# Topological Combinatorics

Abstract (Summary)

This thesis on Topological Combinatorics contains 7 papers. All of them but paper B are published before. In paper A we prove that ? i dim ˜ Hi(Ind(G); Q) ?|Ind(G[D])| for any graph G and its independence complex Ind(G), under the condition that G\D is a forest. We then use a correspondence between the ground states with i +1 fermions of a supersymmetric lattice model on G and ˜ Hi(Ind(G); Q) to deal with some questions from theoretical physics. In paper B we generalize the topological Tverberg theorem. Call a graph on the same vertex set as a (d + 1)(q ? 1)-simplex a (d, q)-Tverberg graph if for any map from the simplex to Rd there are disjoint faces F1,F2,. . . , Fq whose images intersect and no two adjacent vertices of the graph are in the same face. We prove that if d ? 1, q ? 2 is a prime power, and G is a graph on (d + 1)(q ? 1) + 1 vertices such that its maximal degree D satisfy D(D + 1) Bibliographical Information: