May 28, 2026
Given a finite abelian group \(G\) and a Sylow \(p\)-subgroup \(N_p\), we prove that the \(KU_G/p\)-local sphere spectrum is equivalent to the homotopy fixed points of a \(p\)-complete \(KO_{N_p}\)-module spectrum. Then we compute the \(\mathbb{Z}\)-graded homotopy Mackey functors of the \(KU_G\)-local sphere spectrum. This result generalizes the computation of [1] for finite \(p\)-groups, where \(p\) is an odd prime. Finally, by comparing the Bousfield classes of \(KU_G/p\) and \(G\)-equivariant Morava \(K\)-theory, we prove that the \(KU_G/p\)-local sphere spectrum is equivalent to a wedge sum of equivariant Morava \(K\)-theory localized sphere spectra, and describe the \(RO(G)\)-graded homotopy Mackey functors of the \(KU_G/p\)-local sphere spectrum.
Non-equivariantly, chromatic homotopy theory provides a systematical approach to study the large-scale phenomena in the stable homotopy groups of spectra, organized by different periodicities according to chromatic heights. When \(G\) is a finite abelian group, chromatic homotopy theory admits a natural \(G\)-equivariant refinement. This viewpoint suggests that, for such a group \(G\), one can understand \(G\)-equivariant stable homotopy groups by studying the chromatic filtration in the category of \(G\)-spectra. The foundations of equivariant chromatic homotopy theory were developed in [2]–[9]; see also the survey by Behrens and Carlisle [10].
At chromatic height 1, non-equivariant \(v_1\)-periodicity is reflected in the Bott periodicity of \(KO\) and can be studied through \(L_{KU/p}S\). Both topological \(K\)-theory and \(L_{KU}S\) admit natural \(G\)-equivariant refinements, and several computations of the homotopy groups of \(L_{KU_G}S_G\) have been established in recent years. Balderrama [11] computed the \(RO(C_2)\)-graded homotopy \(C_2\)-Green functor of \(L_{KU_{C_2}/2}S_{C_2}\). When \(G\) is a finite \(p\)-group for an odd prime \(p\), Carawan et al. [1] computed the \(\mathbb{Z}\)-graded homotopy Mackey functors of \(L_{KU_G}S_G\), and Balderrama [12] investigated the norm maps in \(\underline{\pi}_0(L_{KU_G}S)\). Furthermore, the equivariant \(J\)-homomorphism was studied in [13], [14], and periodic self-maps have been investigated for \(G=C_2\) in [15]–[17] and for \(G=C_{p^n}\) in [18].
In this paper, we compute the \(\mathbb{Z}\)-graded homotopy Mackey functors of \(L_{KU_G}S_G\) and the \(RO(G)\)-graded homotopy Mackey functors of \(L_{KU_G/p}S_G\) for finite abelian groups \(G\).
It is worth noting that, for a finite group \(G\), knowledge of the homotopy groups of the \(G\)-equivariant sphere spectrum remains very limited. For \(G=C_2\), the \(RO(C_2)\)-graded ring \(\pi_{\star}^{C_2}S_{\mathbb{Q}}\) was computed by Belmont-Xu-Zhang [19], and \(\pi_\star^{C_2} S_{C_2}\) is computed in a range of degrees in [20]–[26]. For \(G=C_3\), Hou-Zhang [27] carried out partial computations of \(\pi_\star^{C_3} S_{C_3}\). Beyond these cases, very few computations are known.
When \(G\) is a \(p\)-group for an odd prime \(p\), Carawan-Field-Guillou-Mehrle-Stapleton [1] show that, for a generator \(g\) of \((\mathbb{Z}_p^\wedge)^\times\), there is a fiber sequence \[L_{KU_G/p}S_G \longrightarrow (KU_{G})_p^{\wedge}\overset{\psi^g-1}{\longrightarrow} (KU_{G})_p^{\wedge},\] and they use this fiber sequence to compute the homotopy groups of \(L_{KU_G/p}S_G\). However, if \(G\) is not a \(p\)-group, this sequence is no longer a fiber sequence. When \(G\) is a finite nilpotent group, we construct a fiber sequence in the following proposition.
Proposition 1 (11). Let \(G\) be a finite nilpotent group, and let \(Cyc\) be the family of all cyclic subgroups of \(G\). For any prime \(p\), let \(N_p\) be the Sylow \(p\)-subgroup of \(G\), and let \(g\) be a topological generator of \(\mathbb{Z}_p^{\times}/\{\pm 1\}\). Then for any finite \(G\)-spectrum \(X\), there is a fiber sequence \[L_{KU_G/p}X \longrightarrow (ECyc_+\wedge\;\operatorname{Inf}_{N_p}^G KO_{N_p}\wedge X)_p^{\wedge}\overset{\psi^g-1}{\longrightarrow} (ECyc_+\wedge\;\operatorname{Inf}_{N_p}^G KO_{N_p}\wedge X)_p^{\wedge} .\]
As a corollary, we prove that for any finite nilpotent group \(G\), \[L_{KU_G/p}S_G\simeq (ECyc_+\wedge \operatorname{Inf}_{N_p}^G L_{KU_{N_p}/p}S_{N_p})_p^{\wedge}.\] For any subgroup \(H\subset G\), let \(\underline{A}_H\) denote the \(H\)-Mackey functor given by the Burnside ring, and let \(\underline{J}_H\) denote the subfunctor of \(\underline{A}_H\) given by the Brauer relations (see 4). Define \[\underline{A/J}_H:=\underline{A}_H/\underline{J}_H .\] When \(H=G\), we omit the subscript and write \(\underline{A/J}\) for \(\underline{A/J}_G\). In order to compute the homotopy groups of \(L_{KU_G/p}S_G\), we study the fixed points of \(G\)-spectra of \((ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)_p^\wedge\) for an \(N_p\)-spectrum \(E\), and prove the following proposition.
Proposition 2 (14). Let \(G\) be a finite nilpotent group with Sylow \(p\)-subgroup \(N_p\), and let \(N\) be the subgroup of \(G\) such that \(G\cong N_p\times N\). Then for any \(N_p\)-spectrum \(E\) such that \(E\) is \(S/p\)-equivalent to \(ECyc_+ \wedge E\), we have an isomorphism between \(G\)-Mackey functors \[\underline{\pi}_*(ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)_p^\wedge\cong (\underline{\pi}_* E \otimes \underline{A/J}_N)_p^\wedge.\] Here the \(N_p\)-Mackey functor \(\underline{\pi}_* E\) is regarded as a \(G\)-Mackey functor via the canonical quotient map \(G\to N_p\), and the \(N\)-Mackey functor \(\underline{A/J}_N\) is regarded as a \(G\)-Mackey functor via \(G\to N\).
When \(E=L_{KU_{N_p}/p}S_{N_p}\), we have \[\underline{\pi}_*(L_{KU_G/p}S_G)\cong \underline{\pi}_*L_{KU_{N_p}/p}S_{N_p}\otimes_{\mathbb{Z}_p} (\underline{A/J}_N)_p^\wedge.\] Note that when \(p\) is an odd prime, \(\underline{\pi}_* L_{KU_{N_p}/p}S_{N_p}\) is computed in [1], so it remains to treat the case \(p=2\). When \(N_2\) is abelian, we can compute \(\underline{\pi}_k L_{KU_{N_2}/2}S_{N_2}\) via the short exact sequence \[0\longrightarrow \underline{\mathrm{coker}}_2\{k+1\}\longrightarrow \underline{\pi}_k L_{KU_{N_2}/2}S_{N_2} \longrightarrow \underline{\ker}_2\{k\}\longrightarrow 0,\] where \[\underline{\ker}_2\{k\}:=\ker(\underline{\pi}_k(KO_{N_2})_2^{\wedge}\overset{\psi^g-1}{\longrightarrow}\underline{\pi}_k (KO_{N_2})_2^{\wedge}),\] \[\underline{\mathrm{coker}}_2\{k\}:=\mathrm{coker}(\underline{\pi}_k (KO_{N_2})_2^{\wedge}\overset{\psi^g-1}{\longrightarrow}\underline{\pi}_k (KO_{N_2})_2^{\wedge}).\] Extension problems occur in degrees \(0\) and \(8d+1\). Let \(\underline{RO(-;\mathbb{R})}\) be the \(G\)-Mackey functor such that for any \(H\subset G\), \[\underline{RO(-;\mathbb{R})}(G/H):=RO(H;\mathbb{R}),\] here \(RO(H;\mathbb{R})\) is the free abelian group generated by the irreducible real \(H\)-representations whose endomorphism rings are isomorphic to \(\mathbb{R}\). The Hurewicz map \(\underline{\pi}_0 S_{N_2}\longrightarrow \underline{\pi}_0 L_{KU_{N_2}/2}S_{N_2}\) induces a map of Mackey functors \[\theta_{N_2}\colon \underline{J}_{N_2} \longrightarrow \underline{\mathrm{coker}}_2\{1\}\cong \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2.\] We reduce the extension problem in degree \(k=0\) to the computation of \(\theta_{N_2}\), which is carried out in 17. The case \(k=8d+1\) is treated similarly and is summarized in 18. Our computation of \(\underline{\pi}_* L_{KU_G/p}S_G\) is summarized in [lem:extension of 2 group KU local sphere,thm: Z graded homotopy of KU_G/p local sphere]. Then we compute \(\underline{\pi}_\ast L_{KU_G}S_G\) for finite abelian groups \(G\) via the arithmetic fracture square.
