In the paper [A faithful linear-categorical action of the mapping class group of a surface with boundary](http://arxiv.org/abs/1012.1032v2) 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](http://arxiv.org/abs/math/0312449v8), in which the author uses the non-commutative machinery to prove that they are. As I have asked in [this](https://mathoverflow.net/questions/216204/are-all-transversely-oriented-foliations-given-by-closed-forms) 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?



Current status of the linearity of mapping class group
Yes. There [is the paper by Bigelow, and our own Ryan Budney](http://arxiv.org/pdf/math/0010310.pdf), who do this for genus 2. I believe that is still the last word.


