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In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the authors provide a faithful linear-categorical action of the mapping class group of a surface with necessarily non-empty boundary using knot Floer homology that does not seem decategorifiable in a non-trivial way.

On the other hand, there exists the following paper, Mapping class groups are linear, in which the author uses the non-commutative machinery to prove that they are. As I have asked in this question, there might perhaps be a mistake in one of the proof ingredients.

Hence the question: is it known that a mapping class group of some surface (possibly with non-empty boundary or non-zero number of punctures) is linear?

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    $\begingroup$ I changed the arXiv links to point to the abstract pages. $\endgroup$
    – David Roberts ♦
    Commented Sep 2, 2015 at 6:43
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    $\begingroup$ The paper ``Mapping class groups are linear'' was published in 2018 as part of the de Gruyter book Topological Algebras and Their Applications: Proceedings of the 8th International Conference on Topological Algebras and Their Applications, 2014. Is the paper believed to be correct at this point, or is this still under dispute? $\endgroup$
    – Jim Belk
    Commented Nov 14, 2021 at 0:02

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Yes. There is the paper by Bigelow, and our own Ryan Budney, who do this for genus 2. I believe that is still the last word.

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