Ordinary RO(G)RO\left( G \right)RO(G)-graded cohomology (original) (raw)

The paper introduces a framework for ordinary RO(G)RO(G)RO(G)-graded cohomology, extending traditional singular cohomology to G-spaces with a focus on compact Lie groups. It critiques existing methods like Borel and Bredon cohomology for insufficiently addressing obstruction theory and proposes a refined approach that incorporates the interplay among the Burnside ring, the representation ring, and G-homotopy theory. New constructions of cohomological and homological theories are presented, along with various algebraic tools and results relevant to these theories.