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I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory. I have a question.

In the knot theory, the Reidemeister moves play fundamental roles. For instance, to prove the fact that the Jones polynomial is an invariant of knots, we can use the Reidemeister moves.

On the other hand, in the 4-manifold theory, there is the Kirby calculus, which play roles similar to the Reidemeister moves.

So, are there some studies about construction of an invariant of 4-manifolds by using the Kirby calculus?

Thanks for your help.

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    $\begingroup$ Most of the invariants of $4$-manifolds satisfy various surgery formulas showing that they are in some sense compatible with Kirby calculus. However, there is one issue in dimension $4$ that is not present in lower dimensions: many smooth $4$-manifolds admit several non-diffeomorphic smooth structures. This is ahrder to see using Kirby calculus alone. $\endgroup$ – Liviu Nicolaescu Jan 31 '16 at 12:44
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Disclaimer: Shameless self-advertising.

Yes, it can be done, and it's really beautiful! You can define the Crane-Yetter invariant with Kirby calculus, and possibly other TQFTs ("dichromatic models"). I've written this down in this article:

https://www.manuelbaerenz.de/article/dichromatic-state-sum-models-for-four-manifolds-from-pivotal-functors

If you want a version with more impressions and pictures, and less text, look at the talk slides:

https://www.manuelbaerenz.de/article/understanding-crane-yetter-model

The Broda invariant is a special case of the dichromatic framework, which was developed by Jerome Petit (and probably Alain Bruguières).

The dichromatic and Crane-Yetter invariants are stronger than signature and Euler characteristic. They are sensitive to the fundamental group (not just homology), but still fail to distinguish $S^2 \times S^2$ and $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$. They are probably still homotopy invariants, but this is a conjecture.

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    $\begingroup$ @nsmath, you're welcome! I'll edit the answer a bit since you were interested in something going beyond Broda's invariant. $\endgroup$ – Manuel Bärenz Feb 27 '17 at 7:27
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The Witten-Reshetikhin-Turaev approach to constructing quantum topological invariants of $3$-manifolds is to define them on framed links and to prove invariance under Kirby moves.

There is a paper of Broda which presents a $4$--manifold version of this strategy to construct "Witten-Reshetikhin-Turaev invariants" for $4$-manifolds.

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  • $\begingroup$ Thank you. But this paper says that the invariants are expressible by classical invariants (signature and Euler character). Do you know another research about construction of "new" invariants? $\endgroup$ – Shinichiro Nakamura Jan 31 '16 at 12:11

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