# Stiefel Whitney class and intersection form

Why is the second Stiefel-Whitney Class of a closed oriented 4-manfifold, $$M^{4}$$, a characteristic element for its intersection form?

• Precisely, why must the following identity hold for closed oriented 4-manifolds, $$\langle w_2(M),\alpha\rangle \equiv \langle \alpha,\alpha\rangle (mod\ 2)\text{ for all } \alpha \in H^{2}(M,\mathbb{Z/2})$$ where $$\langle,\rangle$$ denotes the intersection form.

For a $$\mathbb{Z}$$-module (aka abelian group) $$V$$ with a symmetric, unimodular, bilinear form $$\mu\colon V\otimes V \to V$$, a characteristic element is an element $$c\in V$$ such that
$$\forall v\in V\ \mu(v,v) \equiv \mu(c,v)\mod 2$$
This means you don't want an element with $$\mathbb{Z}/2$$ coefficients, you want an integral class. The correct statement is that
For an oriented closed $$4$$-manifold $$M$$, an element in $$H^2(M;\mathbb{Z})$$ is characteristic iff it is an integral lift of $$w_2(M)$$.
It's not hard to see this algebraically using Wu classes (for a definition and important properties see for example Manifold Atlas or Milnor-Stasheff). In general, for a closed, oriented $$4n$$-manifold, a characteristic element of its intersection form is an integral lift of the $$2n$$-th Wu class $$v_{2n}$$, essentially by definition.
But Wu's formula $$Sq(V) = W$$ tells us that for a closed, oriented manifold we have $$v_2 = w_2$$, so if $$M$$ if $$4$$-dimensional then $$\alpha \in H^2(M;\mathbb{Z})$$ is characteristic iff it is an integral lift of $$v_2$$ iff it is an integral lift of $$w_2$$.