The Whitney Immersion theorem states that any $n$-dimensional manifold can be immersed in $\mathbb{R}^{2n}$.
However, I seem to remember that if $X$ is a compact $n$-dim manifold, then $X$ can be immersed in $\mathbb{R}^{2n-1}$, except at possibly some finite number of places. Does anyone have a reference of where a proof of this result may be found?
Indeed, every smooth compact $n$-manifold ($n\ge 2$) admits an immersion in $R^{2n-1}$. This result was proven by H.Whitney in "The singularities of a smooth $n$-manifold in $(2n − 1)$-space", Ann. of Math. (2), 45 (1944) pp. 247-293.
This result was improved rather dramatically by R.Cohen in 1982 (see an outline here http://www.pnas.org/content/79/10/3390.full.pdf), full proof is in
Ralph L. Cohen "The Immersion Conjecture for Differentiable Manifolds", Annals of Mathematics, Vol. 122, No. 2 (1985), pp. 237-328.
Cohen proved that every smooth compact $n$-manifold admits an immersion in $R^{2n -a(n)}$, where $a(n)$ is the number of 1's in the binary expansion of $n$, $n>1$.