Theorem 3 (21). Let \(G\) be a finite abelian group, let \(N_p\) be its Sylow \(p\)-subgroup, and let \(G/N_p\) denote the product of the Sylow \(q\)-subgroups of \(G\) for \(q\neq p\). \[\underline{\pi}_kL_{KU_G}S_G\cong \begin{cases} \underline{A/J}_{G/N_2} \otimes \frac{\underline{A}_{N_2} \oplus \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2}{\{j-\theta_{N_2}(j):j\in \underline{J}_{N_2}\}} & \quad k=0\\ 0 & \quad k=-1\\ \mathbb{Q}/\mathbb{Z}\otimes (\prod_p \underline{\operatorname{coker}}_p\{0\}\otimes \underline{A/J}_{G/N_p}) & \quad k=-2\\ \prod_p \underline{\pi}_kL_{KU_G/p}S_G & \quad \text{otherwise} \end{cases}\] The computation of \(\theta_{N_2}\) is carried out in 17. The functors \(\underline{\operatorname{coker}}_p\{0\}\) are listed in [lem:psi-1,lem:psi-1 for odd p].
Finally, for any subgroup \(H\subset G\), let \(K(H,1)\) be the equivariant Morava \(K\)-theory defined by Strickland [3]. The comparison of the Bousfield classes of \(KU_G/p\) and equivariant Morava \(K\)-theories, together with the study of \(K(H,1)\)-local \(G\)-spectra, leads to the following theorem.
Theorem 4 (24). Let \(G\) be a finite abelian group, let \(Cyc\) be the family of all cyclic subgroups of \(G\), and let \(N_p\) be the Sylow \(p\)-subgroup of \(G\). For any prime \(p\) and any \(G\)-spectrum \(X\), there is an equivalence of \(G\)-spectra \[L_{KU_G/p}X \simeq L_{\bigvee_{H\in Cyc, H\cap N_p=e}K(H,1)}X \simeq \bigvee_{H\in Cyc, H\cap N_p=e} L_{K(H,1)}X.\]
As a corollary, we can describe \(\underline{\pi}_VL_{KU_G/p}S_G\) for any \(V\in RO(G)\) in terms of certain \(\mathbb{Z}\)-graded homotopy groups, \[\underline{\pi}_VL_{KU_G/p}S_G\cong \bigoplus_{H\in Cyc, p\nmid |H|}\underline{\pi}_{n_{V,H}} L_{KU_{N_p}/p}S_{N_p},\] where \(n_{V,H}\) is the dimension of \(V^H\). The restriction and transfer homomorphisms are described in 2.
In 2, we recall background on equivariant stable homotopy theory and Bousfield localization that will be used throughout the paper. In 3, we study the Bousfield classes of \(KU_G/p\) and \(KO_G/p\) for a finite nilpotent \(G\), and prove 1. In 4, for a finite nilpotent group \(G\) and its Sylow subgroup \(N_p\), we study the fixed points of \(ECyc_+\wedge \operatorname{Inf}_{N_p}^G E\) for an \(N_p\)-spectrum \(E\), and prove 2. In 5, we compute \(\underline{\pi}_\ast L_{KU_{N_2}/2}S_{N_2}\) for a finite abelian \(2\)-group \(N_2\). In 6, by combining 2 with the computation in 5, we obtain \(\underline{\pi}_\ast L_{KU_G/2}S_G\) for finite abelian groups \(G\). Then we prove 3. In 7, we study the \(K(H,1)\)-local sphere spectrum for finite abelian groups \(G\). We prove 4 and compute \(\underline{\pi}_V L_{KU_G/p}S_G\) for \(V\in RO(G)\).
This paper is based on part of my PhD thesis. I would like to thank my advisor, Xu-an Zhao, for his guidance throughout this project and for carefully proofreading the manuscript. I am grateful to Markus Hausmann for his helpful advice and for inspiring my initial interest in equivariant chromatic homotopy theory. I would also like to thank William Balderrama for his valuable suggestions regarding the extension problems for Mackey functors. I also thank Zezhou Zhang and Yifei Zhu for helpful discussions and comments.
In this section, we briefly review some results that will be used throughout the paper. We refer the reader to [28], [29] for background on equivariant homotopy theory, and to [30]–[32] for background on Bousfield localization.
For a finite group \(G\), let \(\mathcal{S}^G\) be the category of \(G\)-spaces, and let \(Sp^G\) be the category of genuine \(G\)-spectra. For a \(G\)-space \(X\), we write \(\Sigma_G^\infty X\) for the suspension \(G\)-spectrum of \(X\). Let \(S_G\) be the \(G\)-equivariant sphere spectrum. For any \(G\)-spectrum \(X\), let \(\underline{\pi}_* X\) be the homotopy Mackey functors of \(X\), where \[\underline{\pi}_n X (G/H)=\pi_n^H(X) \cong [G/H_+\wedge S^n, X]_G\] for any \(G\)-orbit \(G/H\).
Let \(\alpha\colon H\to G\) be a homomorphism of finite groups, any \(G\)-spectrum can be regarded as an \(H\)-spectrum via \(\alpha\), yielding a symmetric monoidal functor \(\alpha^*\colon \mathrm{Sp}^G \to \mathrm{Sp}^H\). In particular:
If \(\alpha:H\subset G\) is inclusion, we denote \(\alpha^*\) by \(\operatorname{Res}_H^G\).
If \(N\trianglelefteq G\) is a normal subgroup and \(\alpha\colon G\to G/N\) is the quotient map, we denote \(\alpha^*\) by \(\operatorname{Inf}_{G/N}^G\). The inflation functor \(\operatorname{Inf}_{G/N}^G\) is left adjoint to the \(N\)-fixed point functor \((-)^N\colon \mathrm{Sp}^G \to \mathrm{Sp}^{G/N}\). For \(X\in \mathrm{Sp}^{G/N}\) and \(Y\in \mathrm{Sp}^G\), there is an equivalence \[(\operatorname{Inf}_{G/N}^G(X)\wedge Y)^N\simeq X\otimes Y^N.\]
There is also a commonly used functor called the geometric fixed point functor. Let \(\mathcal{F}\) be a family of subgroups of \(G\), there is a unbased \(G\)-space \(E\mathcal{F}\) such that \[(E\mathcal{F})^H=\begin{cases} pt & \text{if }\;H\in\mathcal{F}\\ \emptyset & \text{if }\;H\not\in\mathcal{F}, \end{cases}\] and let \(\widetilde{E\mathcal{F}}\) be the cofiber of \(E\mathcal{F}_+\to S^0_G\). For any subgroup \(H\subset G\), let \(\mathcal{F}_{H\not\subset}\) denote the family of subgroups of \(G\) that do not contain \(H\). The \(H\)-geometric fixed point of a \(G\)-spectrum \(X\) is given by \[\Phi^H(X):=(X\wedge \widetilde{E\mathcal{F}_{H\not\subset}})^H.\] If \(H\) is a normal subgroup of \(G\), \(\Phi^H(X)\) has a residual action of \(G/H\), one can regard the \(H\)-geometric fixed point as a functor \(\Phi^H:Sp^G\to Sp^{G/H}\). Moreover, there is an equivalence of functors \(\Phi^H\circ \operatorname{Inf}_{G/H}^G\simeq \operatorname{Id}_{Sp^{G/H}}\).
For any finite group \(G\), topological \(K\)-theory admits a natural \(G\)-equivariant refinement. Let \(KU_G\) (resp., \(KO_G\)) be the \(G\)-equivariant complex (resp., real) \(K\)-theory. Let \(RU(G)\) (resp., \(RO(G)\)) be the complex (resp., real) representation ring of \(G\), and let \(\underline{RU}\) be the \(G\)-Green functor such that \(\underline{RU}(G/H):=RU(H)\). By [33], we have \[\underline{\pi}_* KU_G\cong \underline{RU}[\beta^\pm].\] We denote by \(RO(G;\mathbb{R})\) (resp., \(RO(G;\mathbb{C})\), \(RO(G;\mathbb{H})\)) the free abelian group generated by irreducible real \(G\)-representations whose endomorphism ring is isomorphic to \(\mathbb{R}\) (resp., \(\mathbb{C}\), \(\mathbb{H}\)). Then \[\underline{\pi}_* KO_G \cong \bigoplus_{\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}}\pi_*K\mathbb{F}\otimes RO(G;\mathbb{\mathbb{F}}),\] where \[K\mathbb{F}=\begin{cases} KO & \mathbb{F}=\mathbb{R},\\ KU & \mathbb{F}=\mathbb{C},\\ KSp & \mathbb{F}=\mathbb{H}. \end{cases}\] The restriction and transfer homomorphisms in \(\underline{\pi}_* KO_G\) can be computed via the complexification map \(KO_G\to KU_G\); see [34] for details.
