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Nov 25, 2010 at 16:27 comment added Noz I agree that Stiefel-Whitney classes in general are designed for far more general problem. However in my answer I only talked about the first Stiefel-Whitney class which, correct me if I'm wrong, is specifically designed to address the orientability of vector bundle and is easily defined.
Nov 25, 2010 at 16:04 comment added Ryan Budney Stiefel-Whitney classes are designed for a more general question than the one you ask -- the issue of constructing linearly-independent sections of vector bundles. Your questions have answers that don't need "big machines" like that, but of course they are convieniently answered by big machines.
Nov 25, 2010 at 12:00 comment added Pengfei It seems that the Stiefel-Whitney class was designed for my question (^-^). All answers here are great!
Nov 25, 2010 at 11:57 vote accept Pengfei
Nov 25, 2010 at 11:00 history answered Noz CC BY-SA 2.5