Pontrjagin numbers and exotic spheres

Question

Hi everyone, im reading Milnor's article "On manifolds homeomorphic to the 7-sphere", 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 n-k-2 linearly independent sections on the tangent bundle? References are much appreciated! Thanx in advance

Answer 1

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 (cobordism-invariant) 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 $(4n-1)$-manifolds which bound $4n$-manifolds. This is a useful principle, a variant of which also underlies Chern-Simons theory.

We can define an invariant for homotopy 7-spheres $S$ 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 homotopy-sphere may not be (h-cobordant 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 7-sphere which bounds a parallelizable 8-manifold is in fact standard, but this is not true of homotopy 8-spheres. The invariant $\lambda$ of homotopy 7-spheres is an obstruction to bounding a parallelizable 8-manifold; not a complete invariant, since Kervaire-Milnor show that there are exactly 28 h-cobordism classes.