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edited Jul 1, 2011 at 18:06

I would like to add a few points:

You can define $RO(G)$-graded homotopy groups of $G$-spectra, see for example Stefan Schwede's course notes on equivariant homotopy theory.Stefan Schwede's course notes on equivariant homotopy theory.
These groups are interesting. For example the inclusion of the 0-skeleton $S^0\rightarrow S^{\sigma}$ of the sign representation stabilizes to a non-trivial (and non-nilpotent!) element in $\pi_0^{C_2} S^{\sigma}\cong \pi_{-\sigma}^{C_2} S^0.$ This iscan be proved by showing its Hurewicz image in Bredon homology is non-trivial.
At least for finite groups $G$-representation spheres can be triangulated and constructed out of the standard $G$-cells ($G/H_+\wedge S^n,$ $G/H_+\wedge D^n$). So ordinary equivariant weak equivalences between cell complexes induce $RO(G)$-graded equivalences.
One reason to put representation spheres into your theory is so that you have a reasonable form of equivariant Spanier-Whitehead duality. At the very least we want finite G-sets to be dualizable, so we want them to equivariantly embed into a G-sphere, which should be one of the spaces in the sphere spectrum. This will not work if we only allow trivial spheres.

I would like to add a few points:

You can define $RO(G)$-graded homotopy groups of $G$-spectra, see for example Stefan Schwede's course notes on equivariant homotopy theory.
These groups are interesting. For example the inclusion of the 0-skeleton $S^0\rightarrow S^{\sigma}$ of the sign representation stabilizes to a non-trivial (and non-nilpotent!) element in $\pi_0^{C_2} S^{\sigma}\cong \pi_{-\sigma}^{C_2} S^0.$ This is proved by showing its Hurewicz image in Bredon homology is non-trivial.
At least for finite groups $G$-representation spheres can be triangulated and constructed out of the standard $G$-cells ($G/H_+\wedge S^n,$ $G/H_+\wedge D^n$). So ordinary equivariant weak equivalences between cell complexes induce $RO(G)$-graded equivalences.
One reason to put representation spheres into your theory is so that you have a reasonable form of equivariant Spanier-Whitehead duality. At the very least we want finite G-sets to be dualizable, so we want them to equivariantly embed into a G-sphere, which should be one of the spaces in the sphere spectrum. This will not work if we only allow trivial spheres.

I would like to add a few points:

You can define $RO(G)$-graded homotopy groups of $G$-spectra, see for example Stefan Schwede's course notes on equivariant homotopy theory.
These groups are interesting. For example the inclusion of the 0-skeleton $S^0\rightarrow S^{\sigma}$ of the sign representation stabilizes to a non-trivial (and non-nilpotent!) element in $\pi_0^{C_2} S^{\sigma}\cong \pi_{-\sigma}^{C_2} S^0.$ This can be proved by showing its Hurewicz image in Bredon homology is non-trivial.
At least for finite groups $G$-representation spheres can be triangulated and constructed out of the standard $G$-cells ($G/H_+\wedge S^n,$ $G/H_+\wedge D^n$). So ordinary equivariant weak equivalences between cell complexes induce $RO(G)$-graded equivalences.
One reason to put representation spheres into your theory is so that you have a reasonable form of equivariant Spanier-Whitehead duality. At the very least we want finite G-sets to be dualizable, so we want them to equivariantly embed into a G-sphere, which should be one of the spaces in the sphere spectrum. This will not work if we only allow trivial spheres.

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created Jul 1, 2011 at 17:56

I would like to add a few points:

You can define $RO(G)$-graded homotopy groups of $G$-spectra, see for example Stefan Schwede's course notes on equivariant homotopy theory.
These groups are interesting. For example the inclusion of the 0-skeleton $S^0\rightarrow S^{\sigma}$ of the sign representation stabilizes to a non-trivial (and non-nilpotent!) element in $\pi_0^{C_2} S^{\sigma}\cong \pi_{-\sigma}^{C_2} S^0.$ This is proved by showing its Hurewicz image in Bredon homology is non-trivial.
At least for finite groups $G$-representation spheres can be triangulated and constructed out of the standard $G$-cells ($G/H_+\wedge S^n,$ $G/H_+\wedge D^n$). So ordinary equivariant weak equivalences between cell complexes induce $RO(G)$-graded equivalences.
One reason to put representation spheres into your theory is so that you have a reasonable form of equivariant Spanier-Whitehead duality. At the very least we want finite G-sets to be dualizable, so we want them to equivariantly embed into a G-sphere, which should be one of the spaces in the sphere spectrum. This will not work if we only allow trivial spheres.