= Degree constraint from intersection forms
{title2=$P^{\mathsf T}Q_MP=(\deg f)Q_N$}
For a map $f:M\to N$ of connected closed oriented four-manifolds, naturality of the <cup product> gives
$$
Q_M(f^*u,f^*v)=(\deg f)Q_N(u,v).
$$
When $\deg f\ne0$, nondegeneracy makes $f^*:H^2(N;\mathbb Q)\to H^2(M;\mathbb Q)$ injective. Consequently the target <intersection form>, multiplied by the <mapping degree>, must embed into the source form. For equal second <Betti numbers>, a matrix $P$ for pullback satisfies $P^{\mathsf T}Q_MP=(\deg f)Q_N$; taking determinants often rules out nonzero degrees. In particular, a two-dimensional indefinite source form cannot realize a nonzero multiple of a two-dimensional definite target form.
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