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On the equivalence of geometric and analytic $$K$$-homology. (English) Zbl 1146.19004
The homology theory dual to Atiyah-Hirzebruch $$K$$-theory is called $$K$$-homology; this theory can be abstractly defined in terms of an appropriate spectrum. Work of M. F. Atiyah [Proc. Int. Conf. Funct. Anal. Rel. Topics, Tokyo 1969, 21–30 (1970; Zbl 0193.43601)] highlighted the need for a more concrete formulation of $$K$$-homology, and the work of L. G. Brown, R. G. Douglas, and P. A. Fillmore [Ann. Math. (2) 105, 265–324 (1977; Zbl 0376.46036)] and G. G. Kasparov [Math. USSR, Izv. 9(1975), 751–792 (1976; Zbl 0337.58006)] independently gave analytical definitions of $$K$$-homology. A few years later, P. Baum and R. G. Douglas [Proc. Symp. Pure Math. 38, Part 1, Kingston/Ont. 1980, 117–173 (1982; Zbl 0532.55004)] introduced what they called a geometric definition of $$K$$-homology.
Baum and Douglas constructed a homomorphism from geometric $$K$$-homology to analytical $$K$$-homology; this map turns out to be an isomorphism, but the details of the proof have never been published. The paper under review gives an detailed proof that geometric and analytical $$K$$-homology are isomorphic for finite $$CW$$-complexes.

##### MSC:
 19K33 Ext and $$K$$-homology 55N22 Bordism and cobordism theories and formal group laws in algebraic topology
K-homology
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