Proposition 5. [35]For any finite group \(G\), \[\Phi^{G} KU_{G}\;\simeq\; \begin{cases} KU\otimes \mathbb{Z}\left[\frac{1}{n},\zeta_{n}\right], & \text{if } G \cong C_{n}\\[6pt] \ast, & \text{otherwise} \end{cases},\] where \(\zeta_n\) is a primitive \(n\)-th root of unity.
Let \(G\) be a finite group. For \(E,X\in Sp^G\), the Bousfield localization of \(X\) with respect to \(E\) is an \(E\)-equivalence \(f:X\to L_EX\) such that \(L_EX\) is \(E\)-local. For \(E,F\in Sp^G\), we say that \(E\) and \(F\) are Bousfield equivalent, and write \(\langle E\rangle=\langle F\rangle\), if \[\forall X\in Sp^G, \quad X\wedge E\simeq \ast \;\Longleftrightarrow \; X\wedge F\simeq \ast.\] If \(\langle E\rangle=\langle F\rangle\), then \(L_EX\simeq L_FX\) for every \(X\in Sp^G\). For any prime \(p\), a \(G\)-spectrum \(X\) is called \(p\)-local if \(X\simeq L_{S_G\mathbb{Z}{(p)}}X\), and \(p\)-complete if \(X\simeq L_{S_G/p}X\).
By [32], for any subgroup \(H\subset G\) and any \(X\in Sp^G\), there is an equivalence \[\operatorname{Res}_H^G L_EX\simeq L_{\operatorname{Res}_H^G E} \operatorname{Res}_H^G X.\] When \(p\nmid |H|\), by the splitting of the category of \(p\)-local \(H\)-spectra, it follows from [36] that, for any \(X,E\in Sp^H_{(p)}\), \[\Phi^H L_E X\simeq L_{\Phi^H E} \Phi^H X\] as non-equivariant spectra. As a result, for any \(H\subset G\) such that \(p\nmid |H|\) and for any \(X, E\in Sp^G_{(p)}\), we have \[\Phi^H L_EX\simeq \Phi^H \operatorname{Res}_H^GL_EX\simeq \Phi^H L_{\operatorname{Res}_H^G E}\operatorname{Res}_H^G X\simeq L_{\Phi^H E}\Phi^H X \in Sp_{(p)}.\] In particular, we can show that \(\Phi^H: Sp^G\to Sp\) preserves \(p\)-completion if \(p\nmid |H|\).
Proposition 6. For any \(X\in Sp^G\) and any subgroup \(H\subset G\) with \(p\nmid |H|\), there is an equivalence \(\Phi^H (X_p^\wedge)\simeq (\Phi^H X)_p^\wedge\) of non-equivalence spectra.
Proof. Since every \(S\mathbb{Z}_{(p)}\)-acyclic \(G\)-spectrum is \(S/p\)-acyclic, we have \(L_{S/p}L_{S\mathbb{Z}_{(p)}}\simeq L_{S/p}\). If \(p\nmid |H|\), we have \[\begin{align} \Phi^H L_{S/p}X & \simeq \Phi^H L_{S/p} L_{S\mathbb{Z}_{(p)}}X_{(p)} \simeq L_{\Phi^H S/p} \Phi^H(L_{S\mathbb{Z}_{(p)}}X)\\ & \simeq L_{S/p} L_{S\mathbb{Z}_{(p)}}\Phi^H X\simeq L_{S/p} \Phi^H X, \end{align}\] here \(\Phi^H L_{S\mathbb{Z}_{(p)}}X\simeq L_{S\mathbb{Z}_{(p)}}\Phi^H X\) since \(L_{S\mathbb{Z}_{(p)}}\) is smashing. ◻
Proposition 7. [37]Let \(E,F,X\in Sp^G\). If \(E\wedge L_F X\simeq \ast\), the following diagram is a pullback. \[\begin{tikzcd} L_{E\vee F}X \arrow[r] \arrow[d] & L_E X \arrow[d] \\ L_F X \arrow[r] & L_F L_E X. \end{tikzcd}\]
In particular, let \(E=\bigvee_p S_G/p\), \(F=S_G\otimes\mathbb{Q}\), we have the arithmetic fracture square \[\begin{tikzcd} X \arrow[r] \arrow[d] & \prod_{p} X_{p}^{\wedge} \arrow[d] \\ X_{\mathbb{Q}} \arrow[r] & (\prod_{p} X_{p}^{\wedge})_{\mathbb{Q}} . \end{tikzcd}\] For any \(E,X\in Sp^G\), we have \(L_{E/p}X\simeq (L_E X)_p^\wedge\) and \(L_{E\otimes \mathbb{Q}}X\simeq (L_E X)_{\mathbb{Q}}\). Therefore, \(L_E X\) can be recovered from \(L_{E/p}X\) and \(L_{E\otimes \mathbb{Q}}X\) via the arithmetic fracture square.
Proposition 8. [36] Let \(R\) be a \(G\)-equivariant ring spectrum. Then for any finite \(G\)-spectrum \(X\), \((R\wedge X)_p^{\wedge}\) is \(R/p\)-local.
Proof. The case \(X=S_G\) is treated in [36], and the same proof applies to general finite \(G\)-spectra. For any \(R/p\)-acyclic \(G\)-spectrum \(M\), \(M \wedge R\) is \(S/p\)-acyclic, hence \[[M,(R\wedge X)_p^{\wedge}]_G\cong [M\wedge R,(R\wedge X)_p^{\wedge}]_{G}^{R-\mathrm{alg}}\subset [M\wedge R,(R\wedge X)_p^{\wedge}]_G=0.\] ◻
Finally, we give two examples of Bousfield localization required for this paper.
Example 1. Let \(G\) be a finite group, and let \(H\subset G\) be a subgroup. Let \(\mathcal{F}_H\) denote the smallest family of subgroups of \(G\) that contains \(H\) as an element. Then \[L_{G/H_+}X \simeq F((E\mathcal{F}_H)_+, X)\simeq F((EG/H)_+, X).\] Indeed, \(F((E\mathcal{F}_H)_+, X) \to F((E\mathcal{F}_H)_+, Y)\) is a \(G\)-equivalence if and only if \(X \to Y\) is an \(H\)-equivalence. Thus, a \(G\)-spectrum \(X\) satisfies \(X \simeq F((E\mathcal{F}_H)_+, X)\) if and only if \(X\) is \(G/H_+\)-local, and \(F((E\mathcal{F}_H)_+, X) \simeq L_{G/H_+} X\).
Example 2. Let \(\mathcal{F}\) be a family of subgroups of \(G\), and let \(X \in Sp^G\). We have \[L_{\widetilde{E\mathcal{F}}}X\simeq X \wedge \widetilde{E\mathcal{F}}.\] Indeed, define \(X[\mathcal{F}^{-1}] := X \wedge \widetilde{E\mathcal{F}}\). Then the natural map \(X\to X[\mathcal{F}^{-1}]\) is a \(\widetilde{E\mathcal{F}}\)-equivalence, and \(X[\mathcal{F}^{-1}]\) is \(\widetilde{E\mathcal{F}}\)-local, since \(\widetilde{E\mathcal{F}}\wedge \widetilde{E\mathcal{F}}\simeq \widetilde{E\mathcal{F}}\).
Proposition 9. [3]Let \(G\) be a finite abelian group and let \(H\subset G\) be a subgroup. Let \(E\) be a \(G/H\)-equivariant ring spectrum, and define \(E_G := \bigl(\operatorname{Inf}^{G}_{G/H} E\bigr)\bigl[\mathcal{F}_{H\not\subset}^{-1}\bigr]\). Then for every \(G\)-spectrum \(X\), there are natural isomorphisms \[E_G^{*}(X)\cong E^{*}(\Phi^{H}X), \qquad (E_G)_{*}(X)\cong E_{*}(\Phi^{H}X).\]
In this section, let \(G\) be a finite nilpotent group. For each prime \(p\), let \(N_p\) be the Sylow \(p\)-subgroup of \(G\). Then \[G\cong \prod_p N_p,\] and the projection onto the \(p\)-factor gives a canonical quotient map \(\alpha_p:G\to N_p\). This induces the inflation functor \(\operatorname{Inf}_{N_p}^G=\alpha_p^*:Sp^{N_p}\to Sp^G\). We identify \(L_{KU_G/p}S_G\) as a homotopy fiber in 11, which is related to the Adams operation \(\psi^g\) on \(KO_{N_p}\).
Lemma 1. Let \(Cyc\) be the family of all cyclic subgroups of \(G\). For any prime \(p\), \(KU_G/p\) is Bousfield equivalent to \(ECyc_+ \wedge \operatorname{Inf}_{N_p}^G KU_{N_p}/p\).
