Separating components of real algebraic varieties by cohomological invariants

Let $X$ be a smooth algebraic variety over $\mathbb{R}$. We study the separation of the connected components of $X(\mathbb{R})$ by equivariant cohomology, unramified cohomology, and signatures of unramified quadratic forms. Our results highlight how the geometry of $X$ influences these questions. We apply them to cubic hypersurfaces, complete intersections of two quadrics, abelian varieties, and Enriques surfaces. We also obtain new bounds on the exponent of the cokernel of higher signature morphisms in $\mathbf{I}$-cohomology and investigate the optimality of these bounds for smooth projective varieties.