The Hartree-Fock-Bogoliubov approximation of superconducting ground states can be naturally described using Araki's self-dual canonical anti-commutation relation (CAR) algebra. Using this framework, we first review the K-theoretic classification of free fermions with an emphasis on unique gapped ground states. Ideas from index theory and coarse geometry then allow us to refine this procedure to obtain topological obstructions between gapped ground states with a prescribed locality. These methods also give us hints towards approaches to understand topological invariants of more general (possibly interacting) gapped fermionic ground states.