Proof. By [31], it suffices to show that for any subgroup \(H\subset G\), \(\langle \Phi^H KU_G\rangle = \langle \Phi^H (ECyc_+\wedge \operatorname{Inf}_{N_p}^G KU_{N_p}/p)\rangle\). By 5 we have \[\langle \Phi^H KU_G/p\rangle=\begin{cases} \langle KU/p\rangle & H\in Cyc, \text{ and } p\nmid |H|.\\ \langle \ast \rangle & \text{otherwise}. \end{cases}\] On the other hand, since \(G\) is nilpotent, there is a subgroup \(N\subset G\) such that \(G\cong N_p\times N\). For any subgroup \(H\subset G\), we have the following diagram of groups \[\begin{tikzcd} H\arrow[r, hook] & N\times (H\cap N_p) \arrow[r, twoheadrightarrow]\arrow[d, hook] & H\cap N_p\arrow[d, hook]\\ & G\arrow[r, twoheadrightarrow] & N_p . \end{tikzcd}\] Therefore, \[\begin{align} \Phi^H \operatorname{Inf}_{N_p}^G KU_{N_p}/p & \simeq \Phi^H \operatorname{Inf}_{H\cap N_p}^{N\times (H\cap N_p)}\operatorname{Res}_{H\cap N_p}^{N_p}KU_{N_p}/p\\ &\simeq \Phi^H \operatorname{Inf}_{H\cap N_p}^{H}KU_{H\cap N_p}/p. \end{align}\]
If \(H\cap N_p\neq \{e\}\), \(\Phi^{H\cap N_p} KU_{H\cap N_p}/p\simeq \ast\) since \(p\) is invertible in \(\Phi^{H\cap N_p}KU_{H\cap N_p}/p\). The group homomorphism \(H\cong (H\cap N)\times (H\cap N_p)\to H\cap N_p\) induces a map of ring spectra \[\ast \simeq \Phi^{H\cap N_p} KU_{H\cap N_p}/p\to \Phi^H \operatorname{Inf}_{H\cap N_p}^H KU_{H\cap N_p}/p,\] thus \(\Phi^H\operatorname{Inf}_{H\cap N_p}^H KU_{H\cap N_p}/p\simeq \ast\).
If \(H\cap N_p=\{e\}\), then \(\Phi^H\operatorname{Inf}_{N_p}^G KU_{N_p}/p\simeq \Phi^H\operatorname{Inf}_e^H KU/p\simeq KU/p\), and \[\Phi^H(ECyc_+\wedge \operatorname{Inf}_{N_p}^G KU_{N_p}/p)\simeq \begin{cases} KU/p & H\in Cyc, \text{ and } H\cap N_p=\{e\}.\\ \ast & \text{otherwise.} \end{cases}\] Note that for the Sylow \(p\)-subgroup \(N_p\), \(H\cap N_p=e\) if and only if \(p\nmid |H|\), therefore \(\langle KU_G/p\rangle=\langle ECyc_+\wedge \operatorname{Inf}_{N_p}^G KU_{N_p}/p \rangle\). ◻
Remark 10. Let \(N\) be a normal subgroup of \(G\). The \(N\)-fixed point \((\widetilde{E\mathcal{F}_{G\not\subset}})^{N}\in \mathcal{S}^{G/N}\) can be regarded as a model for \(\widetilde{E\mathcal{F}_{G/N \not\subset}}\). This gives a canonical map of \(G\)-spaces \[i:\operatorname{Inf}_{G/N}^G \widetilde{E\mathcal{F}_{G/N \not\subset}} \to \widetilde{E \mathcal{F}_{G\not\subset}}\] adjoint to the identity of \(\widetilde{E\mathcal{F}_{G/N \not\subset}}\). For any \(G/N\)-ring spectrum \(E\), we can define the map of ring spectra \(\Phi^{G/N}E\to \Phi^{G}\operatorname{Inf}_{G/N}^G E\) as the composition \[\begin{align} \Phi^{G/N}E&=(\widetilde{E\mathcal{F}_{G/N\not\subset}}\wedge E)^{G/N}\to (\widetilde{E\mathcal{F}_{G/N\not\subset}}\wedge (\operatorname{Inf}_{G/N}^G E)^N)^{G/N}\\ &\simeq (\operatorname{Inf}_{G/N}^G \widetilde{E\mathcal{F}_{G/N\not\subset}} \wedge \operatorname{Inf}_{G/N}^G E)^G\\ &\overset{i}{\to} (\widetilde{E \mathcal{F}_{G\not\subset}}\wedge \operatorname{Inf}_{G/N}^G E)^G=\Phi^G(\operatorname{Inf}_{G/N}^G E). \end{align}\] This construction gives the map \[\Phi^{H\cap N_p} KU_{H\cap N_p}/p\to \Phi^H \operatorname{Inf}_{H\cap N_p}^H KU_{H\cap N_p}/p,\] which is used in the proof of 1.
Lemma 2. \(KU_G\) is Bousfield equivalent to \(KO_G\).
Proof. The proof follows the proof of the nonequivariant case in [38]. The equivariant Wood theory [34] says that there is a cofibration \[\Sigma KO_G \overset{\eta}{\longrightarrow} KO_G \longrightarrow KU_G\] where \(\eta\in \pi_1(S)\) is the hopf element. Then \(KO_G\wedge X\simeq \ast\) implies \(KU_G\wedge X\simeq \ast\). Conversely, if \(KU_G\wedge X\simeq \ast\), then multiplication by \(\eta\) induces an isomorphism on \((KO_G)_*(X)\). Since \(\eta\) is nilpotent, it follows that \((KO_G)_*(X)=0\). ◻
Lemma 3. For any closed subgroup \(H\subset G\), \(\Phi^H KU_G/p\simeq \ast\) if and only if \(\Phi^H KO_G/p\simeq \ast\). Furthermore, \[\langle KU_G/p\rangle=\langle ECyc_+\wedge \operatorname{Inf}_{N_p}^G KO_{N_p}/p \rangle.\]
Proof. Apply the geometric fixed point functor \(\Phi^H\), we have a cofibration \[\Phi^H KO_G/p\wedge S^1\overset{1\wedge\eta}{\longrightarrow} \Phi^H KO_G/p \wedge S^0\longrightarrow \Phi^H KU_G/p.\] By the same argument in 2, since \(\eta\) is nilpotent, \(\Phi^H KU_G/p\simeq \ast\) if and only if \(\Phi^H KO_G/p\simeq \ast\).
Therefore, \[\Phi^H(ECyc_+\wedge \operatorname{Inf}_{N_p}^G KO_{N_p}/p)\simeq \begin{cases} KO/p & H\in Cyc, \text{ and } H\cap N_p=\{e\}.\\ \ast & \text{otherwise.} \end{cases}\] Since \(\langle KU/p\rangle=\langle KO/p \rangle\), we have \(\langle KU_G/p\rangle=\langle ECyc_+\wedge \operatorname{Inf}_{N_p}^G KO_{N_p}/p \rangle\). ◻
Proposition 11. Let \(G\) be a finite nilpotent group, and let \(Cyc\) be the family of all cyclic subgroups of \(G\). For any prime \(p\), let \(N_p\) be the Sylow \(p\)-subgroup of \(G\), and let \(g\) be a topological generator of \(\mathbb{Z}_p^{\times}/\{\pm 1\}\). Then for any finite \(G\)-spectrum \(X\), there is a fiber sequence \[L_{KU_G/p}X \longrightarrow (ECyc_+\wedge\;\operatorname{Inf}_{N_p}^G KO_{N_p}\wedge X)_p^{\wedge}\overset{\psi^g-1}{\longrightarrow} (ECyc_+\wedge\;\operatorname{Inf}_{N_p}^G KO_{N_p}\wedge X)_p^{\wedge} .\]
When \(p\) is odd, the \(KO_{N_p}\) appearing in the fiber sequence above can be replaced by \(KU_{N_p}\). In this case, \(g=(\zeta_{p-1},p+1)\) is a topological generator of \(\mathbb{Z}_p^{\times}\), where \(\zeta_{p-1}\) is a primitive \((p-1)\)-th root of unity.
Proof. Let \(I=ECyc_+\wedge \operatorname{Inf}_{N_p}^G KO_{N_p}\), and let \(F_G\) be the fiber of \(\psi^g-1\). \(I\) is a \(G\)-equivariant ring spectrum since the inflation functor is monoidal. By 8, \((I\wedge X)_p^{\wedge}\) is \(I/p\)-local; hence it is \(KU_G/p\)-local by 2. It follows that \(F_G\) is \(KU_G/p\) local.
We claim that \(\psi^g:I_p^\wedge \to I_p^\wedge\) is a map of ring spectra. Therefore, \[S_G\longrightarrow I_p^\wedge \overset{\psi^g-1}{\longrightarrow} I_p^\wedge\] is trivial. Smashing with \(X\) and \(p\)-completing, we see that the composite \[X_p^\wedge \longrightarrow (I\wedge X)_p^\wedge \overset{\psi^g-1}{\longrightarrow} (I\wedge X)_p^\wedge\] is also trivial, which induces a map \(\iota: X_p^\wedge \to F_G\). It suffices to show that \(\iota\) is a \(I/p\)-equivalence, or equivalently, \[f_H:\Phi^H(I/p\wedge \iota): \Phi^H (I/p\wedge X_p^\wedge)\to \Phi^H (I/p\wedge F_G)\] is an equivalence for every subgroup \(H\subset G\). By 5 and 3, \(\Phi^H \operatorname{Inf}_{N_p}^G KO_{N_p}/p\simeq \ast\) if \(H\) is not cyclic or \(H\cap N_p\) is nontrivial, and \(f_H\) is a equivalence between trivial spectra. If \(H\) is cyclic and \(H\cap N_p=\{e\}\), then we have \[\Phi^H I/p\simeq \Phi^H\operatorname{Inf}_e^H KO/p\simeq KO/p.\] By 6, \[\begin{align} \Phi^H (I/p\wedge F_G) & \simeq \Phi^H I/p \wedge \operatorname{fib}(\Phi^H(\psi^g-1))\\ &\simeq KO/p \wedge \operatorname{fib}((\Phi^H (I\wedge X))_p^\wedge\longrightarrow (\Phi^H(I\wedge X))_p^\wedge)\\ &\simeq KO/p \wedge \operatorname{fib}((KO\wedge \Phi^H X)_p^\wedge\overset{\psi^g-1}{\longrightarrow} (KO\wedge \Phi^H X)_p^\wedge)\\ &\simeq KO/p \wedge L_{KU/p}\Phi^H X \simeq \Phi^H(I/p\wedge X). \end{align}\] Here the first equivalence follows from the fact that \(\Phi^H\) preserves fiber sequences and smash products.
