Integrality of the $L$-genus for a smooth manifold For a compact, oriented, smooth manifold $M^{4k}$, the Hirzebruch signature theorem gives the signature $\sigma(M)$ in terms of a polynomial $P_k$ in the Pontryagin numbers of $M$ whose coefficients are rational but have very large denominators. In particular, $P_k$ is integer-valued because $\sigma(M^{4k})$ is. The usual proof (at least the ones I've seen) of the result is to compute $\Omega^{SO}_* \otimes \mathbb{Q}$, note that signature is invariant under (oriented) cobordism, and then show that $P_k$ gives the right result on the generators $\mathbb{CP}^{2m}$. Is there a direct proof, though, or even some sort of reason why the integrality should hold? 
To take a more narrow example: In dimension 4, the class $p_1(M^4) = 3\sigma(M) \in 3\mathbb{Z}$ if $M$ is smooth. Is there any other significance to $p_1(M^4)\in \mathbb{Z}/3\mathbb{Z}$ beyond just being an obstruction to smoothability?
 A: You want $M$ to be compact and oriented. Integrality holds because the signature is manifestly integral; the real question is why the signature is given by some particular rational combination of characteristic numbers. 
From a modernish point of view results of this form should be thought of as consequences of variations on the Atiyah-Singer index theorem, which gives a very general recipe for showing that certain integer-valued invariants (indexes of elliptic operators) can be computed in terms of certain characteristic numbers. Some proofs of the index theorem proceed by using a cobordism argument to reduce the theorem to checking particular cases, but many others don't; there are a startling variety of proofs, really. 
One reason things are particularly mysterious without the index theorem is that more sophisticated integrality arguments of this form depend on the manifold having more sophisticated extra structure. From the perspective of index theory the point of this extra structure is to allow you to define extra interesting elliptic operators; otherwise it's pretty unclear what's going on. For example, the A-hat genus is integral on spin manifolds because spin structures let you write down an operator called the Dirac operator whose index is the A-hat genus.
