On the planarity of infinite 2-complexes (Q1378242)

Kuratowski, in 1930, characterized the finite graphs which are planar as those not containing a homeomorphic copy of \(K_{3,3}\) or \(K_5\). This idea has since been generalized, by various researchers, to characterize planarity for other types of structure, such as Peano continua, polyhedra, and infinite graphs. The authors of this paper give several characterizations of planarity for infinite 2-complexes, and also some characterizations of proper planarity (that is planar embedding without accumulation points.) These characterizations draw upon and unify several existing results in the area, yielding a neat synthesis of (on the one hand) results for compact polyhedra and (on the other) results for infinite graphs.