The signature {\sigma(M)} of a {4k}-dimensional compact, oriented manifold {M} is a classical cobordism invariant of {M}; the so-called Hirzebruch signature formula states that {\sigma(M)} can be computed as a complicated polynomial in the Pontryagin classes of the tangent bundle {TM} (evaluated on the fundamental class of {M}). When {M} is four-dimensional, for instance, we have

\displaystyle \sigma(M) = \frac{p_1}{3}.

This implies that the Pontryagin number {p_1} 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 {M} is a four-dimensional spin-manifold, then {\sigma(M)} is divisible by {16} (and so {p_1} by {48}).

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.  (more…)