There are a good number of quantum topology papers in which a TQFT-like set-up is constructed as a functor to the category of vector spaces from some category of cobordisms which satisfy some "Lagrangian" condition. For example, Cheptea-Habiro-Massuyeau consider a category whose morphisms are cobordisms $M$ between closed oriented surfaces $F_+$ and $F_-$, where we choose Lagrangian subgroups $A_{\pm}$ of $H_1(F_\pm)$ correspondingly, and where we require that $H_1(M)=m_-(A_-)+m_+H_1(F_+)$ and that $m_+(A_+)\subseteq m_-(A_-)$ in $H_1(M)$ (the $m_\pm$ are inclusion maps). Similar conditions are imposed in many other papers.

I have never understood why such conditions are imposed. One half-thought I have is that it is related to Wall's result that the kernel of the inclusion of $H_1(\partial M)$ in $H_1(M)$ is a Lagrangian subgroup of $H_1(\partial M)$. Another half-idea is that it might be some weak 2-framing condition for the cobordism or something.

What is the conceptual explanation for this Lagrangian condition and its variants? Does it have anything to do with framing? (or orientation?) Why do "symplectic" and "Lagrangian" have anything to do with TQFT? (is it all just Wall's result in some guise?)

Every time I see a Lagrangian condition I feel very stupid for not knowing what it's doing there.