The fundamental group of the clique graph

Given a finite connected bipartite graph B=(X,Y) we consider the simplicial complexes of complete subgraphs of the square B2 of B and of its induced subgraphs B2[X] and B2[Y]. We prove that these three complexes have isomorphic fundamental groups. Among other applications, we conclude that the fundamental group of the complex of complete subgraphs of a graph G is isomorphic to that of the clique graph K(G), the line graph L(G) and the total graph T(G).