A remark about weak fillings

Abstract

Let $L$ be a closed manifold of dimension $n\ge 2$ which admits a totally real embedding into ${\mathbb{C}}^n$. Let $S{T}^{*}L$ be the space of rays of the cotangent bundle $T^{*}L$ of $L$, and let $D{T}^{*}L$ be the unit disk bundle of $T^{*}L$ defined by any Riemannian metric on $L$. We observe that $S{T}^{*}L$ endowed with its standard contact structure admits weak symplectic fillings $W$ which are diffeomorphic to $D{T}^{*}L$ and for which any closed Lagrangian submanifold $N\subset W$ has the property that the map $H_1(N,\mathbb{R})\to H_1(W,\mathbb{R})$ has a nontrivial kernel. This relies on a variation on a theorem by Laudenbach and Sikorav.