Rigidity of K - theory under deformation quantization 3

Quantization, at least in some formulations, involves replacing some algebra of observables by a (more non-commutative) deformed algebra. In view of the fundamental role played by K-theory in non-commutative geometry and topology, it is of interest to ask to what extent K-theory remains \rigid" under this process. We show that some positive results can be obtained using ideas of Gabber, Gillet-Thomason, and Suslin. From this we derive that the algebraic K-theory with nite coeecients of a deformation quantization of the functions on a compact symplectic manifold, forgetting the topology, recovers the topological K-theory of the manifold. Notation. If A is a ring, K (A) will denote its (connective) K-theory spectrum , the spectrum associated to the innnite loop space K 0 (A) BGL(A) + , where BGL(A) + is the result of applying the Quillen +-construction to the classifying space of the innnite general linear group over A. By deenition, the (algebraic) K-groups K i (A) of A are (at least in positive degrees) the homo-topy groups of K (A), and the K-groups of A with nite coeecients Z=(m), K i (A; Z=(m)), are deened (at least in positive degrees) to be the homotopy groups of S(Z=(m)) ^ K (A), where S(Z=(m)) is the Z=(m) Moore spectrum. (This is almost, but not quite, the deenition of Browder in 1]; for an explanation of the diierence between the two deenitions, see 11], pp. 285{286.) In the one case below where confusion might be possible between algebraic and topological K-groups, we denote these by K alg j and K top j , respectively.