Existing proofs of Rokhlin's theorem for PL manifolds - MathOverflow

Existing proofs of Rokhlin's theorem for PL manifolds

2012-02-09T20:27:42Z

I'm looking for a comprehensive reference to existing proofs of Rokhlin's theorem that a 4-dimensional closed spin PL manifold has signature divisible by 16. I'm specifically interested in direct proofs (if any such exist) which do not rely on the fact that $\pi_i(PL/O)=0$ for small $i$.

The most commonly cited reference seems to be the book by Kirby "The Topology of 4-manifolds". But the proof there is for smooth manifolds and I'm not sure why it works for PL manifolds although I've seen it claimed in various places that it does. The same is said about Rokhlin's original proof but I don't know why that's true either. I would also like to know if other proofs for PL manifolds exist. I'm particularly interested to know if there is a PL proof based on the Atiyah-Singer index theorem.

Answer by Bruno Martelli for Existing proofs of Rokhlin's theorem for PL manifolds

2012-02-09T23:48:48Z

There is a proof which uses quantum invariants. Since these invariants are typically defined using state-sums and are combinatorial in nature, I suppose that they work in the PL setting. A nice introduction is Justin Roberts' PhD thesis where Rohlin's theorem is proved as Corollary 5.14 at page 55.

Answer by Andy Putman for Existing proofs of Rokhlin's theorem for PL manifolds

2012-02-10T01:18:43Z

Another approach to the theorem that could probably be rewritten to work in the PL category is the approach of Kirby and Melvin in Appendix C of the following paper:

MR1117149 (92e:57011) Kirby, Robion(1-CA); Melvin, Paul(1-BRYN) The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C). Invent. Math. 105 (1991), no. 3, 473–545.

See Corollary C6.

The idea of this approach is as follows. There is a famous $\mathbb{Z}/2$-invariant of homology $3$-spheres called the Rokhlin invariant. The usual definition of this invariant is as follows. Letting $M^3$ be a homology $3$-sphere, there exists a compact spin $4$-manifold $W^4$ with $\partial W^4 = M^3$. Let $\sigma$ be the signature of $W^4$. Rokhlin's theorem implies that modulo $16$, the value of $\sigma$ is independent of $W^4$. Since $\sigma$ is divisible by $8$ for number-theoretic reasons (namely, van der Blij's lemma about quadratic forms), the value of $\sigma/8$ is well-defined modulo $2$.

Using the Kirby calculus, Kirby and Melvin give a ``$3$-dimensional'' construction of the Rokhlin invariant, avoiding all mention of $4$-manifolds. They then go backwards and use this to prove Rokhlin's theorem about $4$-manifolds.

Looking at their proof, Kirby and Melvin use smoothness in two ways. The first is to prove that the Rokhlin invariant is well-defined. But this is harmless since (by work of Moise) all PL $3$-manifolds can be smoothed in a unique way. The second use of smoothness is to obtain a handlebody decomposition of the $4$-manifold. But this should be easier in the PL category!