Saturday, 9 February 2008

dg.differential geometry - Pontrjagin numbers and exotic spheres

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 pa11cdotspakk[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(p1,dots,pn)[X]. When n=2,
sigma(X)=(p21+7p2)[X]/45.



One striking thing about this formula is that sigma(X) is an integer, while (p21+7p2)[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 Chern-Simons theory.



We can define an invariant for homotopy 7-spheres S by taking kappa(S)=45sigma(Y)+p21[Y,partialY]mod7; if Y is another such bounding manifold, the difference between their invariants will be 45sigma(X)+p21(X) for the closed manifold X=(Y)cupSY, hence a multiple of 7 by the signature theorem. (Milnor prefers the invariant lambda=2kappamod7.)



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.

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