My doubt is simple:
How can I prove that in dimension 4, the Euler Characteristic of a Riemannian Manifold is given by $$\chi(M^4)=\frac{1}{8\pi^2}\int\limits_{M^4}\left(|W|^2+\frac{R^2}{24}-\frac{1}{2}|\stackrel{\circ}{Ric}|^2\right)\,d\mu$$
I know that the Rimannian curvature tensor can decompose into irreducilble components: $$Rm=\frac{R}{24}g\circ g+\stackrel{\circ}{Ric}\circ g+W$$
where $``\circ"$ is the classical Kulkarni-Nomizu product
Someone can help me with this? thanks