The signature of a -dimensional compact, oriented manifold is a classical cobordism invariant of ; the so-called Hirzebruch signature formula states that can be computed as a complicated polynomial in the Pontryagin classes of the tangent bundle (evaluated on the fundamental class of ). When is four-dimensional, for instance, we have
This implies that the Pontryagin number must be divisible by three.
There are various further divisibility conditions that hold in special cases. Here is an important early example:
Theorem 1 (Rohlin) If is a four-dimensional spin-manifold, then is divisible by (and so by ).
I’d like to describe the original proof of Rohlin’s theorem, which relies on a number of tools from the 1950s era of topology. At least, I have not gotten a copy of Rohlin’s paper; the proof is sketched, though, in Kervaire-Milnor’s ICM address, which I’ll follow.
1. Obstruction theory
The first observation is that:
Proposition 2 If is a (connected) four-dimensional spin manifold, then restricted to is trivial for any . (This is equivalent to the spin condition.)
To see this, note that the tangent bundle comes from a principal -bundle (which is not unique). The claim is that there is a section over for any , which will imply that is parallelizable.
There are a number of obstructions to finding a section of over . They live in , and . These groups are all zero; the spin groups are simply connected and have zero (like all Lie groups). Also, has no .
What does this mean? If we take a small neighborhood of which is isomorphic to the four-disk, then is trivial on and , and is uniquely determined by an element of . The stable normal bundle is, similarly, determined by an element of .
The strategy of Rohlin’s theorem is the following:
- Show that is mapped to zero in the stable 3-stem (which is ) under the J-homomorphism.
- Deduce that is divisible by in .
- Deduce that of the stable normal bundle is divisible by , which is equivalent to Rohlin’s theorem.
The first observation is actually mostly formal in retrospect, but it is a hugely important idea, because the image of J-homomorphism has been completely determined. (Incidentally, Rohlin’s theorem and its generalizations can be used to bound below the image of the J-homomorphism, by reversing the argument.) Since the -homomorphism in dimension three is a surjection
the second step is then clear. (This fact is not obvious, but it follows from Adams’s and Mahowald’s work; apparently Rohlin had to prove it himself, though.) Computing the Pontryagin classes as a function of an element of is classical.
2. The J-homomorphism and the Pontryagin-Thom construction
The classical J-homomorphism is the map
which can be obtained as follows: there is a natural map since every element of determines an element of by the action of on the sphere (as the one-point compactification of ). Some (rather crude) notes on the J-homomorphism, describing a paper of Adams, can be found here.
Another way to think of it, which is equivalent (up to a sign), is the following. Suppose . There is a canonical trivialization on the normal bundle of (the trivialization that extends to ) in a high-dimensional imbedding . Any element determines a “twist” of this trivialization, a different isomorphism
By the Pontryagin-Thom construction, this determines an element of the stable homotopy group .
Proposition 3 The element determined by the Pontryagin-Thom construction as above is precisely applied to .
I won’t prove this; if you unwind the definition of the Pontryagin-Thom collapse map and the J-homomorphism, it is visually clear.
As a result, we can give a criterion for when an element in goes to zero under the J-homomorphism: this happens precisely when there exists a manifold (of dimension ) such that together with a framing of its stable normal bundle which restricts to the framing of . This is the Pontryagin-Thom theorem, essentially.
Consequently, we have:
Proposition 4 Let be any manifold with boundary . Choose a trivialization of the stable normal bundle of which restricts to an element (regarded as a framing of ). Then this is annihilated by the J-homomorphism.
In fact, this is now essentially tautological: the manifold , with the framing given by the element , is the (framed) boundary of . Consequently, goes to zero under .
3. Rohlin’s theorem
We now have essentially all the ingredients to prove Rohlin’s theorem, at least if we believe a certain fact about the J-homomorphism.
Theorem 5 The -homomorphism is a surjection .
In general, the order of the image of the J-homomorphism in dimensions (the cases when , by Bott periodicity) is given by the denominator of ; this is a theorem of Mahowald and Adams.
Proof: Let be a four-dimensional spin-manifold. If , we have seen that we can describe the stable normal bundle of by an element by trivializing away from . Alternatively, we choose a trivialization of on minus a small disk around , and restrict the trivialization to the boundary to get an element of . By the discussion in the previous section, must map to zero under , so it is divisible by 24.
Now the Pontryagin class of a vector bundle given by a clutching function in this way are linear in . In fact, I claim that up to sign, is given by . If we prove this, it will follow that
which is Rohlin’s theorem.
To see this, note that we can collapse to a point (for a small neighborhood of ) and using the trivialization , collapse to a vector bundle on . This (stable) vector bundle is given precisely by the clutching function .
So, to complete the argument, we may as well prove:
Lemma 6 The Pontryagin class of a stable vector bundle on as a function of the clutching function is .
To see this lemma, we note that is linear in , and the image of is exactly . To see this, note that the image of the second Chern class for complex vector bundles on is precisely (this is a special case of a theorem of Bott) and complexification
has image of index two. Since of a real vector bundle is of the complexification, this completes the argument.
The Kervaire-Milnor talk is not actually about deducing Rohlin’s theorem; it is about reversing the argument. Rohlin’s theorem was later generalized by the statement that the –genus of a spin manifold of dimension is an even integer; Atiyah and Singer, for instance, deduced this by interpreting it as an index. As a result, Kervaire and Milnor showed that if an element mapped to zero under , one could use the same Pontryagin-Thom argument to construct a manifold with only its top Pontryagin class nontrivial and determined by ; the integrality of the -genus now gives a strong divisibility condition on the top Pontryagin class and thus on .