Graph homology and cohomology by Ian Dewan

By Ian Dewan

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A graph is planar if it has an embedding into R2 . For the purposes of this section, we will consider only embeddings that are nice. A nice embedding into a 2-manifold X is one in which the realization of every edge lies in the boundary of one or two faces, is disjoint from the boundary of every other face, and for every face F of the embedding there is a continuous function f : D → X from the closed disk to X satisfying (1) (2) (3) (4) f (D) = F , the restriction of f to the open disk is a homeomorphism onto F , the realization of a vertex in the boundary of F has a finite preimage, and the realization of a point in an edge in the boundary of F has a preimage consisting of one point if that edge is in the boundary of multiple faces, and two points if it is in the boundary only of F .

The edges of Γ† are the edges of Γ: for each edge e between faces F and G, M contains a path from fF ((0, 0)) to xe , and from xe to fG ((0, 0)): then the concatenation of these paths realizes e (that the interior of the path is homeomorphic to (0, 1) follows from the fact that fF and fG are homeomorphisms). The realizations of edges and vertices clearly partition M . It remains only to show that a subset of M that meets the closure of every edge in a closed set in closed. Let Q ⊆ M be such a set, and x a limit point of Q.

Similarly, f is surjective on edges, since there is always a path from v0 to one endpoint of the edge, which can be extended to a walk ending in the edge. Moreover, f preserves the degree of vertices: if a vertex of ∆ is a walk ending at a vertex v of Γ, then it has one incident edge for the edge leading into it, and one for each edge leading out, which does not include the edge leading in, since the walks represented by vertices in ∆ never use the same edge twice in a row. Since loops in Γ are duplicated in Γ , they are counted twice in the degree of corresponding vertex of ∆.

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