Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated ["A tour of bordered Floer theory"](http://arxiv.org/abs/1107.5621). To set the stage let me give two quotes from this paper.



> 
> 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.
> 
> 
> [...]
> 
> 
> 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.
> 
> 
> 


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:


**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 $\widehat{HF}$?**


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.


Any information is very much appreciated.



Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?
A biased answer, based on Auroux's work <http://arxiv.org/abs/1003.2962>. 


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: <http://arxiv.org/abs/1102.3160>.


**A general set-up:** 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. 


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 $z\in F$. 


**Forms of Floer cohomology**: There are various forms of Floer cohomology one can consider.


(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.


(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 $D$. 


(iii) One can consider non-compact Lagrangians $L\_i\subset M$ which go to infinity nicely (following the Liouville flow). These have *wrapped* 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 $X$.


**Invariants for 3-manifolds with boundary.** 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 $Y$. 


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. 


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. 


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.


