Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T04:30:37Z http://mathoverflow.net/feeds/question/71794 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71794/where-are-and-infty-in-bordered-heegaard-floer-theory Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory? Stefan Behrens 2011-08-01T10:56:21Z 2011-08-01T18:48:44Z <p>Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated <a href="http://arxiv.org/abs/1107.5621" rel="nofollow">"A tour of bordered Floer theory"</a>. To set the stage let me give two quotes from this paper.</p> <blockquote> <p>Heegaard Floer homology has several variants; the technically simplest is $\widehat{HF}$, which is sufficient for most of the 3-dimensional applications discussed above. Bordered Heegaard Floer homology, the focus of this paper, is an extension of $\widehat{HF}$ to 3-manifolds with boundary.</p> <p>[...]</p> <p>the Heegaard Floer package contains enough information to detect exotic smooth structures on 4-manifolds. For closed 4-manifolds, this information is contained in $HF^+$ and $HF^-$; the weaker invariant $\widehat{HF}$ is not useful for distinguishing smooth structures on closed 4-manifolds.</p> </blockquote> <p>Since I am mainly interested in closed 4-manifolds, I have not paid too much attention to the developments in bordered Heegaard-Floer thoery. But right from the beginning I have wondered why only $\widehat{HF}$ appears in the bordered context. So my question is:</p> <p><strong>Why are there no $^+$, $^-$ or $^\infty$ flavors of bordered Heegaard-Floer theory? Are the reasons of technical nature or is there an explanation that the theory cannot give more than $\widehat{HF}$?</strong></p> <p>I assume there are issues with the moduli spaces of holomorphic curves that would be relevant to defining bordered versions of the other flavors of Heegaard-Floer theory, but I am neither enough of an expert on holomorphic curves to immediately see the problems nor could I find anything in the literature that pins down the problems.</p> <p>Any information is very much appreciated.</p> http://mathoverflow.net/questions/71794/where-are-and-infty-in-bordered-heegaard-floer-theory/71819#71819 Answer by Tim Perutz for Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory? Tim Perutz 2011-08-01T18:42:52Z 2011-08-01T18:48:44Z <p>A biased answer, based on Auroux's work <a href="http://arxiv.org/abs/1003.2962" rel="nofollow">http://arxiv.org/abs/1003.2962</a>. </p> <p>Auroux makes a connection between bordered Floer theory and an alternative approach, due to Lekili and myself, which is (still) under development, but which should include the $\pm$ and $\infty$ versions. We do have a preliminary paper out: <a href="http://arxiv.org/abs/1102.3160" rel="nofollow">http://arxiv.org/abs/1102.3160</a>.</p> <p><b>A general set-up:</b> Say you have a compact symplectic manifold $(X,\omega_X)$; and a codim 2 symplectic submanifold $D$, whose complement $M$ is exact: ${\omega_X}|_M=d\theta$, say. </p> <p>Key example: $X=Sym^g(F)$, where $F$ is a compact surface of genus $g$, and $\omega_X$ a suitable Kaehler form; $M=Sym^g(F-z)$, where $z\in F$. </p> <p><b>Forms of Floer cohomology</b>: There are various forms of Floer cohomology one can consider.</p> <p>(i) As in $\widehat{HF}$ Heegaard theory, one can consider $HF^\ast_M(L_0,L_1)$, the Floer cohomology in $M$ of a pair of (exact) compact Lagrangian submanifolds of $M$. When $L_0$ and $L_1$ are spin, this can be defined as a $\mathbb{Z}$-module.</p> <p>(ii) As in $HF^-$ Heegaard theory, one can consider the filtered Floer cohomology $HF^\ast_{X,D}(L_0,L_1)$ of a pair of compact Lagrangians $L_i\subset M$ as before. The coefficients are in $\mathbb{Z}[[U]]$. The differential counts holomorphic bigons in $X$, weighted by $U^n$ where $n$ is intersection number with $D$. </p> <p>(iii) One can consider non-compact Lagrangians $L_i\subset M$ which go to infinity nicely (following the Liouville flow). These have <i>wrapped</i> Floer cohomology $HW^\ast(L_0,L_1)$, as well as "partially wrapped" variants. Wrapping concerns how one chooses to perturb $L_0$ at infinity. This version takes place in $M$, and (AFAIK) can't naturally be extended to something that takes place in $X$.</p> <p><b>Invariants for 3-manifolds with boundary.</b> A basic idea is that a 3-manifold $Y$ bounding $F$ should define a (generalized) Lagrangian submanifold $L_Y$ where $X=Sym^{g(F)}F$, as in the "key example" above. The collection of filtered Floer modules $HF^*_{X,D}(\Lambda, L_Y)$ as $\Lambda$ ranges over Lagrangian submanifolds of $M$ (more precisely, the module, over the compact filtered Fukaya category of $(X,D)$, defined by $L_Y$) should be an invariant of $Y$. </p> <p>If one is interested only in the simpler groups $HF^*_M(\Lambda,L_Y)$, one can (in principle) determine these by looking at the finite collection of (partially wrapped) groups $HW^*(W_i,L_Y)$, where $W_i$ ranges over the thimbles for a certain Lefschetz fibration $M\to \mathbb{C}$. That is, one thinks of $L_Y$ as defining a module over the algebra $A_{LOT}$ formed by the sum of groups $HW^*(W_i,W_j)$. This follows from a deep theorem of Seidel about generating Fukaya categories by thimbles, adapted by Auroux. </p> <p>The algebra $A_{LOT}$ is (part of) what Lipshitz-Ozsvath-Thurston assign to a parametrized surface, and the module is what they call $\widehat{CFA}(Y)$. They arrived at it by a quite different route. They don't bother with constructing $L_Y$ itself, only the module it defines. Because they use the groups of type (iii) to form their algebra, their approach only works in $M$, not $X$. For that reason, they only capture the hat-theory. </p> <p>The great advantage of LOT's approach is its finiteness and computability. Lekili and I do construct $L_Y$. We can guess at finite collections of "test Lagrangians" sufficient to compute the module $HF^*_{X,D}(\cdot, L_Y)$, but have not yet proved that they are sufficient.</p>