Local coefficients and Euler class groups

In this paper, we establish an isomorphism between the Euler class group E(R(X),L) for a real smooth affine variety X=Spec(A) and the 0-th homology group H0(Mc;G) with local coefficients in a bundle G of groups constructed from the line bundle L over M corresponding to the orientation rank-1 projective module L, where Mc is the compact part of the manifold M of real points in X. Then by Steenrod's Poincaré duality between homology and cohomology groups with local coefficients, this isomorphism is identified with the Whitney class homomorphism.