We show that the action of the mapping class group on bordered Floer homology in the second to extremal spin-structure is faithful. This paper is designed partly as an introduction to the subject, and much of it should be readable without a background in Floer homology.
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Peter S. Ozsváth, Dylan P. Thurston, Robert Lipshitz, A faithful linear-categorical action of the mapping class group of a surface with boundary. J. Eur. Math. Soc. 15 (2013), no. 4, pp. 1279–1307DOI 10.4171/JEMS/392