I have a question about Characteristic cohomology class of 4-manifolds. $X^4$ denotes the compact 4-manifold with boundary. I'm mainly concerned with $\partial X$ is nonempty.
If $X^4$ is closed, we define $w\in H^2(X;\mathbb{Z})$ is characteristic cohomology class if $Q_X(w,x)\equiv Q_X(x,x)$ modulo 2 for all $x\in H^2(X;\mathbb{Z})$, where $Q_X\colon H^2(X;\mathbb{Z})\times H^2(X;\mathbb{Z})\to \mathbb{Z}$.
What is the analogue definition of characteristic cohomology class for 4-manifold with boundary?