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of with this normal bundle is equivalent to a codimension 0 immersion of the total space of this bundle, which is an open manifold.

The stable normal bundle is the class of normal bundles plus trivial bundles, and thus if the stable normal bundle has cohomological dimension , it cannot come from an (unstable) normal bundle of dimension less than . Thus, the cohomology dimension of the stable normal bundle, as detected by its highest non-vanishing characteristic class, is an obstruction to immersions.

Since characteristic classes

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