Embedding-dependent unstable normal Thom data

At the point its tangent space is zero, so [F1] gives the normal fibers $\mathbb{R}$ and ${\mathbb{R}}^2$. These bundles have different ranks and are not isomorphic; rank is preserved by a fiberwise linear isomorphism.

Their Thom spaces are $S^1$ and $S^2$ by [F2]. They are not homeomorphic: removing any point from $S^1$ gives an open interval, and removing any further point disconnects it; removing a point from $S^2$ gives ${\mathbb{R}}^2$, which remains path connected after removing any further point (polygonal paths may be detoured around that point). A putative sphere homeomorphism would preserve these deletion properties. Nevertheless adding one trivial line to the first normal fiber gives the second and suspends its Thom sphere, in accordance with [F3]. This explicitly exhibits why only the stable normal class is intrinsic.