# Immersions of manifolds into Euclidean space

Recall the well-known result of Whitney that any (compact) smooth $$n$$-manifold admits an immersion into $$\mathbb{R}^{2n-1}$$. Today there was a preprint posted on the arXiv (arXiv:2011.00974) which mentioned in its introduction the following result of Cohen, which strengthens Whitney’s result as follows:

Any (compact) smooth $$n$$-manifold admits an immersion into $$\mathbb{R}^{2n-\alpha(n)}$$, where $$\alpha(n)$$ is the number of ones in the dyadic expression of $$n$$.

That is to say, writing $$n = 2^{i_1} + \cdots + 2^{i_l}$$ with $$i_1 < \ldots < i_l$$, then $$\alpha(n) = l$$.

Cohen’s paper was published in 1985, but up to now I was not aware of his result …