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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?

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    Just to amplify the point of Henry Horton's reply below: a characteristic element is something defined for a bilinear form (http://ncatlab.org/nlab/show/characteristic+element+of+a+bilinear+form). So for every choice of bilinear form on cohomology that you come up with, there is a concept of characteristic cohomology class. The intersection in cohomology-relative-boundary that Henry Horton gives is the most obvious choice, but depending on your applications there might be a different one.2014-06-02

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