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Just curious if someone knows of any Characteristic class used to determine if a manifold allows

a Complex structure? It seems strange that Complex Space C^n is topologically Identical to R^{2n}

yet I believe not all R^{2n}s ( if any) allow Complex structures. Thanks for any comments, refs.

EDIT: I believe we would be working with classifying spaces associated to , respectively,

almost-complex structures and GL(2n) , and then using the inclusion of almost-complex

into GL(2n) and then working with the associated inclusion i of the classifying spaces

and lifts for the associated classifying maps.

But we would then need to know about the cohomology associated to each, in order to

figure out the obstructions to the existence . Am I on the right track, and, if not, please correct me,

if so, p-lease help me take the next step on the details.

Thanks.

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# A Existence of Complex Structures and Characteristic Classes

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