The Level of a Topological Space

The level of a Z/2Z-space X is defined to be the minimum n such that there is a Z/2Z equivariant map from X to S^n-1 (with the antipodal action). The level of a unital ring is defined to be the minimum n such that -1=e₁²+e₂²+...+e_n² where e_n are elements of the ring. We explore an intimate relationship between the level of a Z/2Z space and the level of commutative R algebras. We also compute the level of even dimensional real projective spaces, spheres, and mention results on the computations of odd dimensional projective spaces.