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In this comprehensive study of Kitaev's abelian models defined on a graphembedded on a closed orientable surface, we provide complete proofs of thetopological ground state degeneracy, the absence of local order parameters,compute the entanglement entropy exactly and characterise the elementaryanyonic excitations. The homology and cohomolgy groups of the cell complex playa central role and allow for a rigorous understanding of the relations betweenthe above characterisations of topological order.

Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons