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Feb 24, 2010 at 10:15 comment added Daniel Moskovich So your answer is that it is important to classify vector bundles because it leads to invariants of vector bundles, which are, in turn, useful invariants of the manifold? If so, why not just take a shortcut and calculate characteristic classes without bothering about the classification of vector bundles? (playing devil's advocate)
Feb 24, 2010 at 8:59 history answered Pascal Lambrechts CC BY-SA 2.5