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Is the equivariant cohomology an equivariant cohomology?

A reference is "Equivariant homotopy and cohomology theory" by J. P. May et al. … Instead of defining $H^*_G$ let me only say that there is a Hurewicz homomorphism $\omega^*_G(X)\to H^*_G(X)$, and the equivariant stable cohomotopy $\omega_G^{V-W}(X)$ can be defined as the equivariant
Sergey Melikhov's user avatar