Lagrangian cobordisms and K-theory of symplectic bielliptic surfaces

Abstract

We consider a family of closed symplectic manifolds 4-manifolds which we call symplectic bielliptic surfaces and study its Lagrangian cobordism group of weakly-exact Lagrangian G-branes (that is, Lagrangians equipped with a grading, a Pin structure and a G-local system); relations come from Lagrangian cobordisms satisfying a tautologically unobstructedness-type condition, also equipped with G-brane structures. Our first theorem computes its subgroup generated by tropical Lagrangians. When G is the unitary group of the Novikov field, we use homological mirror symmetry to compute the Grothendieck group of the Fukaya category and show it agrees with our computation for the cobordism group. This leads us to conjecture that tropical Lagrangians generate the whole cobordism group.

Figures (3)

• Figure 1: The shaded region (including the two horizontal lines) depicts a typical projection of a two-ended Lagrangian cobordism $V \subset X \times \C$ to $\C$. We have included (in dashed lines) an example of a compact region $K \subset\C$ outside which $V$ is product type: in $\C \setminus K$ the projection looks like two straight lines, and living over them in $X \times \C$ we have the cylidrical Lagrangians $L_- \times \R_{<-a_-}$ and $L_+ \times \R_{>a_+}$.
• Figure 2: Schematic picture to show that $2\mathbb F_1 \sim 0$. The left picture shows the (projection to $K$ of the) class $\mathbb F_1 \in \mathop{\mathrm{\mathrm{Cob}}}\nolimits_{fib}(\mathcal{K})_{\hom}$ as well as its preimage $p^{-1}(\mathbb F_1) \in \mathop{\mathrm{\mathrm{Cob}}}\nolimits_{fib}(T^2 \times T^2)_{\hom}$. The right picture shows the decomposition of $p^{-1}(\mathbb F_1)$ in the product $T^2 \times T^2$; namely, the top part shows the class $(F_\theta^1 - F_{\theta + 1/2}^1)\times F^2_t$, whereas the bottom part shows $(F_\theta^1 - F_{\theta + 1/2}^1)\times F^2_{1-t}$. Note that while the torus in the left picture represents a tropical affine base, the $2$-tori in the right picture represent symplectic tori living above the tropical base (that is, the symplectic manifold $T^2 \times T^2$ in the right picture is the total space of the Lagrangian torus fibration over the tropical $2$-torus $T^2$ in the left picture).
• Figure 3: Decomposition of the Klein bottle to run Mayer-Vietoris. Note that both $V_1$ and $V_2$ retract to a circle, whereas $V_1 \cap V_2$ retracts to the disjoint union of two circles

Theorems & Definitions (29)

• Theorem A: $=$\ref{['th:cobK']}
• Theorem B: $=$\ref{['th:isoK']}
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