THE SECOND SKEW-SYMMETRIC COHOMOLOGY GROUP AND SPATIAL EMBEDDINGS OF GRAPHS

Let L(G) be the second skew-symmetric cohomology group of the residual space of a graph G. We determine L(G) in the case G is a 3-connected simple graph, and give the structure of L(G) in the case of G is a complete graph and a complete bipartite graph. By using these results, we determine the Wu invariants in L(G) of the spatial embeddings of the complete graph and those of the complete bipartite graph, respectively. Since the Wu invariant of a spatial embedding is a complete invariant up to homology which is an equivalence relation on spatial embeddings introduced in [12], we give a homology classification of the spatial embeddings of such graphs.