geometric representation theory
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The sphere spectrum in (global) equivariant stable homotopy theory.
Its RO(G)-graded homotopy groups are the equivariant version of the stable homotopy groups of spheres.
The $G$-equivariant sphere spectrum is the equivariant suspension spectrum of the 0-sphere $S^0 = \ast_+$
for $S^0$ regarded as equipped with the (necessarily) trivial $G$-action. It follows that for $V$ an orthogonal linear $G$-representation then in RO(G)-degree $V$ the equivariant sphere spectrum is the corresponding representation sphere $\mathbb{S}(V) \simeq S^V$.
(e.g. Schwede 15, example 2.10)
Just as for the plain sphere spectrum, the equivariant homotopy groups of the equivariant sphere spectrum in ordinary integer degrees $n$ are all torsion, except at $n = 0$:
But in some RO(G)-degrees there may appear further non-torsion groups, see the examples below.
In degree 0, the tom Dieck splitting applied to the equivariant suspension spectrum $\mathbb{S} = \Sigma^\infty_G S^0$ gives that $\pi_0^G(\mathbb{S})$ is the free abelian group on the set of conjugacy classes of subgroups of $G$:
(e.g. Schwede 15, p. 64)
Consider $G= \mathbb{Z}_2$ the cyclic group of order 2 and write $\pi^S_{p,q}$ for the homotopy group in RO(G)-degree given by the representation on $\mathbb{R}^{p+q}$ with $\mathbb{Z}_2$ acts by reflection on the first $p$ coordinates, and trivially on the remaining $q$ coordinates:
The following groups contain $\mathbb{Z}$-summands:
$\pi^S_{0,0} = \mathbb{Z} \times \mathbb{Z}$ (by the equivariant Hopf degree theorem, this Example)
$\pi^S_{1,0}$ (Araki-Iriye 82, theorem 8.7, p. 25) generated by the complex Hopf fibration $\hat \eta$ (Araki-Iriye 82, p. 24)
$\pi^S_{2,0}$ generated by $\hat \eta^2$ (Araki-Iriye 82, theorem 9.6, p. 27)
$\pi^S_{3,0}$ generated by $\hat \eta^3$ (Araki-Iriye 82, theorem 10.11, p. 31)
$\pi^S_{4,0}$ generated by $\hat \eta^4$ (Araki-Iriye 82, theorem 11.3, p. 32)
$\pi^S_{5,0}$ (Araki-Iriye 82, theorem 12.4, p. 34)
$\pi^S_{6,0}$ (Araki-Iriye 82, theorem 13.12, p. 38)
$\pi^S_{7,0}$ (Araki-Iriye 82, theorem 14.18, p. 45)
$\pi^S_{8,0}$ (Araki-Iriye 82, theorem 15.26, p. 54)
In addition we have
In summary and more generally we have the following $\mathbb{Z}/2$-equivariant stable homotopy groups of spheres in low bidegree:
The table shows the $\mathbb{Z}/2$-equivariant stable homotopy groups of spheres $\pi^S_{p,q}$ with $p+q$ increasing horizontally to the right, and $p$ increasing vertically upwards. The origin is the double-circled $\pi^S_{0,0} = \mathbb{Z}^2$. The complex Hopf fibration $\hat\eta$ generates $\pi^S_{1,0} = \mathbb{Z}$, and the quaternionic Hopf fibration generates $\pi^S_{2,1} = \mathbb{Z}/24$
graphics grabbed from Dugger 08, based on Araki-Iriye 82
(beware that Dugger 08 uses a different bi-degree labeling convention: the $(p,q)$ here is $(p+q,p)$ in Dugger 08, matching the coordinates of the above table)
The global equivariant sphere spectrum for all the cyclic groups over the circle group is canonically a cyclotomic spectrum and as such is the tensor unit in the monoidal (infinity,1)-category of cyclotomic spectra (see there).
See at quaternionic Hopf fibration – Class in equivariant stable homotopy theory
General lecture notes include
Discussion in rational equivariant stable homotopy theory includes
The sphere spectrum in global equivariant homotopy theory is discussed in
John Rognes, Galois extensions of structured ring spectra (arXiv:math/0502183)
Markus Hausmann, Dominik Ostermayr, Filtrations of global equivariant K-theory (arXiv:1510.04011)
Discussion of $G$-equivariant homotopy groups for $G = \mathbb{Z}/2$ is in
Peter Landweber, On Equivariant Maps Between Spheres with Involutions, Annals of Mathematics Second Series, Vol. 89, No. 1 (Jan., 1969), pp. 125-137 (jstor)
Shôrô Araki, Kouyemon Iriye, Equivariant stable homotopy groups of spheres with involutions. I, Osaka J. Math. Volume 19, Number 1 (1982), 1-55. (Euclid:1200774828)
Kouyemon Iriye, Equivariant stable homotopy groups of spheres with involutions. II, Osaka J. Math. Volume 19, Number 4 (1982), 733-743 (euclid:1200775536)
Daniel Dugger, Daniel Isaksen, $\mathbb{Z}/2$-equivariant and R-motivic stable stems, Proceedings of the American Mathematical Society 145.8 (2017): 3617-3627 (arXiv:1603.09305)
with exposition in
Discussion for $G = \mathbb{Z}/4$ is in
General background includes
Last revised on March 3, 2019 at 08:30:37. See the history of this page for a list of all contributions to it.