It remains to prove the claim. Recall that for any \(k\in\mathbb{Z}\), there is an Adams operation \(\psi^k\) on equivariant \(K\)-theory of a \(G\)-space defined in [39]. Hirata-Kono [40] shows that the Adams operation \(\psi^k\) induces a stable operation after inverting \(k\) if and only if \((k,|G|)=1\). Then the Adams operation induces a map of ring spectra \[\widetilde{\psi^k}: (KO_{N_p})_p^\wedge\to (KO_{N_p})_{p}^\wedge\] for any \(p\)-group \(N_p\) and \(k\in \mathbb{Z}_p^\times/\{\pm 1\}\), and \[\psi^g=(ECyc_+\wedge\;\operatorname{Inf}_{N_p}^G \widetilde{\psi^g})_p^{\wedge}: I_p^\wedge \to I_p^\wedge\] is a map of ring spectra since \((ECyc_+\wedge\;\operatorname{Inf}_{N_p}^G -)_p^{\wedge}\) is a monoidal functor. ◻
Remark 12. When \(G=N_p\) is a \(p\)-group, \(ECyc_+\wedge KO_{N_p}\) is \(S_{N_p}/p\)-equivalent to \(KO_{N_p}\), so we have \[(ECyc_+\wedge KO_{N_p})_p^{\wedge}\simeq (KO_{N_p})_p^\wedge.\] In this case, the fiber sequence in 11 agrees with the fiber sequences in [12] and [36].
Corollary 1. For any prime \(p\), \(L_{KU_G/p}S_G\simeq (ECyc_+\wedge \operatorname{Inf}_{N_p}^G L_{KU_{N_p}/p}S_{N_p})_p^\wedge.\)
Proof. Since \(\operatorname{Inf}_{N_p}^G KO_{N_p}/p\simeq \operatorname{Inf}_{N_p}^G (KO_{N_p})_p^\wedge/p\), there is an equivalence of \(G\)-spectra \[f:(ECyc_+\wedge \operatorname{Inf}_{N_p}^G KO_{N_p})_p^\wedge \simeq (ECyc_+\wedge \operatorname{Inf}_{N_p}^G (KO_{N_p})_p^\wedge)_p^\wedge.\] So we can rewrite the fiber sequence in 11 by applying the functor \((ECyc_+\wedge \operatorname{Inf}_{N_p}^G(-))_p^\wedge\) to the fiber sequence \[L_{KU_{N_p}/p}S_{N_p}\to (KO_{N_p})_p^\wedge \to (KO_{N_p})_p^\wedge .\] ◻
Remark 13. When \(N_p=e\) for some prime \(p\), let \[F_G=ECyc_+\wedge \operatorname{Inf}_{e}^G L_{KU/p}S.\] For any subgroup \(H\subset G\), \(\Phi^H F_G\) is either trivial or \(L_{KU/p}S\), thus \(\Phi^H F_G\) is \(p\)-complete for all \(H\subset G\). Since \(p\nmid |G|\), by 6, there is an equivalence of non-equivariant spectra \[\Phi^H F_G\simeq (\Phi^H F_G)_p^\wedge \simeq \Phi^H (F_G)_p^\wedge,\] i.e. \(F_G\simeq (F_G)_p^\wedge\) is \(p\)-complete. It follows that for \(p\nmid |G|\), \[L_{KU_G/p}S_G\simeq (F_G)_p^\wedge\simeq ECyc_+\wedge \operatorname{Inf}_{e}^G L_{KU/p}S,\] which agrees with the result of [36].
In order to compute \(\underline{\pi}_* L_{KU_G/p }S_G\), we need to study the fixed points of \((ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)_p^\wedge\) for an \(N_p\)-spectrum \(E\). The main result of this section is 14, which allows us to compute \(\underline{\pi}_* L_{KU_G/p}S_G\) for a finite nilpotent group \(G\).
Lemma 4. Let \(H\) and \(K\) be finite groups with coprime orders, let \(G=H\times K\), and let \(X\in \mathcal{S}^K_*\) be a pointed \(K\)-space. Regard \(X\) as a \(G\)-space via the quotient map \(G\to K\). Then, as an \(H\)-spectrum, \((\Sigma^\infty_G X)^K\simeq \operatorname{Inf}_e^H (\Sigma_K^\infty X)^K\).
Proof. Since \((|H|,|K|)=1\), for any subgroup \(N\subset G\), the Weyl group satisfies \(W_G N\cong W_K L_1\times W_H L_2\), where \(L_1=N\cap K\) and \(L_2=N\cap H\). By tom-Dieck splitting, \[\begin{align} (\Sigma^\infty_{G} X)^{G} & \simeq \bigoplus_{N\subset G}\Sigma^\infty EW_GN_+\wedge_{W_G N} X^N \\ & \simeq \bigoplus_{L_1\subset K, L_2\subset H} \Sigma^\infty ((E W_H L_2)_+\wedge (EW_K L_1)_+)\wedge_{W_H L_2\times W_K L_1} X^{L_1} \\ & \simeq \bigoplus_{L_1\subset K, L_2\subset H} \Sigma^\infty (B W_H L_2)_+\wedge ((EW_K L_1)_+\wedge_{W_K L_1} X^{L_1}) \\ & \simeq \bigoplus_{L_2\subset H}\Sigma^\infty ((BW_H L_2)_+\wedge (\bigoplus_{L_1\subset K} (EW_K L_1)_+\wedge_{W_K L_1} X^{L_1}))\\ & \simeq \bigoplus_{L_2\subset H}(BW_H L_2)_+\wedge (\Sigma_K^\infty X)^K\simeq (S_H)^H\wedge (\Sigma_K^\infty X)^K. \end{align}\] There is a canonical inclusion \[g: (\Sigma^\infty_K X)^K \hookrightarrow (S_H)^H\wedge (\Sigma_K^\infty X)^K \simeq (\Sigma^\infty_G X)^G.\] Let \[f:\operatorname{Inf}_e^H (\Sigma_K^\infty X)^K\to (\Sigma^\infty_G X)^K\] be the map of \(H\)-spectra adjoint to \(g\). We can show that \(f\) is an \(H\)-equivalence. Indeed, since \(S_H=\operatorname{Inf}_e^H \Sigma S^0\), the \(H\)-fixed poinf \(f^H\) is precisely the equivalence \[\begin{align} (\operatorname{Inf}_e^H (\Sigma_K^\infty X)^K)^{H}&\simeq (S_{H}\wedge \operatorname{Inf}_e^H (\Sigma_K^\infty X)^K)^{H} \\ & \simeq (S_{H})^{H}\wedge (\Sigma^\infty_K X)^K \simeq (\Sigma^\infty_{H\oplus K} X)^{H\oplus K}, \end{align}\] and the same argument shows that \(f^L\in Sp\) is an equivalence for every subgroup \(L\subset H\). ◻
Throughout the rest of this section, let \(G\) be a finite nilpotent group with Sylow \(p\)-subgroup \(N_p\), and let \(N\) denote the product of the Sylow \(q\)-subgroups of \(G\) for \(q\neq p\). Then we have \(G=N_p\times N\), and \((|N_p|, |N|)=1\).
Lemma 5. For any subgroup \(H\subset G\), let \(P=H\cap N_p\) and \(L=H\cap N\), then \[(ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)^H\simeq (ECyc^P_+ \wedge \operatorname{Res}_P^{N_p} E)^P \wedge (\bigvee_{T\in Cyc^L}BW_L T_+).\] Here \(Cyc^P\) (resp., \(Cyc^L\)) on the right-hand side of the equivalence is the family of all cyclic subgroups of \(P\) (resp., \(L\)).
Proof. For any \(K\subset G\), let \(Cyc^K\) be the family of all cyclic subgroups of \(K\), then we have \[\operatorname{Res}_K^G ECyc_+\simeq ECyc_+^K.\] Since \(p\nmid |N|\), for any cyclic subgroup \(P_0\subset N_p\) and \(L_0\subset N\), \(P_0\oplus L_0\) is also a cyclic subgroup. By the definition of \(ECyc\), we have \[ECyc_+\simeq ECyc^{N_p}_+\wedge ECyc^N_+.\] Since \(H=P\times L\), there is an equivalence of spectra \(X^H\simeq (X^L)^P\) for any \(G\)-spectrum \(X\).
