at.algebraic topology - cell complexes and higher graph theory

Suppose that, on an intuitive basis, one defines a "2-graph" $(V,E,F,partial)$ as a collection of vertices, oriented edges and oriented faces, all of which should be considered as abstract objects whose relationships are determined by an incidence relation $partial$ such that

• $partial_0$ is the usual incidence matrix of graph $(V,E)$


• $partial_1$ is defined in an obvious way as the edge-face incidence matrix with the only requirement that the boundary of a face be an oriented cycle, so that $partial^2 = 0$.

Let a realization of a 2-graph be an embedding of vertices, lines and surfaces in $mathbb{R}^4$, such that surfaces only meet at lines and vertices, and lines only meet at vertices. Generalizing planarity, one can ask, for example, when is a 2-graph "spatial", meaning that it can be embedded in $mathbb{R}^3$.

I gather that 3-dimensional CW-complexes should correpond to such realizations of spatial 2-graphs, but find it hard to visualize the situation. Moreover, for practical porpouses I would prefer to work with the more intuitive notion of a 2-graph as a purely combinatorial object, and use their representations just as a tool for visualizing incidences, instead of working with topological spaces. So here are my questions:

• do you think the sloppy definitions above are substantially correct? In particular, is it too unfair to impose $partial^2 = 0$ rather then derive it?


• do realizations of planar graphs correspond to 3-dim cell complexes?


• is there any result such as Kuratowsky's theorem?


• can you help me find a reference where the combinatorial properties of cell-complexes (expecially 3-dim) are illustrated in detail?

Thanks