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Report on $$K$$-theory for convenient algebras. II. (English) Zbl 0830.46062
Bureš, J. (ed.) et al., The proceedings of the Winter school Geometry and topology, Srní, Czechoslovakia, January 1992. Palermo: Circolo Matemático di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 32, 175-184 (1994).
In part I [ibid. 30, 55-63 (1993; Zbl 0790.19007)] I gave an overview over the first steps of a generalization of $$K$$-theory for Banach algebras to a much more general class of algebras, the so called convenient algebras. In this paper I continue this overview with the discussion of the two fundamental long exact sequences in $$K$$-theory, the one induced by a smooth map and the other induced by a bounded algebra homomorphism. Throughout this paper we use the notions, notations and results of part I.
This paper splits into two parts: In the first part we develop some basic homotopy theory for smooth spaces, in particular the theory of fibrations and cofibrations. The main result is that there are smooth versions of the Puppe sequences, long exact sequences of certain sets of smooth homotopy classes of smooth mappings. In the second part we discuss higher $$K$$-groups and relative $$K$$-groups and interpret these groups in terms of homotopy theory. Then we derive the fundamental long exact sequences from the Puppe sequences.
For the entire collection see [Zbl 0794.00022].

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 46H05 General theory of topological algebras 19K99 $$K$$-theory and operator algebras