\documentclass[11 pt]{article} \usepackage{amsmath,amsthm,amsfonts,amssymb,titlesec} \usepackage{hyperref} \usepackage{tikz} \usepackage{verbatim} \usepackage{accents} \usepackage[citestyle=alphabetic,bibstyle=alphabetic,backend=bibtex]{biblatex} \usepackage{todonotes} \usepackage[american]{babel} \usepackage{fancyhdr} \hypersetup{colorlinks=false} \usetikzlibrary{calc, decorations.pathreplacing,shapes.misc} \usetikzlibrary{decorations.pathmorphing} \usepackage[left=1in,top=1in,right=1in]{geometry} \usepackage[capitalize]{cleveref} \newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}} \newcommand{\xxx}{T base with combinatorial potential data } \newcommand{\Xxx}{T base with combinatorial potential data } \newcommand{\xxxc}{combinatorial potential stratified space } \newcommand{\Xxxc}{combinatorial potential stratified space } \newcommand{\argument}{symplectic character } \newcommand{\arguments}{symplectic characters } \newcommand{\snip}[2]{#1} 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\Crefname{maincor}{Corollary}{Corollaries} \renewcommand*{\themainthm}{\Alph{mainthm}} \makeatletter \def\namedlabel#1#2{\begingroup \def\@currentlabel{#2}% \label{#1}\endgroup } \makeatother \fancypagestyle{firstpage}{% \fancyhf{} \renewcommand\headrulewidth{0pt} \fancyfoot[R]{Original text at \texttt{ \href{http://jeffhicks.net/snippets/index.php?tag=con_heegaardFloer}{snippets/con\_heegaardFloer}}} } \newcommand{\wt}{\widetilde} \newcommand{\wh}{\widehat} \newcommand{\wb}{\overline} \newcommand{\bb}{\mathbb} \newcommand{\scr}{\mathscr} \newcommand{\RR}{\mathbb R} \newcommand{\ZZ}{\mathbb Z} \newcommand{\CC}{\mathbb C} \newcommand{\TT}{\mathbb T} \newcommand{\NN}{\mathbb N} \newcommand{\PP}{\mathbb P} \newcommand{\LL}{\mathbb L} \newcommand{\II}{\mathbb I} \newcommand{\CP}{\mathbb{CP}} \newcommand{\del}{\nabla} \newcommand{\pp}{\mathbf{m}} \newcommand{\into}{\hookrightarrow} \newcommand{\emprod}{m} \newcommand{\tensor}{\otimes} \renewcommand{\Re}{\text{Re}} 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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{\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}, year={2012}, publisher={OUP} } @article{craw2007explicit, title={Explicit methods for derived categories of sheaves}, author={Craw, Alastair}, publisher={Citeseer} } @article{weinstein1971symplectic, title={Symplectic manifolds and their {L}agrangian submanifolds}, author={Weinstein, Alan}, journal={Advances in Mathematics}, volume={6}, number={3}, pages={329--346}, year={1971}, publisher={Academic Press} } @article{hanlon2022aspects, title={Aspects of functoriality in homological mirror symmetry for toric varieties}, author={Hanlon, A and Hicks, J}, journal={Advances in Mathematics}, volume={401}, pages={108317}, year={2022}, publisher={Elsevier} } @article{biran2013lagrangian, title={{L}agrangian cobordism. {I}}, author={Biran, Paul and Cornea, Octav}, journal={Journal of the American Mathematical Society}, volume={26}, number={2}, pages={295--340}, year={2013} } @article{tanaka2016fukaya, title={The Fukaya category pairs with Lagrangian cobordisms}, author={Tanaka, Hiro Lee}, journal={arXiv preprint arXiv:1607.04976}, year={2016} } @book{seidel2008fukaya, title={Fukaya categories and Picard-Lefschetz theory}, author={Seidel, Paul}, volume={10}, year={2008}, publisher={European Mathematical Society} } @article{seidel2003long, title={A long exact sequence for symplectic {F}loer cohomology}, author={Seidel, Paul}, journal={Topology}, volume={42}, pages={1003--1063}, year={2003} } @article{da2001lectures, title={Lectures on symplectic geometry}, author={da Silva, Ana Cannas}, journal={Lecture Notes in Mathematics}, volume={1764}, year={2001}, publisher={Springer} } @article{polterovich1991surgery, title={The surgery of {L}agrange submanifolds}, author={Polterovich, Leonid}, journal={Geometric \& Functional Analysis GAFA}, volume={1}, number={2}, pages={198--210}, year={1991}, publisher={Springer} } @misc{perutz2008handleslide, doi = {10.48550/ARXIV.0801.0564}, url = {https://arxiv.org/abs/0801.0564}, author = {Perutz, Timothy}, keywords = {Symplectic Geometry (math.SG), Geometric Topology (math.GT), FOS: Mathematics, FOS: Mathematics, 53D12; 53D40; 57M27; 32U40}, title = {Hamiltonian handleslides for {H}eegaard {F}loer homology}, publisher = {arXiv}, year = {2008}, copyright = {Assumed arXiv.org perpetual, non-exclusive license to distribute this article for submissions made before January 2004} } @incollection{audin1994symplectic, title={Symplectic rigidity: {L}agrangian submanifolds}, author={Audin, Mich{\`e}le and Lalonde, Fran{\c{c}}ois and Polterovich, Leonid}, booktitle={Holomorphic curves in symplectic geometry}, pages={271--321}, year={1994}, publisher={Springer} } @phdthesis{oancea2003suite, title={La suite spectrale de {L}eray-{S}erre en homologie de {F}loer des vari{\'e}t{\'e}s symplectiques compactes {\`a} bord de type contact}, author={Oancea, Alexandru}, year={2003}, school={Universit{\'e} Paris Sud-Paris XI} } @article{abouzaid2010geometric, title={A geometric criterion for generating the {F}ukaya category}, author={Abouzaid, Mohammed}, journal={Publications Math{\'e}matiques de l'IH{\'E}S}, volume={112}, pages={191--240}, year={2010} } @article{viterbo1999functors, title={Functors and computations in {F}loer homology with applications, I}, author={Viterbo, Claude}, journal={Geometric \& Functional Analysis GAFA}, volume={9}, number={5}, pages={985--1033}, year={1999}, publisher={Springer} } @misc{stacks-project, author = {The {Stacks project authors}}, title = {The Stacks project}, howpublished = {\url{https://stacks.math.columbia.edu}}, year = {2022}, } @article{wendlbeginner, title={A beginner’s overview of symplectic homology}, author={Wendl, Chris}, journal={Preprint. www. mathematik. hu-berlin. de/wendl/pub/SH. pdf} } @article{seidel2006biased, title={A biased view of symplectic cohomology}, author={Seidel, Paul}, journal={Current developments in mathematics}, volume={2006}, number={1}, pages={211--254}, year={2006}, publisher={International Press of Boston} } @article{arnol1980lagrange, title={{L}agrange and {L}egendre cobordisms. I}, author={Arnol'd, Vladimir Igorevich}, journal={Funktsional'nyi Analiz i ego Prilozheniya}, volume={14}, number={3}, pages={1--13}, year={1980}, publisher={Russian Academy of Sciences} } @article{fukaya2007lagrangian, title={{L}agrangian intersection {F}loer theory-anomaly and obstruction, chapter 10}, author={Fukaya, K and Oh, YG and Ohta, H and Ono, K}, journal={Preprint, can be found at http://www. math. kyoto-u. ac. jp/\~{} fukaya/Chapter10071117. pdf}, year={2007} } @article{biran2014lagrangian, title={Lagrangian cobordism and Fukaya categories}, author={Biran, Paul and Cornea, Octav}, journal={Geometric and functional analysis}, volume={24}, number={6}, pages={1731--1830}, year={2014}, publisher={Springer} } @article{bourgeois2009symplectic, title={Symplectic homology, autonomous {H}amiltonians, and {M}orse-{B}ott moduli spaces}, author={Bourgeois, Fr{\'e}d{\'e}ric and Oancea, Alexandru}, journal={Duke mathematical journal}, volume={146}, number={1}, pages={71--174}, year={2009}, publisher={Duke University Press} } @incollection{auroux2014beginner, title={A beginner’s introduction to {F}ukaya categories}, author={Auroux, Denis}, booktitle={Contact and symplectic topology}, pages={85--136}, year={2014}, publisher={Springer} } @article{singer1933three, title={Three-dimensional manifolds and their {H}eegaard diagrams}, author={Singer, James}, journal={Transactions of the American Mathematical Society}, volume={35}, number={1}, pages={88--111}, year={1933}, publisher={JSTOR} } @article{ozsvath2004holomorphic, title={Holomorphic