\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: invariance of Heegaard-Floer cohomology

invariance of Heegaard-Floer cohomology

SECTIONS
Download .tex

theorem 0.0.1

Suppose that \((\Sigma_g,\underline \alpha, \underline \beta,z)\) and \((\Sigma_{g'},\underline \alpha', \underline \beta',z')\) are pointed admissible Heegaard diagrams for \(M\). Then \[\HHeF(\Sigma, \underline{\alpha}, \underline{\beta},z)\cong\HHeF(\Sigma, \underline{\alpha}, \underline{\beta},z).\] The proof of this theorem is beyond the scope of these notes. However, we can outline some of the steps in the proof.
  1. Show that a similar theorem to (Heegaard moves generate equivalences) holds for pointed Heegaard diagrams.
  2. Invariance of \(\HHeF(\Sigma_g, \underline \alpha, \underline \beta, z)\) under isotopies of almost complex structure.
  3. 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.
  4. 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 \(L_{\underline \alpha}\).
  5. Invariance of \(\HHeF(\Sigma_g,\underline \alpha, \underline \beta,z)\) under stabilizations: see (invariance of Heegaard Floer cohomology).
  6. 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.