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Báo cáo toán học: "Mapping cones and exact sequences in KK-theory "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Lập bản đồ tế bào hình nón và trình tự chính xác trong KK-lý thuyết. | J. OPERATOR THEORY 15 1986 163-180 Copyright by INCREST 1986 MAPPING CONES AND EXACT SEQUENCES IN KK-THEORY J. CUNTZ and G. SKANDALIS INTRODUCTION Let J. and 2 be c -algebras and p A - B a -homomorphism. The mapping cone for p is a c -algebra Cy. As the equivalence relation defining Kasparov s KK-functor is homotopy we get for all c -algebras D a long exact sequence KK Ỡ X 0 1 --4 KK .5 0 1 - KK D c p - - KK D X KK 2 B . Assuming moreover that the algebras A and B are separable we also obtain using Bott periodicity an exact sequence KK 5 27 KK X Ũ KK C 2 - KK 5 0 I D KK X 0 1 . These two exact sequences are the mapping cone or Puppe exact sequences. Of course the Puppe sequences can be obtained from the exact sequences associated with an ideal established by Kasparov 12 7 cf. 14 for the Z 2 graded case since we have a short exact sequence 0 - S 0 1 - c . - A - 0. On the other hand the exactness of the Puppe sequences is an immediate consequence of the definition of the Kasparov groups and can as we shall show in fact be used to give a much simpler proof for the existence of the long exact sequences associated with an ideal. For this let 0 - 1 A A I - 0 be a short exact sequence admitting a completely positive cross-section and let e I - Cq be the natural embedding. We construct an inverse for e KK Cfl namely the element of KK CỢ Z corresponding to the short exact sequence 0 - 2 0 1 X 0 1 - cq - 0. 164 J. CUNTZ and G. SKANDALIS Therefore in the Puppe sequences associated with q we can replace Cq by I and obtain the long exact sequences associated with the ideal I. The fact that the second Puppe sequence is less obvious than the first one is due to the unsymmetry in A and B of the equivalence relation that defines the functor KK L B namely homotopy. We introduce here an equivalence relation that we call cobordism and which reverses the roles played by A and B. We then prove that for separable A cobordism and homotopy coincide. We illustrate this dual equivalence .