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Hi!

Novikov's additivity theorem states that if you glue together two compact oriented 4n-manifolds along a connected component of their boundaries, the signature of the resulting manifold is simply the sum of the signatures of the pieces. It is proved for example in Kirby's The Topology of 4-manifolds and Kauffman's On Knots.

However, I haven't found any direct reference to an original article of Novikov's on this topic. Sometimes, The index of elliptic operators III by Atiyah and Singer is given as a reference.

So my questions are : what is the original reference? Has Novikov written this result in any of his papers? What has happened between Novikov's first proof and the one by Atiyah and Singer?

Any comment on this topic will be appreciated.

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up vote 7 down vote accepted

Here is as Novikov himself describes it (in russian):

Rokhlin in 1965 drew my attention repeatedly to the fact that for prime p (large enough for a given dimension), the definition of combinatorial Pontryagin-Hirzebruch classes modulo p is unknown, and this issue is not trivial. I thought about it and found an interesting "additive" property of the signature of manifolds with boundary by gluing along connected boundary components. From this we with Rokhlin extracted right definition of the classes mod p. This additive has been used by Yanih and others after the Moscow Congress, where I talked about it much, for the axiomatization of the signature. We now know that this property of the signature is equivalent in modern terminology to building an "abelian" nontrivial topological quantum field theory.

In 1965-1966 we with Rokhlin sat down to write the joint work, but could not finish it. None of us wanted to submit to another as how to write it. Work broke up, Rokhlin refused. In my preprint of the International Congress of Mathematicians in Moscow in 1966 I placed the information on this as a joint result. This preprint was published with the amendments in the collection in honor of de Rham in 1970.

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Thanks a lot for your answer (and your translation)! –  Maxime Bourrigan Jul 3 '11 at 20:20
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Novikov's 1970 paper Pontrjagin classes, the fundamental group and some problems of stable algebra is available online. A proof of Novikov additivity was first published on pages 587-589 of the Atiyah-Singer paper "The index of elliptic operators: III." (Annals of Maths. 87, 546-604 (1968)) mentioned in the question. There is a footnote on page 587 "We are indebted to Hirzebruch for drawing our attention to Novikov's result."

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