Consider the \(L\) fixed point of \(ECyc_+\wedge \operatorname{Inf}_{N_p}^G E\) as a \(P\)-spectrum, we have \[\begin{align} (ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)^L & \simeq (ECyc^L_+\wedge \operatorname{Inf}_{P}^{H} (ECyc^P_+\wedge \operatorname{Res}_{P}^{N_p}E))^L \\ & \simeq (ECyc^P_+\wedge \operatorname{Res}_{P}^{N_p}E) \wedge (\Sigma_H^\infty ECyc_+^L)^L. \end{align}\] By 4, \[(\Sigma_H^\infty ECyc_+^L)^L\simeq \operatorname{Inf}_e^{P}(\Sigma_L^\infty ECyc_+^L)^L,\] thus \[\begin{align} (ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)^H&\simeq ((ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)^L)^P \\ &\simeq ((ECyc^P_+\wedge \operatorname{Res}_{P}^{N_p}E) \wedge (\Sigma_H^\infty ECyc_+^L)^L)^P\\ &\simeq ((ECyc^P_+\wedge \operatorname{Res}_{P}^{N_p}E) \wedge \operatorname{Inf}_e^{P}(\Sigma_L^\infty ECyc_+^L)^L)^P\\ &\simeq (ECyc^P_+\wedge \operatorname{Res}_{P}^{N_p}E)^P \wedge (\Sigma_L^\infty ECyc_+^L)^L, \end{align}\] where the last equivalence follows from the formula \[(\operatorname{Inf}_e^P x\cdot y)^P=x\cdot y^P, \quad \forall \;x\in Sp, \;y\in Sp^P.\] By tom-Dieck splitting formula, \[(\Sigma_L^\infty ECyc_+^L)^L\simeq \bigvee_{T\subset L}\Sigma^\infty EW_L T_+\wedge_{W_L T} (ECyc^L_+)^T.\] If \(T\subset L\) is not cyclic, \((ECyc^L_+)^T\) is \(W_L T\)-equivariant contractible, and \[EW_L T_+\wedge_{W_L T} (ECyc^L_+)^T\simeq \ast.\] If \(T\subset L\) is cyclic, then for any subgroup \(K \subset W_L T\), \[(EW_L T_+\wedge (ECyc_+^L)^T)^{K}\simeq\begin{cases} S^0 & K=\{e\},\\ \ast & K\neq \{e\}, \end{cases}\] which implies that there is an equivalence of \(W_L T\)-spaces \(EW_L T_+\wedge (ECyc_+^L)^T\simeq EW_L T_+\), so \[EW_L T_+\wedge_{W_L T} (ECyc^L_+)^T\simeq BW_L T_+.\] Therefore, we have \((\Sigma_L^\infty ECyc_+^L)^L\simeq \bigvee_{T\in Cyc^L}BW_L T_+\), and \[(ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)^H \simeq (\operatorname{Res}_P^{N_p} E\wedge ECyc_+)^P \wedge (\bigvee_{T\in Cyc^L}BW_L T_+).\] ◻
To state our results, we first fix some notation. For any finite group \(G\), let \(A(G)\) be the Burnside ring of \(G\), let \(R\mathbb{Q}(G)\) (resp., \(RU(G)\)) be the rational (resp., complex) representation ring of \(G\). For any subgroup \(H\subset G\), we denote by \(\underline{A}_H\) the \(H\)-Green functor with \(\underline{A}_H(H/K)=A(K)\) for each orbit \(H/K\), and define \(\underline{R\mathbb{Q}}_H\) and \(\underline{RU}_H\) similarly. When \(H=G\), we omit the subscript and write \(\underline{A}\) for \(\underline{A}_G\). There is a natural homomorphism of \(H\)-Green functors \[\mathcal{R}_H: \underline{A}_H\longrightarrow \underline{R\mathbb{Q}}_H\longrightarrow \underline{RU}_H\] sends a finite \(H\)-set to the free rational (complex) vector space on the underlying set, and let \(\underline{J}_H=\ker \mathcal{R}_H\), which is called the Brauer relations. Let \(\underline{A/J}_H := \underline{A}_H/\underline{J}_H\).
By [41], the ideal \(J(G)\) is generated by those elements \(S\in A(G)\) such that \(|S^C|=0\) for every cyclic subgroup \(C\subset G\), hence the number of additive generators of \(A(G)/J(G)\) equals the number of conjugacy classes of cyclic subgroups of \(G\). If \(G\) is a \(p\)-group, the map \(A(G)\to R\mathbb{Q}(G)\) is surjective, and \(\underline{A/J}\cong \underline{R\mathbb{Q}}\). If \(p\nmid |G|\), then after \(p\)-completion, \[(\underline{A/J})_p^\wedge \cong \bigoplus_{H\in \underline{\mathrm{Cyc}}}\mathbb{Z}_p^\wedge,\] where \(\underline{\mathrm{Cyc}}(G/K)=\{\,H\subset K \mid H \text{ is cyclic}\,\}\). The restriction maps in the Mackey functor on the right-hand side are the natural projections, and the transfer maps are the natural inclusions.
Now we can compute the coefficients of \((ECyc_+ \wedge \operatorname{Inf}_{N_p}^G E)_p^\wedge\) for an \(N_p\)-spectrum \(E\) satisfying \(E/p\simeq ECyc^{N_p}_+ \wedge E/p\).
Proposition 14. Let \(G=N_p\oplus N\) be a finite nilpotent group with Sylow \(p\)-subgroup \(N_p\), and let \(E\) be an \(N_p\)-spectrum such that \(E/p\simeq ECyc^{N_p}_+ \wedge E/p\). There is an isomorphism of \(G\)-Mackey functors \[\underline{\pi}_*(ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)_p^\wedge\cong (\underline{\pi}_* E \otimes \underline{A/J}_N)_p^\wedge.\] Here the \(N_p\)-Mackey functor \(\underline{\pi}_* E\) is regarded as a \(G\)-Mackey functor via the canonical quotient map \(G\to N_p\), and the \(N\)-Mackey functor \(\underline{A/J}_N\) is regarded as a \(G\)-Mackey functor via \(G\to N\).
In particular, when \(E=L_{KU_{N_p}/p}S_{N_p}\), there is an isomorphism \[\underline{\pi}_*(L_{KU_G/p}S_G)\cong \underline{\pi}_*L_{KU_{N_p/p}}S_{N_p}\otimes_{\mathbb{Z}_p} (\underline{A/J}_N)_p^\wedge.\]
Proof. For any subgroup \(H\subset G\), let \(P=H\cap N_p\) and \(L=H\cap N\), then \(H=P\times L\). By 5, there is an equivalence \[(ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)^H \simeq (ECyc^P_+\wedge E)^P \wedge (\bigvee_{T\in Cyc^L}BW_L T_+).\] After \(p\)-completion, \((BW_L T)_p^\wedge\simeq \ast\) since \(p\nmid |L|\). Moreover, since \(E/p\simeq E/p\wedge ECyc^{N_p}_+\), we have \(E_p^\wedge\simeq (E\wedge ECyc^P_+)_p^\wedge\). Thus \[L_{S/p}(ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)^H \simeq \bigvee_{T\in Cyc^L} (E^P)_p^\wedge ,\] and \[\pi_*^H (ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)_p^\wedge\cong ((A/J)(L) \otimes \pi_*^P E)_p^\wedge.\]
We also need to determine the restriction and transfer homomorphism. For any \(P_1\subset P_2\subset N_p\), without loss of generality, we can assume that \(N=\{e\}\), then \(\operatorname{Res}_{P_1}^{P_2}\) and \(\operatorname{Tr}_{P_1}^{P_2}\) are inherited from those in \(\underline{\pi}_* E\). For \(N_1\subset N_2\subset N\), assume that \(N_p=\{e\}\), \(\operatorname{Res}_{N_1}^{N_2}\) is the natural projection, and \(\operatorname{Tr}_{N_1}^{N_2}\) is the natural inclusion. Thus \[\underline{\pi}_*(ECyc_+\wedge \operatorname{Inf}_{N_p}^G E)_p^\wedge\cong (\underline{\pi}_* E \otimes \underline{A/J}_N)_p^\wedge.\]
In particular, when \(E=L_{KU_{N_p}/p}S_{N_p}\), 1 implies that \(L_{KU_{N_p}/p}S_{N_p}\simeq (ECyc^{N_p}_+\wedge L_{KU_{N_p}/p}S_{N_p})_p^\wedge\), and hence \(L_{KU_{N_p}/p}S_{N_p}\) satisfies the assumption of this proposition. Therefore, \[\begin{align} \underline{\pi}_*(L_{KU_G/p}S_G) & \cong \underline{\pi}_*(ECyc_+\wedge \operatorname{Inf}_{N_p}^G L_{KU_{N_p}/p}S_{N_p})_p^\wedge \\ & \cong \underline{\pi}_*L_{KU_{N_p}/p}S_{N_p}\otimes_{\mathbb{Z}_p} (\underline{A/J}_N)_p^\wedge . \end{align}\] ◻
By 12, \(KO_{N_p}\) and \(KU_{N_p}\) satisfy the assumption of 14, which leads to the following isomorphism \[\underline{\pi}_*(ECyc_+\wedge \operatorname{Inf}_{N_p}^G KU_{N_p})_p^\wedge\cong (KU_*\otimes \underline{RU}_{N_p}\otimes \underline{A/J}_N)_p^\wedge.\]
Remark 15. If \(G\) is a finite cyclic group, \(L_{KU_G/p}S_G\simeq (\operatorname{Inf}_{N_p}^G L_{KU_{N_p}/p}S_{N_p})_p^\wedge\). for any \(N_p\)-spectrum \(E\), we can compute the \(\mathbb{Z}\)-graded homotopy Mackey functor of \((\operatorname{Inf}_{N_p}^G E)_p^\wedge\) by 4 in a same way. In this case, \[(\operatorname{Inf}_{N_p}^G E)^{P\oplus L}\simeq E^P \wedge (\Sigma_L^\infty S)^L.\] After \(p\)-completion, \(\underline{\pi}_* (\operatorname{Inf}_{N_p}^G E)_p^\wedge\cong \underline{\pi}_* E_p^\wedge \otimes \underline{A}_N\). This is compatible with 14, since \(J(G)=0\) for every cyclic group \(G\).
