Integral Hodge classes on fourfolds fiberd by quadric bundles

As we think that our geometric approach might still be interesting, we make it available. There will be a future work related to this paper. Some typos have been fixedInternational audienceWe discuss the space of sections and certain bisections on a quadric surfaces bundle $X$ over a smooth curve. The Abel-Jacobi from these spaces to the intermediate Jacobian will be shown to be dominant with rationally connected fibers. As an application, we prove that the integral Hodge conjecture holds for degree four integral Hodge classes of fourfolds fibered by quadric bundles over a smooth curve. This gives an alternative proof of a result of Colliot-Th{\'e}l{\`e}ne and Voisin