To get back into the flow of posting, I’ll talk about a curiosity in differential topology that is most organically proven using the machinery of characteristic classes.
An between smooth manifolds is a smooth map whose differential on tangent spaces is injective on each fiber. The Whitney Immersion Theorem states that any smooth n dimensional manifold admits an immersion into (and an embedding into , which can be improved nontrivially to ). We may ask whether the dimension is optimal: that is, whether or not every smooth manifold of dimension embeds into for some . This turns out not to be the case if is not a power of , but using Stiefel-Whitney classes, we may show that the number is optimal if .
. For , real projective space does not immerse into .
. Suppose there existed an immersion . Then, is a morphism of vector bundles which covers and which factors through the pullback . Because is an immersion, its differential is fiberwise injective, and the same is true for the induced map . Since is a trivial vector bundle, it follows that , i.e. is a subbundle of a trivial bundle of rank .
Now consider the normal bundle , defined as the orthogonal complement of in . By this definition, we have , implying that . Moreover, by the axioms for Stiefel-Whitney classes, we have . However, whose multiplicative inverse in the ring is . This contradicts the fact that
Comments
I had to follow along with Milnor-Stasheff since it’s been a while since I did algebraic topology, but really nice writeup! I like your other articles as well. Could you give an example of a manifold for which is not optimal?
Thank you, and good question! For a low-hanging class of examples,
$S^n$embeds into$\mathbb{R}^{n+1}$in the usual way, which beats the general Whitney bound for$n > 2$. I think (and I’ll try to find a reference) that every smooth, closed$3$-manifold immerses into$\mathbb{R}^4$, although augmenting “immersion” to “embedding” is far more subtle and unresolved except for special classes, e.g. for$3$-manifolds fibered over surfaces. I think there are also general results for$\mathbb{CP}^n$immersing into$\mathbb{R}^{4n-4}$for suitable constraints on$n$.