Communications in Mathematics and Applications Vol. 5, No. 2, pp. 73–81, 2014 ISSN 0975-8607 (online); 0976-5905 (print) Published by RGN Publications http://www.rgnpublications.com An Explicit Isomorphism in R/Z-K-Homology Research Article Adnane Elmrabty * and Mohamed Maghfoul Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco *Corresponding author: adnane_elmrabty@yahoo.com Abstract. In this paper, we construct an explicit isomorphism between the flat part of differential K-homology and the Deeley R/Z-K-homology. Keywords. Spin c -manifold; Chern character; R/Z-K-homology MSC. 19K33; 19L10 Received: September 29, 2014 Accepted: October 12, 2014 Copyright © 2014 Adnane Elmrabty. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction K-homology is the homology theory dual to topological K-theory. A geometric model for K-homology was introduced by Baum-Douglas (see [1]), and proved to be an extremely important tool in index theory and physics (see [5]). Motivated by generalizing the pairings between K-theory and K-homology to the case of R/Z-coefficients, Deeley defined in [2] a model for geometric K-homology with R/Z-coefficients using approach of operators algebras. Let X be a finite CW-complex and N be a II 1 -factor. A cycle in the Deeley R/Z-K-homology (which we call R/Z-K-cycle) over X is a triple (W , ( H, ε, α) (∇ H ,∇ ε ) , g) where W is a smooth compact Spin c - manifold, H is a fiber bundle over W with fibers are finitely generated projective Hermitian Hilbert N-modules, with a Hermitian connection ∇ H , ε is a Hermitian vector bundle over ∂W with a Hermitian connection ∇ ε , α is an isomorphism from H| ∂W to ε ⊗ N, and g : W → X is a continuous map. The Deeley R/Z-K-homology group K * ( X , R/Z) is the quotient of the set of isomorphism classes of R/Z-K-cycles over X by the equivalence relation generated by bordism and vector bundle modification (Definition 3.6).