Stable Normal Bundle - Construction Via Classifying Spaces

Construction Via Classifying Spaces

An n-manifold M has a tangent bundle, which has a classifying map (up to homotopy)

Composing with the inclusion yields (the homotopy class of a classifying map of) the stable tangent bundle. The normal bundle of an embedding ( large) is an inverse for, such that the Whitney sum $\tau_M\oplus \nu_M \colon M \to BO(n+k)$ is trivial. The homotopy class of the composite is independent of the choice of inverse, classifying the stable normal bundle .

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