| 1. | This is an extraordinary result that does not generalize to other characteristic classes.
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| 2. | The theory of characteristic classes generalizes the idea of obstructions to our extensions.
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| 3. | Equivalently, all finite characteristic classes pull back from a stable class in BG.
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| 4. | The Euler class, in turn, relates to all other characteristic classes of vector bundles.
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| 5. | Because of this, the tautological bundle is important in the study of characteristic classes.
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| 6. | The result is expressed as an integral of certain characteristic classes over the infinite fiber.
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| 7. | Such a rational version, in fact a characteristic class of twisted K-theory, is already known.
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| 8. | Stated another way, the only characteristic class of a 3-manifold is the obstruction to orientability.
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| 9. | The fundamental characteristic class in cyclic cohomology, the JLO cocycle, generalizes the classical Chern character.
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| 10. | In this way foundational cases for the theory of characteristic classes depend only on line bundles.
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