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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 (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. –  Urs Schreiber Jun 2 at 16:11

1 Answer 1

The concept of a characteristic element is purely algebraic. Let $G$ be a finitely generated free abelian group and let $$Q: G \times G \longrightarrow \Bbb Z$$ be a symmetric bilinear form on $A$. Then $x \in G$ is a characteristic element of $G$ for $Q$ if $$Q(x, \alpha) \equiv Q(\alpha, \alpha) \pmod 2 \text{ for all } \alpha \in G.$$ So to define characteristic cohomology classes for a $4$-manifold with boundary, we just need to define the notion of an intersection form for a $4$-manifold with boundary.

If $X$ is a compact, oriented $4$-manifold with boundary $\partial X$, then it has an orientation class $[X, \partial X] \in H_4(X, \partial X; \Bbb Z)$. Then we define the intersection form $Q_X$ of $X$ by $$Q_X: H^2(X, \partial X; \Bbb Z) \times H^2(X, \partial X; \Bbb Z),$$ $$(\alpha, \beta) \mapsto \langle \alpha \smile \beta, [X, \partial X] \rangle.$$ Then a characteristic cohomology class of $X$ is a class $x \in H^2(X, \partial X; \Bbb Z)$ such that $$Q_X(x, \alpha) \equiv Q_X(\alpha, \alpha) \pmod 2 \text{ for all } \alpha \in H^2(X, \partial X; \Bbb Z).$$

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