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.