disks and three-manifold invariants: properties and applications}, author={Ozsv{\'a}th, Peter and Szab{\'o}, Zolt{\'a}n}, journal={Annals of Mathematics}, pages={1159--1245}, year={2004}, publisher={JSTOR} } @article{ozsvath2004introduction, title={An introduction to {H}eegaard {F}loer homology}, author={Ozsv{\'a}th, Peter and Szab{\'o}, Zolt{\'a}n}, journal={{F}loer homology, gauge theory, and low-dimensional topology}, volume={5}, pages={3--27}, year={2004} } @article{fet1952variational, title={Variational problems on closed manifolds}, author={Fet, Abram Il'ich}, journal={Matematicheskii Sbornik}, volume={72}, number={2}, pages={271--316}, year={1952}, publisher={Russian Academy of Sciences, Steklov Mathematical Institute of Russian~…} }\end{filecontents} \addbibresource{references.bib}\begin{document} \title{construction of Heegaard Floer complex} \maketitle \thispagestyle{firstpage}Let $(\Sigma_g, \underline{\alpha}, \underline{\beta})$ be a Heegaard diagram. Let $X=\Sym^g(\Sigma_g)$. Consider the submanifolds of $X$ \begin{align*} L_{\underline \alpha}:=\{[(z_1, \ldots, z_g)]\in X\st z_i\in \alpha_i\} && L_{\underline \beta}:=\{[(z_1, \ldots, z_g)]\in X\st z_i\in \beta_i\} \end{align*} Since the $\alpha_i$ are disjoint from one another, this gives a smooth submanifold whose topology is $T^g$. Furthermore, since each of the $\alpha_i$ is a real subspace of $\Sigma_g$, the submanifold $L_\alpha$ is a real submanifold of $\Sigma_g$. A similar statement holds for $L_\beta$. \Cite{perutz2008handleslide} shows that $\Sym^g(\Sigma_g)$ can be equipped with a symplectic form which makes $L_{\underline \alpha},L_{\underline \beta}$ Lagrangian submanifolds. We wish to define the Heegaard-Floer cohomology as the Lagrangian intersection Floer cohomology of $L_{\underline \alpha},L_{\underline \beta}$. However, this is not an invariant of Heegaard diagrams --- note that isotopies of Heegaard diagrams give isotopies of Lagrangian submanifolds, while Lagrangian intersection Floer cohomology is only invariant under \emph{Hamiltonian isotopies} of Lagrangian submanifolds. We given an example exhibiting some of the problems with this preliminary approach. \begin{example} \label{exm:nonadmissibleHeegaardDiagram} We look at the example of $M=S^2\times S^1$. Observe that $S^2=D^2\cup_{S^1} D^2$, so we can write $M=D^2\times S^1 \cup_{\Sigma_1} D^2\times S^1$. The Heegaard diagram $(\Sigma_1, \alpha, \beta)$ consists of a torus with two meridional cycles. \begin{figure} \label{fig:nonadmissibleHeegaardDiagram} \centering \begin{tikzpicture} \begin{scope}[] \begin{scope}[] \draw[fill=gray!20] (-2,-1.5) .. controls (-3.5,-1.5) and (-4,0) .. (-4,1) .. controls (-4,2) and (-3.5,3.5) .. (-2,3.5) (-2,1.5) .. controls (-2.5,1.5) and (-2.5,0.5) .. (-2,0.5); \end{scope} \begin{scope}[xscale=-1, shift={(-0.5,0)}]] \draw[fill=gray!20] (-2,-1.5) .. controls (-3.5,-1.5) and (-4,0) .. (-4,1) .. controls (-4,2) and (-3.5,3.5) .. (-2,3.5) (-2,1.5) .. controls (-2.5,1.5) and (-2.5,0.5) .. (-2,0.5); \end{scope} \fill[gray!20] (-2,0.5) rectangle (2.5,-1.5); \fill[gray!20] (-2,3.5) rectangle (2.5,1.5); \end{scope} \begin{scope}[] \fill[orange!20] (1,-0.5) ellipse (0.5 and 1); \fill[orange!20] (-0.5,0.5) rectangle (1,-1.5); \fill[gray!20] (-0.5,-0.5) ellipse (0.5 and 1); \end{scope} \draw (-2,-1.5) -- (2.5,-1.5) (-2,0.5) -- (2.5,0.5) (-2,1.5) -- (2.5,1.5) (-2,3.5) -- (2.5,3.5); \begin{scope}[] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[red, shift={(1.5,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[blue, shift={(3,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-0.5,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(4.5,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(4.5,3)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(0,3)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \node[left, red] at (0,-0.5) {$\beta$}; \node[right, blue] at (1.5,-0.5) {$\alpha$}; \node at (0.5,0) {$\mathcal D$}; \node[circle, fill=black, scale=.2] at (0.5,2.5) {}; \node[right] at (0.5,2.5) {$z$}; \end{tikzpicture}\caption{A non-admissible Heegaard diagram} \end{figure} If the diagram is chosen so that $\alpha, \beta$ are disjoint, then the Lagrangian intersection Floer cohomology $\HF(\alpha, \beta)$ vanishes. \begin{figure} \label{fig:admissibleHeegaardDiagram} \centering \begin{tikzpicture} \begin{scope}[] \begin{scope}[] \draw[fill=gray!20] (-2,-1.5) .. controls (-3.5,-1.5) and (-4,0) .. (-4,1) .. controls (-4,2) and (-3.5,3.5) .. (-2,3.5) (-2,1.5) .. controls (-2.5,1.5) and (-2.5,0.5) .. (-2,0.5); \end{scope} \begin{scope}[xscale=-1, shift={(-0.5,0)}]] \draw[fill=gray!20] (-2,-1.5) .. controls (-3.5,-1.5) and (-4,0) .. (-4,1) .. controls (-4,2) and (-3.5,3.5) .. (-2,3.5) (-2,1.5) .. controls (-2.5,1.5) and (-2.5,0.5) .. (-2,0.5); \end{scope} \fill[gray!20] (-2,0.5) rectangle (2.5,-1.5); \fill[gray!20] (-2,3.5) rectangle (2.5,1.5); \end{scope} \begin{scope}[] \fill[orange!20] (1,-0.5) ellipse (0.5 and 1); \fill[orange!20] (-0.5,0.5) rectangle (1,-1.5); \fill[gray!20] (-0.5,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[red, shift={(1.5,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (0.5,-1.5); \draw[fill=gray!20] (-2,0.5) .. controls (-1.5,0.5) and (-1.5,0.5) .. (-1.5,0) .. controls (-1.5,-0.5) and (0.5,0) .. (0.5,-0.5) .. controls (0.5,-1) and (-1.5,-0.5) .. (-1.5,-1) .. controls (-1.5,-1.5) and (-1.5,-1.5) .. (-2,-1.5); \clip (-0.5,0) rectangle (0.5,-1); \draw[fill=orange!20] (-1.5,0) .. controls (-1.5,-0.5) and (0.5,0) .. (0.5,-0.5) .. controls (0.5,-1) and (-1.5,-0.5) .. (-1.5,-1); \clip (-1.5,0) .. controls (-1.5,-0.5) and (0.5,0) .. (0.5,-0.5) .. controls (0.5,-1) and (-1.5,-0.5) .. (-1.5,-1); \draw[fill=gray!20] (-0.5,-0.5) ellipse (0.5 and 1); \draw (-1.5,0) .. controls (-1.5,-0.5) and (0.5,0) .. (0.5,-0.5) .. controls (0.5,-1) and (-1.5,-0.5) .. (-1.5,-1); \end{scope} \begin{scope}[blue, shift={(3,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-0.5,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(4.5,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(4.5,3)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(0,3)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \node[right, red] at (2,-0.5) {$\beta'$}; \node[left, blue] at (1.5,-0.5) {$\alpha$}; \node at (0.5,0) {$\mathcal D$}; \node[circle, fill=black, scale=.2] at (0.5,2.5) {}; \node[right] at (0.5,2.5) {$z$}; \draw (-2,-1.5) -- (2.5,-1.5) (-2,0.5) -- (2.5,0.5) (-2,1.5) -- (2.5,1.5) (-2,3.5) -- (2.5,3.5); \end{tikzpicture}\caption{An admissible Heegaard diagram} \end{figure} However, if the diagram is chosen so that $\alpha, \beta'$ intersect transversely, the Lagrangian intersection Floer cohomology (with $\ZZ/2\ZZ$ coefficients) is $\ZZ/2\ZZ\oplus \ZZ/2\ZZ$. Note that $\beta'$ can be chosen so that it is Hamiltonian isotopic to $\beta$. The discrepancy between these two answers comes from the non-convergence of the homotopy between the composition of continuation maps \[f\circ g:\CF(\alpha, \beta')\to \CF(\alpha, \beta) \to \CF(\alpha, \beta')\] \[\id: \CF(\alpha, \beta')\to \CF(\alpha, \beta')\] over $\ZZ/2\ZZ$ coefficients. The presence of an annulus between $\alpha, \beta$ is the culprit for the non-convergence. One can make the quantities converge by using \underline{\href{https://jeffhicks.net/snippets/index.php?tag=def:novikovRing}{ Novikov coefficients}} instead of $\ZZ/2\ZZ$ coefficients. In that setting, the differential in \cref{fig:nonadmissibleHeegaardDiagram} will be exact unless the areas of the two strips agree --- that is, the Lagrangians $\alpha, \beta$ are Hamiltonian isotopic. \end{example} We desire the best of both worlds: invariance under Lagrangian isotopies, and