A bordism-type description of homology
This paper develops a representation theorem for generalized homology theories which is ``more geometric'' than the usual representation of such theories in terms of spectra. Let \(h^*(\;)\) be a generalized multiplicative cohomology theory and \(X\) a topological space. The associated geometric homology group, \(h'_q(X)\), is the quotient of the free abelian group generated by triples \((M,x,f)\), with M an \(h^*\) oriented \(n+q\) manifold, \(x\in h^n(X)\), and \(f:M\rightarrow X\) by an equivalence relation combining bordism of the manifold \(M\) and ``vector bundle modification''. It is shown that for \(X\) having the homotopy type of a finite CW complex the map \(\phi:h'_q(X)\rightarrow h_q(X)\) given by \(\phi([M,x,f])=f_*(x\cap[M])\) is an isomorphism. The cap product and homology cross product are described in terms of this representation and the special case of complex \(K\)-theory is considered where the results are closely related to work of \textit{P. Baum} and \textit{R. G. Douglas} [K-Theory 5, No. 1, 1-46 (1991; Zbl 0755.46035)].
scientific article; zbMATH DE number 1160800