Homology groups, symmetry representations and polyhedral clusters

Physical properties of cages and clusters obey symmetry rules that are extensions of the celebrated Euler-Poincare theorem on polyhedra. A connection is established between this result and a fundamental topological relationship in the theory of homology groups. The connection allows us to assign symmetry representations to physically relevant topological invariants. The results are illustrated by a derivation of the symmetries of the low-lying empty orbitals in leapfrog fullerenes.