Can one give an immersion of exotic sphere $S^7$ in a standard sphere $S^8$ of radius $1$?
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Yes. By SmaleHirsch theory it is enough to find a bundle injection $T\Sigma \to \epsilon^8$, so it is enough to find a trivialisation of $T\Sigma \oplus \epsilon^1$. It is a theorem of Kervaire and Milnor that every exotic sphere is stably framable, so $T\Sigma \oplus \epsilon^N$ is trivial for some large N, and the connectivity of $BO(8) \to BO$ means you can destabilise this to trivialise $T\Sigma \oplus \epsilon^1$. 