By 14, we need to study \(\underline{\pi}_* L_{KU_{N_p}/p}S_{N_p}\) for all primes \(p\). When \(p\) is odd, \(\underline{\pi}_*L_{KU_{N_p}/p}S_{N_p}\) is computed in [1]. In this section, we compute \(\underline{\pi}_*L_{KU_{N_2}/2}S_{N_2}\) via the fiber sequence \[L_{KU_{N_2}/2}S_{N_2} \to (KO_{N_2})_2^\wedge \xrightarrow{\;\psi^g-1\;} (KO_{N_2})_2^\wedge\] when \(N_2\) is an abelian \(2\)-group.
Throughout this section, \(N_2\) is abelian. Let \(g\) is a generator of \(\mathbb{Z}_2^\times/\{\pm 1\}\). Building on the study of the \(\psi^g\)-action on \(\underline{RU}\) in [1], we obtain the following lemma.
Lemma 6. \(\psi^g\) acts on \(\underline{RO}_{N_2}\) as a homomorphism of \(N_2\)-Green functor. For any \(K\subset N_2\), \(\psi^g\) acts trivially on \(RO(K;\mathbb{R})\) and permutes the generators of \(RO(K;\mathbb{C})\).
Proof. The first statement follows from the facts that \(\psi^g\) acts on \(\underline{RU}\) as a homomorphism of \(N_2\)-Green functors [1] and that \(\psi^g\) commutes with the complexification map \(c\colon \underline{RO}\to \underline{RU}\).
For any \(K\subset N_2\), every generator \(\tau\in RO(K;\mathbb{R})\) is of \(1\)-dimensional since \(K\) is abelian, which is a sign representation, thus \(\psi^g(\tau)=\tau^g=\tau\).
For any irreducible \(x\in RO(K;\mathbb{C})\), let \(X\in RU(K)\) be the irreducible complex \(K\)-representation whose underlying real representation is \(x\), then \(\psi^g(x)\) is the underlying real representation of \(\psi^g(X)\), which is irreducible by [1]. ◻
For any \(K\subset N_2\), let \(M_K=\hom(K,O(1))\) be the group of irreducible real \(K\)-representation of real type, and let \(C_K\) be the set of all irreducible real \(N_2\)-representation of complex type. Then \[RO(K;\mathbb{R})\cong \mathbb{Z}[M_K],\quad RO(K;\mathbb{C})\cong \mathbb{Z}\{C_K\},\] and \[\pi_* KO_{K} \cong \mathbb{Z}[\eta,\alpha,u^{\pm}]/(2\eta, \eta^3, \eta\alpha, \alpha^2-4u)\{\tau:\tau\in M_K\}\bigoplus \mathbb{Z}[\beta^{\pm}]\{x:x\in C_K\}.\] The Adams operation \(\psi^g\) on \(\underline{\pi}_* KO_{K}\) is given by
\(\psi^g(\eta)=\eta\), \(\psi^g(\alpha)=g^2\alpha\), and \(\psi^g(u)=g^4u\).
for any \(\tau\in M_K\), \(\psi^g \tau=\tau\).
for any \(x\in C_K\), \(\psi^g(x\beta^i)=g^i\psi^g(x)\beta^i\).
For any \(k\in \mathbb{Z}\), define \[\underline{\ker}_2\{k\}:=\ker(\underline{\pi}_k(KO_{N_2})_2^{\wedge}\overset{\psi^g-1}{\longrightarrow}\underline{\pi}_k (KO_{N_2})_2^{\wedge}),\] \[\underline{\mathrm{coker}}_2\{k\}:=\mathrm{coker}(\underline{\pi}_k (KO_{N_2})_2^{\wedge}\overset{\psi^g-1}{\longrightarrow}\underline{\pi}_k (KO_{N_2})_2^{\wedge}).\] These Mackey functors can be computed by following the method of [1].
Lemma 7. As \(N_2\)-Mackey functors, \(\underline{\ker}_2{k}\) and \(\underline{\mathrm{coker}}_2{k}\) are determined by the following values on orbits.
(1) For \(k=0\), and for any \(K\subset N_2\), \[\underline{\ker}_2\{0\}\cong \underline{R\mathbb{Q}}_2^\wedge, \quad \underline{\mathrm{coker}}_2\{0\}(N_2/K)\cong \bigoplus_{\text{cyclic } T\subset K}\mathbb{Z}_2^\wedge.\]
(2) For \(k=8d\) and \(d\neq 0\), \(\underline{\ker}_2\{8d\}\cong 0\), and \[\underline{\mathrm{coker}}_2\{8d\}(N_2/K)\cong RO(K;\mathbb{R})\otimes \mathbb{Z}/2^{4+\nu_2(d)} \oplus (\bigoplus\limits_{C_{2^t}\subset K, t\geq 2} \mathbb{Z}/2^{2+t+\nu_2(d)}),\] where \(\nu_2\) is the \(2\)-adic valuation. (3) For \(k=8d+1\), \[\begin{align} \underline{\ker}_2\{8d+1\} & \cong \underline{RO(-;\mathbb{R})}\otimes \mathbb{Z}/2\{\eta u^d\},\\ \underline{\mathrm{coker}}_2\{8d+1\} & \cong \underline{RO(-;\mathbb{R})}\otimes \mathbb{Z}/2\{[\eta u^d]\}. \end{align}\] Here \([x]\in \underline{\mathrm{coker}}_2\{8d+1\}\) is the equivalent class of the corresponding element \(x\in \underline{\pi}_{8d+1}KO_{N_2}\). For any \(K\subset N_2\), \(\underline{RO(-;\mathbb{R})}(N_2/K)=RO(K;\mathbb{R})\), the restriction and transfer maps are those in \(\underline{RO}\), after quotienting out all elements in \(RO(-;\mathbb{C})\)
(4) For \(k=8d+2\), \[\begin{align} \underline{\ker}_2\{8d+2\}(N_2/K) & \cong RO(K;\mathbb{R})\otimes \mathbb{Z}/2\{\eta^2 u^d\},\\ \underline{\mathrm{coker}}_2\{8d+2\}(N_2/K) & \cong (RO(K;\mathbb{R})\otimes \mathbb{Z}/2\{\eta^2 u^d\}) \oplus (\bigoplus\limits_{C_{2^t}\subset K, t\geq 2} \mathbb{Z}/2^{t}). \end{align}\]
(5) For \(k=8d+4\), \(\underline{\ker}_2\{8d+4\}(N_2/K) \cong 0\), and \[\underline{\mathrm{coker}}_2\{8d+4\}(N_2/K) \cong RO(K;\mathbb{R})/2^3 \bigoplus (\bigoplus_{C_{2^t}\subset K, t\geq 2} \mathbb{Z}/2^{t+1}).\]
(6) For \(k=8d+6\), \(\underline{\ker}_2\{8d+6\}(N_2/K) \cong 0\), and \[\underline{\mathrm{coker}}_2\{8d+6\}(N_2/K) \cong \bigoplus\limits_{C_{2^t}\subset K, t\geq 2} \mathbb{Z}/2^{t}.\]
Proof. Since \(\psi^g\) is a homomorphism of Green functors, the restriction and transfer homomorphisms are inherited from those in \(\underline{\pi}_* (KO_{N_2})_2^\wedge\), and it suffices to compute for any orbit \(N_2/K\).
(1) and (2): When \(k=0\), \(\underline{\ker}_2\{0\}=(\underline{RO}^{\psi^g})_2^\wedge\). For any \(V\in KU(K)\) such that \(\psi^g V=V\), by [36], the character \(\chi_V\) takes values in \(\mathbb{Q}\), i.e., \(V\) is in the image of \(R\mathbb{Q}(K)\to RU(K)\), which factors through \(RO(K)\). Then \[\underline{RO}^{\psi^g}\cong \underline{RU}^{\psi^g}\cong \underline{R\mathbb{Q}_\chi},\] where \[\underline{R\mathbb{Q}_\chi} (N_2/K):=R\mathbb{Q}_\chi(K)=\{V\in RU(K):\chi_V \text{ take values in } \mathbb{Q}\}.\] By [42], the Schur indices for \(N_2\) equal to \(1\) since \(N_2\) is abelian. So \(\underline{R\mathbb{Q}_\chi}\cong \underline{R\mathbb{Q}}\), and \(\underline{\ker}_2\{0\}\cong \underline{R\mathbb{Q}}_2^\wedge\).
