de Rham cohomology groups are mathematical structures used in the field of differential geometry and algebraic topology. They provide a way to study the topology of smooth manifolds by analyzing differential forms, which are mathematical objects that can be integrated over manifolds. The groups capture information about the manifold's shape and features, such as holes and voids. These cohomology groups are defined using the concept of differential forms and the exterior derivative, which measures how these forms change. The de Rham cohomology groups are denoted as H^k_dR(M) , where M is the manifold and k indicates the degree of the forms being considered.