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\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\TropB}{TropB} \DeclareMathOperator{\weight}{wt} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Fuk}{Fuk} \DeclareMathOperator{\str}{star} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\CritVal}{CritV} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\FS}{FS} \DeclareMathOperator{\Vect}{Vect} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Bl}{Bl} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Arg}{\mathbf{M}}\begin{filecontents}{references.bib} @article{ballard2012hochschild, title={Hochschild dimensions of tilting objects}, author={Ballard, Matthew and Favero, David}, journal={International Mathematics Research Notices}, volume={2012}, number={11}, pages={2607--2645}, 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Then \[\HHeF(\Sigma, \underline{\alpha}, \underline{\beta},z)\cong\HHeF(\Sigma, \underline{\alpha}, \underline{\beta},z).\] \end{theorem} The proof of this theorem is beyond the scope of these notes. However, we can outline some of the steps in the proof. \begin{enumerate} \item Show that a similar theorem to \cref{thm:heegaardMoves} holds for pointed Heegaard diagrams. \item Invariance of $\HHeF(\Sigma_g, \underline \alpha, \underline \beta, z)$ under isotopies of almost complex structure. \item Invariance of $\HHeF(\Sigma_g,\underline \alpha, \underline \beta,z)$ under isotopies which do not create/destroy critical points. This can be reduced to the previous step by pulling back the almost complex structure along the isotopy. \item Invariance of $\HHeF(\Sigma_g,\underline \alpha, \underline \beta,z)$ under isotopies which do not create/destroy critical points: these isotopies can always be taken to be Hamiltonian isotopies of the Lagrangian $L_{\underline \alpha}$. \item Invariance of $\HHeF(\Sigma_g,\underline \alpha, \underline \beta,z)$ under stabilizations: see \cref{exr:invarianceOfHHF}. \item Invariance of $\HHeF(\Sigma_g,\underline \alpha, \underline \beta,z)$: by far, the trickiest portion of the proof. This requires looking at the product structure on Lagrangian intersection Floer cohomology. \end{enumerate}\printbibliography \end{document}