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.



Construction of invariants of 4-manifolds with the Kirby calculus

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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://doi.org/10.1007/s00220-017-3012-9>


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.


