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Does the use of Chern classes and other characteristic classes extend to the existence of embeddings? For example, it is known that ## S^n ## , the n-sphere does not embed in ## \mathbb R^n## or ## \mathbb R^m ## for ##m<n ## (I think this is a corollary of Borsuk Ulam). Can this be determined by Chern classes (or some other general theory, e.g., (co)homology, homotopy theory, etc.)? I think there are some results using the normal bundle of the embedded object, which must be trivial, so that the bundle sum of the embedded figure and its normal complement must be trivial in the ambient space. Are there anything other results?