equivariant sphere spectrum



Stable Homotopy theory

Representation theory



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 GG-equivariant sphere spectrum is the equivariant suspension spectrum of the 0-sphere S 0=* +S^0 = \ast_+

𝕊=Σ G S 0 \mathbb{S} = \Sigma^\infty_G S^0

for S 0S^0 regarded as equipped with the (necessarily) trivial GG-action. It follows that for VV an orthogonal linear GG-representation then in RO(G)-degree VV the equivariant sphere spectrum is the corresponding representation sphere 𝕊(V)S V\mathbb{S}(V) \simeq S^V.

(e.g. Schwede 15, example 2.10)


Equivariant homotopy groups

Just as for the plain sphere spectrum, the equivariant homotopy groups of the equivariant sphere spectrum in ordinary integer degrees nn are all torsion, except at n=0n = 0:

π n H(𝕊)={ forn=0 0 otherwise \pi_n^H(\mathbb{S})\otimes \mathbb{Q} = \left\{ \array{ \cdots & for \; n = 0 \\ 0 & otherwise } \right.

(Greenlees-May 95, prop. A.3)

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 𝕊=Σ G S 0\mathbb{S} = \Sigma^\infty_G S^0 gives that π 0 G(𝕊)\pi_0^G(\mathbb{S}) is the free abelian group on the set of conjugacy classes of subgroups of GG:

(1)π 0 G(𝕊)[HG]π 0 W GH(Σ + E(W GH))[conjugacyclassesofsubgroups] \pi_0^G(\mathbb{S}) \simeq \underset{[H \subset G]}{\oplus} \pi_0^{W_G H}(\Sigma_+^\infty E (W_G H)) \simeq \mathbb{Z}[conjugacy\;classes\;of\;subgroups]

(e.g. Schwede 15, p. 64)



Consider G= 2G= \mathbb{Z}_2 the cyclic group of order 2 and write π p,q S\pi^S_{p,q} for the homotopy group in RO(G)-degree given by the representation on p+q\mathbb{R}^{p+q} where 2\mathbb{Z}_2 acts by reflection on the first pp coordinates, and trivially on the remaining qq coordinates:

The following groups contain \mathbb{Z}-summands:

In addition we have

In summary and more generally we have the following /2\mathbb{Z}/2-equivariant stable homotopy groups of spheres in low bidegree:

The table shows the /2\mathbb{Z}/2-equivariant stable homotopy groups of spheres π p,q S\pi^S_{p,q} with p+qp+q increasing horizontally to the right, and pp increasing vertically upwards. The origin is the double-circled π 0,0 S= 2\pi^S_{0,0} = \mathbb{Z}^2. The complex Hopf fibration η^\hat\eta generates π 1,0 S=\pi^S_{1,0} = \mathbb{Z}, and the quaternionic Hopf fibration generates π 2,1 S=/24\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)(p,q) here is (p+q,p)(p+q,p) in Dugger 08, matching the coordinates of the above table)

n sgn+nn_{sgn} + n 00112233
n sgnn_{sgn}π n sgn+n st\pi^{st}_{n_{sgn} + n}
220000=(h ) 2\mathbb{Z} = \langle (h_{\mathbb{C}})^2\rangle 24=h \mathbb{Z}_{24} = \langle h_{\mathbb{H}}\rangle
1100=h \mathbb{Z} = \langle h_{\mathbb{C}}\rangleπ 1 st 2\pi_1^{st} \oplus \mathbb{Z}_2π 2 st 2\pi^{st}_2 \oplus \mathbb{Z}_2
00π 0 st\pi_0^{st} \oplus \mathbb{Z}π 1 st( 2) 2\pi_1^{st} \oplus (\mathbb{Z}_2)^2 π 2 st( 2) 2\pi_2^{st} \oplus (\mathbb{Z}_2)^2π 3 st 8 24\pi_3^{st} \oplus \mathbb{Z}_8 \oplus \mathbb{Z}_{24}

Cyclic (G DSO(2)G_D \hookrightarrow SO(2)-)equivariance

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).

G ADESO(3)G_{ADE} \hookrightarrow SO(3)-equivariance

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

Relation to equivariant framed bordism

Proof that equivariant framed bordism homology theory is co-represented by the equivariant sphere spectrum:

  • Czes Kosniowski, Equivariant Stable Homotopy and Framed Bordism, Transactions of the American Mathematical Society Vol. 219 (1976), pp. 225-234 (jstor:1997591)


Discussion of GG-equivariant homotopy groups for G=/2G = \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, /2\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=/4G = \mathbb{Z}/4 is in

  • M. C. Crabb, Periodicty in /4\mathbb{Z}/4 equivariant stable homotopy theory, in Mark Mahowald and Stewart Priddy (eds.) Algebraic Topology, Proceedings of the International Conference held March 21-24 (1988) Contemporary Mathematics, volume 96, 1989 (pdf, publisher page)

General background includes

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