1}$. In order for this to work, there should be an action of the symmetric group $S_k$ on $M^n_k$ permuting the boundary components.