Let π: E → B be a fiber bundle with fiber having the mod 2 cohomology algebra of a real or a complex projective space and let π’: E’ → B be a vector bundle such that ℤ₂ acts fiber preserving and freely on E and E’-0, where 0 stands for the zero section of the bundle π’: E’ → B. For a fiber preserving ℤ₂-equivariant map f: E → E’, we estimate the cohomological dimension of the zero set $Z_f=x\in E\left|f\right(x)=0$. As an application, we also estimate the cohomological dimension of the ℤ₂-coincidence set $A_f=x\in E\left|f\right(x)=f(T\left(x\right))$ of a fiber preserving...