Dual To Tangent Bundle
The normal bundle is dual to the tangent bundle in the sense of K-theory: by the above short exact sequence,
in the Grothendieck group. In case of an immersion in, the tangent bundle of the ambient space is trivial (since is contractible, hence parallelizable), so, and thus .
This is useful in the computation of characteristic classes, and allows one to prove lower bounds on immersibility and embeddability of manifolds in Euclidean space.
Read more about this topic: Normal Bundle
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