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Equivariant semiprojectivity

We define equivariant semiprojectivity for C*-algebras equipped with actions of compact groups. We prove that the following examples are equivariantly semiprojective: arbitrary finite dimensional C*-algebras with arbitrary actions of compact groups; the Cuntz algebras ${\mathcal{O}}_d$ and extended Cuntz algebras $E_d,$ for finite $d,$ with quasifree actions of compact groups; the Cuntz algebra ${\mathcal{O}}_{\infty}$ with any quasifree action of a finite group. For actions of finite groups, we prove that equivariant semiprojectivity is equivalent to a form of equivariant stability of generators and relations. We also prove that if $G$ is finite, then $C^* (G)$ is graded semiprojective.