Abstract
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 H 0 ( M c ; 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 M c 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.
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