Hi everyone, im reading Milnor's article "On manifolds homeomorphic to the 7sphere", in which he constructs the first example of an exotic structure, id like to know if there's a particular reason why Pontrjagin classes become so relevant in defining the invariant that "detects" exotic smoothness. More generally, is there a way to interpret the Pontrjagin classes that connects them with the way the smooth structure of the manifold behaves? Or are they just happy coincidences and the most "geometric" meaning you can get from them is the obstruction of finding nk2 linearly independent sections on the tangent bundle? References are much appreciated! Thanx in advance
The Pontryagin classes of the tangent bundle are not easy to interpret (witness the Novikov conjecture), but one geometric datum that can be extracted from them is the rational cobordism class of a manifold. According to Thom, the Pontryagin numbers $p_1^{a_1}\cdots p_k^{a_k}[X]$ of a closed, oriented, smooth $4n$manifold $X$ vanish iff there's a compact, oriented $(4n+1)$dimensional manifold bounding a disjoint union of copies of $X$. From Thom's theorem follows Hirzebruch's, expressing the (cobordisminvariant) signature $\sigma(X)$ as a certain Pontryagin number $L(p_1,\dots,p_n)[X]$. When $n=2$, $$\sigma(X)=(p_1^2+7p_2)[X]/45.$$ One striking thing about this formula is that $\sigma(X)$ is an integer, while $(p_1^2+7p_2)[X]/45$ is a priori only a rational number. Milnor's observation is that an integrality theorem for a characteristic class of closed $4n$manifolds gives rise to an invariant of those $(4n1)$manifolds which bound $4n$manifolds. This is a useful principle, a variant of which also underlies ChernSimons theory. We can define an invariant for homotopy 7spheres $S$ by taking $\kappa(S)=45\sigma(Y)+p_1^2[Y,\partial Y] \mod 7$; if $Y'$ is another such bounding manifold, the difference between their invariants will be $45 \sigma(X)+p_1^2(X)$ for the closed manifold $X= (Y') \cup_S Y$, hence a multiple of 7 by the signature theorem. (Milnor prefers the invariant $\lambda=2\kappa \mod 7$.) In his later work with Kervaire ("Groups of homotopy spheres I"), Milnor identifies two different reasons why a homotopysphere may not be (hcobordant to) a standard sphere: (i) it may not bound a parallelizable manifold; or (ii) it may bound a parallelizable manifold, but not one which is also contractible. A homotopy 7sphere which bounds a parallelizable 8manifold is in fact standard, but this is not true of homotopy 8spheres. The invariant $\lambda$ of homotopy 7spheres is an obstruction to bounding a parallelizable 8manifold; not a complete invariant, since KervaireMilnor show that there are exactly 28 hcobordism classes. 

