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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.

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Try section 1.5 of Mandelbaum's survey "Four-dimensional topology" projecteuclid.org/DPubS/Repository/1.0/…. It gives a sketch for PL manifolds with trivial first homology group. The proof does not use Atiyah-Singer theorem though. By the way, your link asks for mathscinet subscription and since I connect to mathscinet through library proxy I cannot use the link, and cannot even guess what paper it points to. –  Igor Belegradek Feb 9 '12 at 22:35
    
It should be Section 1.4. (I own Russian edition where it is 1.5). –  Igor Belegradek Feb 9 '12 at 22:37
    
Thanks, Igor. that paper does look interesting but after looking at it briefly it seems to suffer from the same problems that I found in other proofs that I've seen. That is it uses some results which were only proved in smooth case in PL category (such as that connected sum stabilization eventually turns PL h-cobordant manifolds into PL difeeomorphic ones). Also, to be clear, I don't insist on an Atiyah-Singer index theorem proof. but if there is one, I'd like to see it. Lastly, the link in my post was to Kirby's book that I mentioned. It just got mangled in formatting. I'll try to fix it. –  Vitali Kapovitch Feb 10 '12 at 0:23

2 Answers 2

up vote 11 down vote accepted

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!

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Thanks, I'll certainly take a look at that paper. I really want to see a clean PL proof that makes it clear where exactly the PL structure is used and why the proof fails in the TOP category. From what you are saying it semms that in the approach you describe this happens on the handlebody decomposition step. It's still far from clear to me though because I think topological handlebody decomposition always exists by Freedman, isn't this right? –  Vitali Kapovitch Feb 10 '12 at 4:00
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@Vitali Kapovich : Not always. In fact, if a 4-manifold has a handle decomposition, then it is smoothable. The point is that the attaching maps are homeomorphisms of $3$-manifolds onto their images, and such maps are always isotopic to smooth maps. There's a brief discussion of this in Chapter 9.2 of Freedman and Quinn's book. –  Andy Putman Feb 10 '12 at 4:20
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@Andy Putman: that's a very nice observation! I didn't realize that. Of course this being the case my original question becomes somewhat moot: once one proves that a PL 4-manifold has a handlebody decomposition (which I think is obvious) it is then smoothable and hence the smooth Rokhlin's theorem applies. –  Vitali Kapovitch Feb 10 '12 at 4:52
    
@Vitali : Yes, that's one way to do it. I wasn't sure what you wanted in your question -- I had assumed that you wanted a PL proof for aesthetic reasons or something. But if all you care about is the correctness of the result, then this is all you need. –  Andy Putman Feb 10 '12 at 5:02
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@Vitali Kapovitch : Isn't the handlebody observation I described essentially equivalent to PL=Diff in dimension 4 (the existence part, not the uniqueness part; but all you need is existence). –  Andy Putman Feb 10 '12 at 5:47

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.

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Thanks. I must say that I'm completely unfamiliar with that approach. Is it discussed anywhere that it all really works in PL category? Roberts doesn't mention the issue at all as far as I can tell. –  Vitali Kapovitch Feb 10 '12 at 0:39
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I don't think that this issue is discussed. They define invariants on PL objects (triangulations or handle decompositions), then they prove that they are invariant under the corresponding moves (Pachner moves? and Kirby moves), then they show that the signature and w_2 can be computed from these invariants, and the algebra of the invariants show that 16 divides the signature when w_2=0. –  Bruno Martelli Feb 10 '12 at 2:04
    
I mean, Roberts work with smooth manifolds but probably PL is enough, but one needs to check that carefully. –  Bruno Martelli Feb 10 '12 at 2:05

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