The group \mathcal{O} of Bogoliubov automorphisms of the infinite dimensional Clifford albegra, implementable in a Fock representation, the analogous group of automorphisms of the canonical commutation relations and various generalisations are discussed. Their homotopy type is determined in a topology naturally defined by the spin and metaplectic representations. A theorem of Araki and Evans on a
Z2-index for certain projections is generalised using our “mod 2” index for \mathcal{O}. Connections with
K1 of certain Banach algebras are described.
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