A note on the Lagrangian cobordism group of Weinstein sectors

TL;DR

This note proves that the Lagrangian cobordism group $\Omega(X,\mathfrak f)$ of a stopped Weinstein sector is isomorphic to its middle-dimensional relative cohomology $H_n(X,\partial_\infty X\setminus \mathfrak f)$. The isomorphism $i:\Omega(X,\mathfrak f)\to H_n(X,\partial_\infty X\setminus \mathfrak f)$ is constructed by projecting cobordisms to $X$, and when $X$ is Weinstein with a Weinstein hypersurface stop, cocores generate $\Omega(X,\mathfrak f)$, yielding the isomorphism and a cocore-based description of $i$. The paper also describes a geometric Viterbo restriction for cobordism groups and places $i$ into a commutative diagram involving the partially wrapped Fukaya category $\mathcal{W}(X,\mathfrak f)$, its Grothendieck group $K_0$, the Dennis trace $\mathcal{T}:K_0\to HH_0$, and the open-closed map, connecting to Lazarev's construction. It discusses how these relations behave under restriction to Weinstein subdomains and notes phenomena such as non-injectivity in flexible cases.

Abstract

The aim of this note is to show that the Lagrangian cobordism group of a Weinstein sector is isomorphic to its middle-dimensional singular cohomology. As an application, a geometric description of Viterbo restriction for cobordism groups is obtained.

Figures (2)

• Figure 1: A Lagrangian cobordism $V$ can be visualized by its projection to $\mathbb C$. As $V$ is conical outside a compact set no information is lost when only considering $V\cap \left (K\times X_0\right)$. In green, the projection of $V\cap \left (\mathbb C\times X_0\right)$ to $\mathbb C$ is drawn. The Lagrangian cobordism $V$ has 4 ends $L_0,\ldots, L_3$ parametrized by the rays $\gamma_0, \ldots, \gamma_3$ in $\mathbb C$ in this example. The dotted circle $\partial K$ is the boundary at infinity of $\mathbb C$ stopped at 4 points $\mathfrak f_3\subset \partial K$.
• Figure 2: An $(n-1)$-handle $T^*B^{n-1}\times B^2$ associated to a critical point $x$ projected to $B^2$. In well-chosen coordinates, positive gradient flow lines from the critical point $x$ of index $(n-1)$ to critical points of index $n$ are constant in the first coordinate and point radially outward. Each such flow line gives rise to a linking disk $D_j$ orthogonal to the flow line.

Theorems & Definitions (8)

• Theorem 1
• Corollary 2
• Corollary 3
• Remark 4
• Remark 5
• proof : Proof of Theorem \ref{['thm1']}
• proof : Proof of Corollary \ref{['cor1']}
• proof : Proof of Corollary \ref{['cor2']}