convergence. In \cref{exm:nonadmissibleHeegaardDiagram} this can be achieved by only looking at strips which avoid the marked point $z$. When this marked point is chosen well, we will obtain a criterion (admissible Heegaard diagrams) which precludes the existence of annuli disjoint from the marked point $z$. In general, the admissibility criterion will limit us to configurations of cycles for which the number of holomorphic strips contributing to the Floer differential is finite. The marked point is introduced by slightly generalizing our definition of a Heegaard diagram. \begin{definition} \label{def:pointedHeegaardDiagram} A \emph{pointed Heegaard diagram} $(\Sigma_g, \underline \alpha, \underline \beta, z)$ is a Heegaard diagram $(\Sigma_g, \underline \alpha, \underline \beta)$ along with a choice of point $z$ disjoint from the cycles $\underline \alpha, \underline \beta$. \end{definition} As a vector space, the Heegaard Floer cohomology of the pointed Heegaard diagram $(\Sigma,\underline{\alpha}, \underline{\beta},z)$ is \[\HeF(\Sigma, \underline{\alpha}, \underline{\beta},z):=\bigoplus_{x\in L_{\underline \alpha}\cap L_{\underline{\beta}}} \ZZ/2\ZZ\langle x\rangle.\] The differential is defined in a manner similar to the Lagrangian intersection Floer cohomology, but we will need to overcome the following difficulties. \begin{itemize} \item Gromov-Compactness: Observe that this is defined over $\ZZ/2\ZZ$ coefficients. In general, when we look at the strips connecting two points $x, y$, we are only guaranteed compactness of the moduli space of pseudoholomorphic strips which have bounded energy. To get around this problem --- when strips from $x$ to $y$ have possibly unbounded energy --- we would employ Novikov coefficients in Lagrangian intersection Floer cohomology. Here, we do not use such a coefficient ring. \item If we treat $X$ as a symplectic manifold, the Lagrangian submanifolds $L_{\underline \alpha}, L_{\underline \beta}$ are not tautologically unobstructed (i.e. $\omega(\pi_2(X, L))\neq 0$). We therefore must rule out disk and sphere bubbling to show that the differential squares to zero. \end{itemize} The choice of base point defines a subset $Y_z\subset X$ given by \[ Y_z:=\{[(z, z_2, \ldots, z_g)] \st z_i\in \Sigma_g\} \subset X\] which is disjoint from $L_{\underline \alpha}\cup L_{\underline \beta}$. Both issues above can be avoided by only looking at strips which are disjoint from $Y_z$. For $x, y\in L_{\underline \alpha}\cap L_{\underline \beta}$ let $\mathcal M(x, y)$ denote the moduli space of holomorphic strips with boundary on $L_{\underline \alpha}\cup L_{\underline \beta}$, and ends limiting to $x$ and $y$. Let $\mathcal M(x, y)_{z=0}^0$ be the zero-dimensional component of $\mathcal M(x, y)$ consisting of strips which are disjoint from $Y_z$. The differential of Heegaard Floer cohomology is defined by the structure coefficients \[\langle d_zx , y \rangle = \#\mathcal M(x, y)_{z=0}^0.\] The following lemmas rule out the presence of disk and sphere bubbling. \begin{lemma} \label{lem:heegaardNoSpheres} Suppose that $g>2$. Then $\pi_2(\Sym^g(\Sigma_g))=\ZZ$. The generator of this group intersects $Y_z$. \end{lemma} \begin{lemma} \label{lem:heegaardNoDisks} Suppose that $u: D^2\to L_\alpha$ is a holomorphic disk. Then the image of $u$ intersects $Y_z$. \end{lemma} Provided that we can show that $(d_z)^2=0$, the Heegaard-Floer cohomology is defined to be the homology of the chain complex \[\HHeF(M):= H^\bullet(\HeF(\Sigma, \underline \alpha, \underline \beta ,z), d_z).\] As a result of these two lemmas, our restriction to counting disks which \emph{do not} pass through $Y_z$ gives us moduli spaces whose compactifications, if defined, would be free from disk and sphere bubbling. However, it remains to show that we have Gromov compactness. \printbibliography \end{document}