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 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 orient...

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Bibliographic Details
Published in:Journal of algebra 2009-12, Vol.322 (12), p.4295-4330
Main Authors: Mandal, Satya, Sheu, Albert J.L.
Format: Article
Language:English
Subjects:
Euler classes
Projective modules
Vector bundles
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Summary: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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2009.09.012