Complexifiable characteristic classes

Authors: Alexander D Rahm

Publish Date: 2014/01/21

Volume: 10, Issue: 3, Pages: 537-548

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Abstract

In the theory of characteristic classes in the sense of Milnor and Stasheff 4 whom we follow in terminology and notation in this article it is wellknown how the Chern classes are mapped to even Stiefel–Whitney classes when converting complex vector space bundles to real vector space bundles by forgetting the complex structure In the other direction we have the fibrewise complexification Given a real vector bundle F rightarrow B with fibre mathbb Rn its complexification is the complex vector bundle Fmathbb C = Fotimes mathbb R mathbb C rightarrow B obtained by declaring complex multiplication on F oplus F in each fibre mathbb Rn oplus mathbb Rn by ix y = y x for the imaginary unit i The Pontrjagin classes of a real vector bundle are up to a sign constructed as Chern classes of its complexification Conversely which classes of a real vector bundle can be attributed to its complexification These are the complexifiable characteristic classes which we determine in this article under the request that they are characteristic classes in the sense of 4Not every complex vector bundle is realgenerated as the odd degree Chern classes have the property c 2k+1overlineE = c 2k+1E on the complex conjugate bundle overlineE it is an easy exercise to show that no complex vector bundle with some nonzero and non2torsion odd Chern class can admit a real generator bundle This makes it seem possible that the subcategory of realgenerated vector bundles could admit information additional to its Chern classes in terms of complexifiable classes of the real generator bundles However we will see that the Chern classes already contain all of the relevant information

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