X-Nico

Normal invariant

In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental classes and preserving normal bundle information.

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For example, if $p$ is less than half the dimension of $X$, every map $S^p\backslash to\; X$ is homotopic to an embedding by a theorem of Whitney.

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This kind of cobordism theory is a homology theory whose coefficients have been calculated by Thom: therefore the cobordism classes of such maps are computable at least in theory for all spaces $X$.

see also