Computing equivariant homotopy with a splitting method. (English) Zbl 07690047
This article focuses on the computation of equivariant homotopy groups of \(G\)-spectra, where \(G\) is a finite group. For a \(G\)-spectrum \(X\), the group \(\pi_V^G(X)\) localized at the prime \(p\) is expressed as the limit of a diagram of equivariant homotopy groups for the actions of \(p\)-subgroups and \(p\)-subquotients of \(G\) on \(X\) and its geometric fixed points, respectively. By a local-to-global argument, the group \(\pi_V^G(X)\) is also the limit of a larger diagram with the same equivariant homotopy groups, but now running over all primes dividing the order of \(G\). Equivariant stable homotopy for finite group actions is thus reduced to the case of \(p\)-groups.
This method is illustrated with the full computation of the equivariant homotopy groups of the Eilenberg-MacLane spectrum \(H\underline{\mathbb{Z}}\) for the actions of \(D_{2p}\) and \(A_5\), and some partial computations. Another consequence of this decomposition is a different approach to the evenness conjecture for \(MU_G\). Namely, if the homotopy of \(MU_P \wedge (BU_P)^{\wedge n}_+\) is even for any \(p\)-subquotient of \(G\) and any \(n \geq 0\), then the homotopy of \(MU_G\) is even.
The proof of the decomposition requires the study of the universal \(G\)-space \(E\mathcal{F}\) for actions with isotropy in a family \(\mathcal{F}\) of subgroups of \(G\). The author determines the type of torsion appearing in the Mackey-functor-valued homology of \(E\mathcal{F}\) with coefficients in the Burnside ring of \(G\). This is used to find which primes must be inverted to split the cofiber sequence \(\Sigma^{\infty}E\mathcal{F}_+ \to S \to \Sigma^{\infty}\widetilde{E\mathcal{F}}\), where \(S\) is the sphere spectrum. As a byproduct, one obtains a stable elements formula for the localization of \(\pi_V^G(E\mathcal{F}_+ \wedge X)\) at the prime \(p\), when \(\mathcal{F}\) is the family of \(p\)-subgroups of \(G\).
This method is illustrated with the full computation of the equivariant homotopy groups of the Eilenberg-MacLane spectrum \(H\underline{\mathbb{Z}}\) for the actions of \(D_{2p}\) and \(A_5\), and some partial computations. Another consequence of this decomposition is a different approach to the evenness conjecture for \(MU_G\). Namely, if the homotopy of \(MU_P \wedge (BU_P)^{\wedge n}_+\) is even for any \(p\)-subquotient of \(G\) and any \(n \geq 0\), then the homotopy of \(MU_G\) is even.
The proof of the decomposition requires the study of the universal \(G\)-space \(E\mathcal{F}\) for actions with isotropy in a family \(\mathcal{F}\) of subgroups of \(G\). The author determines the type of torsion appearing in the Mackey-functor-valued homology of \(E\mathcal{F}\) with coefficients in the Burnside ring of \(G\). This is used to find which primes must be inverted to split the cofiber sequence \(\Sigma^{\infty}E\mathcal{F}_+ \to S \to \Sigma^{\infty}\widetilde{E\mathcal{F}}\), where \(S\) is the sphere spectrum. As a byproduct, one obtains a stable elements formula for the localization of \(\pi_V^G(E\mathcal{F}_+ \wedge X)\) at the prime \(p\), when \(\mathcal{F}\) is the family of \(p\)-subgroups of \(G\).
Reviewer: José Maria Cantarero Lopez (Mérida)
MSC:
| 55P91 | Equivariant homotopy theory in algebraic topology |
| 55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |
References:
| [1] | Abram, W. C.; Kriz, I., The equivariant complex cobordism ring of a finite abelian group, preprint · Zbl 1373.55005 |
| [2] | Angeltveit, V., Recovering a cohomological Mackey functor from its restriction to Sylow subgroups, preprint |
| [3] | D.J. Barnes, Rational equivariant spectra, PhD Thesis. · Zbl 1254.55005 |
| [4] | Cole, M.; Greenlees, J. P.C.; Kriz, I., The universality of equivariant complex cobordism, Math. Z., 239, 455-475 (2002) · Zbl 1008.55015 |
| [5] | tom Dieck, T., Bordism of G-manifolds and integrality theorems, Topology, 9, 345-358 (1970) · Zbl 0209.27504 |
| [6] | tom Dieck, T., Orbittypen und äquivariante Homologie, I, Arch. Math., 23, 307-317 (1972) · Zbl 0252.55003 |
| [7] | Elmendorf, A. D., Systems of fixed point sets, Trans. Am. Math. Soc., 277, 1 (1983) · Zbl 0521.57027 |
| [8] | Ellis-Bloor, B., The Klein four homotopy Mackey functor structure of \(H \mathbb{F}_2_{\_} (2020)\), ANU Honours thesis |
| [9] | Georgakopoulos, N., The \(R O( C_4)\)-integral homology of a point, preprint |
| [10] | Greenlees, J. P.C.; May, J. P., Generalized Tate Cohomology (1995), Memoirs of the AMS · Zbl 0876.55003 |
| [11] | Greenlees, J. P.C., The coefficient ring of equivariant homotopical bordism classifies equivariant formal group laws over Noetherian rings, preprint |
| [12] | Hill, M. A.; Hopkins, M. J.; Ravenel, D. C., The slice spectral sequence for the \(C_4\) analog of real K-theory, Forum Math., 29, 2, 383-447 (2017) · Zbl 1362.55009 |
| [13] | Holler, J.; Kriz, I., On \(R O(G)\)-graded equivariant “ordinary” cohomology where G is a power of \(\mathbb{Z} / 2\), Algebraic Geom. Topol., 17, 741-763 (2017) · Zbl 1366.55006 |
| [14] | Kriz, I.; Lu, Y., On the \(R O(G)\)-graded coefficients of dihedral equivariant cohomology, Math. Res. Lett., 27, 4, 1109-1128 (2020) · Zbl 1466.55002 |
| [15] | Lewis, L. G., The theory of Green functors, preprint |
| [16] | Lewis, L. G.; May, J. P.; McClure, J., Ordinary \(R O(G)\)-graded cohomology, Bull. Am. Math. Soc., 4, 208-212 (1981) · Zbl 0477.55009 |
| [17] | Lewis, L. G.; May, J. P.; Steinberger, M., Equivariant Stable Homotopy Theory (1986), Springer-Verlag · Zbl 0611.55001 |
| [18] | Lewis, L. G.; Mandell, M. A., Equivariant universal coefficient and Künneth spectral sequences, Proc. Lond. Math. Soc. (3), 92, 2, 505-544 (2006) · Zbl 1178.55007 |
| [19] | Lu, Y., On the \(R O(G)\)-graded coefficients of \(Q_8\) equivariant cohomology, Topol. Appl., 307, Article 107921 pp. (2022) · Zbl 1490.55005 |
| [20] | May, J. P., Equivariant homotopy and cohomology theory, (Conference Board of the Mathematical Sciences, vol. 91 (1996)) · Zbl 0890.55001 |
| [21] | Willson, S. J., Equivariant homology theories on G-complexes, Trans. Am. Math. Soc., 212, 155-271 (1975) · Zbl 0308.55002 |
| [22] | M. Zeng, Mackey functors, equivariant Eilenberg-Maclane spectra and their slices, PhD thesis. |
| [23] | Y. Zou, \(R O( D_{2 p})\)-graded Slice Spectral Sequence of \(H \mathbb{Z}_{\_} \), PhD Thesis. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
