Subsection 1.4. (0PE8) · ScalingStacks

1.4. Heegaard Floer homology

Heegaard Floer homology, defined by Ozsváth–Szabó [OsSz1, OsSz2, OsSz3], is a set of invariants for 3- and 4-dimensional manifolds. For a 3-manifold $Y$ , the Heegaard Floer invariant of $Y$ (an abelian group) is defined by choosing a Heegaard decomposition of $Y$ as two handlebodies glued along a genus- $g$ surface $\mathcal{H}$ , then computing a Lagrangian intersection Floer homology group between two Lagrangian submanifolds in $\mathrm{Sym}^{g}(\mathcal{H})$ induced by the two handlebodies.

In bordered Heegaard Floer homology [LiOzTh1, Za], there are also extended Heegaard Floer invariants for 2d surfaces and 3d cobordisms. Let $F$ be the data of a surface $(\Sigma,M)$ with a set of points $M\subset\partial\Sigma$ as above, equipped with a choice of arc decomposition. To such a surface, bordered Heegaard Floer associates a differential algebra $A(F)$ . Auroux [Au2] has shown that the algebra $A(F)$ is the endomorphism algebra of a generating object of $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ determined by the arc decomposition.

Our constructions are based on the combinatorics of $A(F)$ and we do not work directly with $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ . Given a component of $\partial\Sigma\setminus M$ , we define a differential bimodule $E$ over $A(F)$ , and a bimodule endomorphism $\tau$ of $E\otimes_{A(F)}E$ , that yield a higher representation of $\mathfrak{gl}(1|1)^{+}$ .

Theorem 1.3.1 follows now from the following result.

0P4J

Theorem 1.4.1. If $F_{1}$ and $F_{2}$ are surfaces with arc decompositions glued as in Theorem 1.3.1 to form $F$ , then

A(F)\cong A(F_{1}){\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}A(F_{2})

as higher representations of $\mathfrak{gl}(1|1)^{+}$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2