Brodered Floer Homology

Bordered Floer Homology

Classical and Bordered Heegaard Floer Homology

For closed, oriented, and connected 3-manifold $Y$, we have finitely generated modules $\widehat{HF},H{F}^{+},H{F}^{-}$.
The cobordism $W$ between two 3-manifolds $Y$ and $Y^{\prime }$ gives a morphism

$${\widehat{HF}}^{\pm }(Y_1)\to {\widehat{HF}}^{\pm }(Y_2).$$

Through this cobordism, we get a structure as a $(3+1)$-dimensional TQFT.

• In dimension 4: we get a number.
• In dimension 3: we get a vector space or module.
• A 4-dimensional cobordism gives a map.

Bordered Setting

In the bordered setting, we can think in $(2+1+1)$-dimensional TQFT.

• A parametrized surface $\Sigma$ gives an algebra $A(\Sigma )$.
• A cobordism between $\Sigma$ and ${\Sigma }^{\prime }$ gives a bimodule over $A({\Sigma }_1)$ and $A({\Sigma }^{\prime })$.
• A 3-manifold $Y$ with $\Sigma$ boundary gives a module over $A(\Sigma )$, such as an $A_{\infty }$ or differential graded module.

Theorem (Pairing Theorem).

$$\widehat{HF}(Y_1{\cup }_{\Sigma }{Y}_2)=H_{\ast }(\text{tensor product of the modules for }{Y}_1\&Y_2).$$

More generally, invariants from more than two surfaces and cobordisms form a tensor product of bimodules.

Bordered Floer homology is defined using bordered Heegaard diagrams. For the special case of torus boundary, the algebra is easier.
This case connects to knot Floer homology and to immersed curves.