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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"][1]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. [1]: http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=1001966

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# ExistngExisting proofs of Rokhlin's theorem for PL manifolds

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I'm looking for a comprehensive reference to all 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"4-manifolds"][1]. 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. [1]: http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=1001966

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