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David White
David White
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There is a subject of equivariant stable homotopy theory. You are referring to $bu$ as what is called a naive $G$-spectrum (a spectrum acted on by a group), and your notation $E^G$ is confusing the categorical fixed point spectrum with the homotopical fixed point spectrum. The difference between the two is, for example, the subject of the Segal conjecture. There is a homotopy fixed point spectral sequence for homotopical fixed point spectra. There is a huge literature on equivariant stable homotopy theory (which was a key input to solution of the Kervaire invariant problem by Hill, HopkindsHopkins, and Ravenel). One expository introduction is: 

Greenlees and May. Equivariant stable homotopy theoryEquivariant stable homotopy theory. Handbook of Algebraic Topology, edited by I.M. James, pp. 279-325. 1995. But

But a lot has been done since then. Naive $G$-spectra are not so interesting. Genuine $G$-spectra have $RO(G)$-graded homology and cohomology.

There is a subject of equivariant stable homotopy theory. You are referring to $bu$ as what is called a naive $G$-spectrum (a spectrum acted on by a group), and your notation $E^G$ is confusing the categorical fixed point spectrum with the homotopical fixed point spectrum. The difference between the two is, for example, the subject of the Segal conjecture. There is a homotopy fixed point spectral sequence for homotopical fixed point spectra. There is a huge literature on equivariant stable homotopy theory (which was a key input to solution of the Kervaire invariant problem by Hill, Hopkinds, and Ravenel). One expository introduction is: Greenlees and May. Equivariant stable homotopy theory. Handbook of Algebraic Topology, edited by I.M. James, pp. 279-325. 1995. But a lot has been done since then. Naive $G$-spectra are not so interesting. Genuine $G$-spectra have $RO(G)$-graded homology and cohomology.

There is a subject of equivariant stable homotopy theory. You are referring to $bu$ as what is called a naive $G$-spectrum (a spectrum acted on by a group), and your notation $E^G$ is confusing the categorical fixed point spectrum with the homotopical fixed point spectrum. The difference between the two is, for example, the subject of the Segal conjecture. There is a homotopy fixed point spectral sequence for homotopical fixed point spectra. There is a huge literature on equivariant stable homotopy theory (which was a key input to solution of the Kervaire invariant problem by Hill, Hopkins, and Ravenel). One expository introduction is: 

Greenlees and May. Equivariant stable homotopy theory. Handbook of Algebraic Topology, edited by I.M. James, pp. 279-325. 1995.

But a lot has been done since then. Naive $G$-spectra are not so interesting. Genuine $G$-spectra have $RO(G)$-graded homology and cohomology.

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Peter May
Peter May
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There is a subject of equivariant stable homotopy theory. You are referring to $bu$ as what is called a naive $G$-spectrum (a spectrum acted on by a group), and your notation $E^G$ is confusing the categorical fixed point spectrum with the homotopical fixed point spectrum. The difference between the two is, for example, the subject of the Segal conjecture. There is a homotopy fixed point spectral sequence for homotopical fixed point spectra. There is a huge literature on equivariant stable homotopy theory (which was a key input to solution of the Kervaire invariant problem by Hill, Hopkinds, and Ravenel). One expository introduction is: Greenlees and May. Equivariant stable homotopy theory. Handbook of Algebraic Topology, edited by I.M. James, pp. 279-325. 1995. But a lot has been done since then. Naive $G$-spectra are not so interesting. Genuine $G$-spectra have $RO(G)$-graded homology and cohomology.