By 6, we can decompose \(C_K\) into orbits under \(\psi^g\). Since irreducible rational \(K\)-representations are in bijection with the cyclic subgroups of \(K\), the orbits in \(C_K\) are in bijection with \(\{ C_{2^t}\subset K:t\geq 2\}\), and elements in \(M_K\) are in bijection with the subgroups \(C_{2^t}\subset K\) for \(t=0,1\). Let \([C_{2^t}\subset K]\subset C_K\) denote the orbit corresponds to the subgroup \(C_{2^t}\subset K\). Since the degree of the \(2^t\)-th cyclotomic polynomial \[\Phi_{2^t}(x)=\prod_{\gcd(k,d)=1,\;1\leq k<d}\left(x-e^{2\pi i k/d}\right)\] is \(2^{t-1}\), the orbit \([C_{2^t}\subset K]\) has \(2^{t-2}\) elements.
On every orbit \([C_{2^t}\subset K]\) with \(t\geq 2\), we can choose a basis such that \(\psi^g\) acts as the matrix \[M= \begin{pmatrix} 0 & 1 & & & \\ & 0 & 1& & \\ & & \ddots& \ddots & \\ & & & 0& 1 \\ 1 & & & & 0 \end{pmatrix}.\] For \(k=8d\), \(\psi^g-1\) acts on \(RO(K)_2^\wedge\{u^d\}\) via \(g^{4d}M-I\) on every orbit, which are equivalent to diagonal matrices \[g^{4d}M-I\simeq \begin{pmatrix} 1 & & & & \\ & 1 & & & \\ & & \ddots & & \\ & & & 1 & \\ & & & & g^{2^{t-2}\cdot 4d}-1 \end{pmatrix}\] using a combination of row and column operations. Thus \(\psi^g-1\) is injective when \(d\neq 0\).
When \(d=0\), every orbit contributes a summand of \(\mathbb{Z}_2^\wedge\) in cokernel, so \[\mathrm{coker}_2\{0\}(N_2/K)\cong \bigoplus_{\text{cyclic }T\subset K} \mathbb{Z}_2^\wedge.\] When \(d\neq 0\), every orbit \([C_{2^t}\subset K]\) contributes a summand of \(\mathbb{Z}_2^\wedge/g^{4d}-1\) in cokernel for \(t\leq 1\), and a summand of \(\mathbb{Z}_2^\wedge/g^{2^td}-1\) in cokernel for \(t\geq 2\). Since \(g\) is a generator of \((\mathbb{Z}_2^\wedge)^\times/\{\pm 1\}\cong 1+4\mathbb{Z}_2^\wedge\), then \(1-g^{2^td}\) is a generator of \(2^{2+t+\nu_2(d)}\mathbb{Z}_2^\wedge\), thus \[\mathbb{Z}_2^\wedge/g^{2^t d}-1\cong \mathbb{Z}/2^{2+t+\nu_2(d)},\] and \[\underline{\mathrm{coker}}_2\{0\}(N_2/K)\cong (RO(K;\mathbb{R})\otimes \mathbb{Z}/2^{4+\nu_2(d)}) \oplus (\bigoplus\limits_{C_{2^t}\subset K, t\geq 2} \mathbb{Z}/2^{2+t+\nu_2(d)}).\]
(3) For \(k=8d+1\), \(\pi_{8d+1} KO_{N_2}\cong RO(N_2;\mathbb{R})\{\eta u^d\}/2\), thus \(\psi^g\) acts trivially on \(\underline{\pi}_{8d+1}KO_{N_2}\).
Finally, (4)-(6) follow from the same computation as above. ◻
Remark 16. When \(N_2\) is nonabelian, \(RO(N_2;\mathbb{H})\) is nontrivial. By [42], for any \(V\in RO(N_2;\mathbb{H})\), the Schur index of the complexification of \(V\) over \(\mathbb{Q}\) equals to \(2\). In this case, for any orbit \(N/K\) \[\underline{\ker}_2\{0\}(N/K)\cong R\mathbb{Q}_\chi (K)_2^\wedge\not\cong R\mathbb{Q}(K)_2^\wedge.\] Therefore, \[\underline{\ker}_2\{0\}\cong (\underline{R\mathbb{Q}_{\chi}}_{N_2})_2^\wedge \not\cong (\underline{R\mathbb{Q}}_{N_2})_2^\wedge.\] Furthermore, for any \(V\in RO(N_2;\mathbb{R})\), the Schur index of the complexification of \(V\) over \(\mathbb{Q}\) equals to \(1\), which implies that \[\begin{align} \underline{\ker}_2\{1\}(N_2/K)&\cong RO(K;\mathbb{R})^{\psi^g}/2 \cong RO(K;\mathbb{R})\cap R\mathbb{Q}(K)/2,\\ \underline{\mathrm{coker}}_2\{1\}(N_2/K) &\cong \mathbb{Z}/2\otimes_{\mathbb{Z}_2}\underline{\mathrm{coker}}_2\{0\}(N_2/K)/(RO(K;\mathbb{C})\oplus RO(K;\mathbb{H})). \end{align}\] Here \(V\in R\mathbb{Q}(K)\) is regarded as a real representation via the canonical inclusion \(R\mathbb{Q}(K)\to RO(K)\).
For example, if \(N_2=Q_8\), then \(N_2\) has four one-dimensional irreducible complex representations \(\rho_i\), \(1\leq i\leq 4\), and one two-dimensional irreducible complex representation \(\theta\). The Schur indices of the \(\rho_i\) are \(1\), and the Schur index of \(\theta\) is \(2\). Therefore, after suitably choosing the \(\rho_i\), we have \[R\mathbb{Q}_\chi (N_2)\cong \mathbb{Z}\{\rho_1,\rho_2,\rho_3+\rho_4,\theta\}, \qquad R\mathbb{Q}(N_2)\cong \mathbb{Z}\{\rho_1,\rho_2,\rho_3+\rho_4,2\theta\}.\] As a result, \(\underline{\ker}_2\{0\}\cong (\underline{R\mathbb{Q}_{\chi}}_{Q_8})_2^\wedge\), and \[\underline{\ker}_2\{1\}(Q_8/Q_8)\cong \underline{\mathrm{coker}}_2\{1\}(Q_8/Q_8) \cong \mathbb{Z}/2\{\rho_1,\rho_2, \rho_3+\rho_4\}.\]
For an abelian \(2\)-group \(N_2\) and any \(k\in \mathbb{Z}\), we can compute \(\underline{\pi}_k L_{KU_{N_2}/2}S_{N_2}\) via the short exact sequence: \[0\longrightarrow \underline{\mathrm{coker}}_2\{k+1\}\longrightarrow \underline{\pi}_k L_{KU_{N_2}/2}S_{N_2} \longrightarrow \underline{\ker}_2\{k\}\longrightarrow 0.\] By 7, we need to solve the extension problems when \(k=0\) and \(k=8d+1\).
When \(k=0\), we need to study the exact sequence \[0\longrightarrow \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2\xrightarrow{i} \underline{\pi}_0 L_{KU_{N_2}/2}S_{N_2}\xrightarrow{\pi} (\underline{R\mathbb{Q}}_{N_2})_2^\wedge\longrightarrow 0.\] Since \(N_2\) is a \(2\)-group, \(\underline{R\mathbb{Q}}_{N_2}\cong \underline{A/J}_{N_2}\), and the Hurewicz map \[(\underline{A}_{N_2})_2^\wedge \to \underline{\pi}_0 L_{KU_{N_2}/2}S_{N_2}\xrightarrow{\pi} (\underline{R\mathbb{Q}}_{N_2})_2^\wedge\] induces a morphism of Mackey functors \[\theta_{N_2}:\underline{J}_{N_2}\longrightarrow \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2.\]
Lemma 8. With notations as above, there is a natural isomorphism of Mackey functors \[\underline{\pi}_0 L_{KU_{N_2}/2}S_{N_2}\cong \frac{(\underline{A}_{N_2})_2^\wedge \oplus \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2}{\{j-\theta_{N_2}(j):j\in (\underline{J}_{N_2})_2^\wedge\}}.\]
Proof. Since \(\underline{A}_{N_2}\) is a representable \(N_2\)-Mackey functor, there is no nontrivial extension of \(\underline{A}_{N_2}\) by \(\underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2\). Consider the following commutative diagram with exact rows: \[\begin{tikzcd}[column sep=small] 0 \arrow[r] & \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2 \arrow[r] \arrow[d, "\cong"'] & (\underline{A}_{N_2})_2^\wedge\oplus \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2 \arrow[r,"p_2"] \arrow[d, "\Phi"] & (\underline{A}_{N_2})_2^\wedge \arrow[r] \arrow[d, "\mathcal{R}_{N_2}"] \arrow[ld,"h"] & 0 \\ 0 \arrow[r] & \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2 \arrow[r,"i"] & \underline{\pi}_0 L_{KU_{N_2}/2}S_{N_2} \arrow[r,"\pi"] & (\underline{R\mathbb{Q}}_{N_2})_2^\wedge \arrow[r] & 0 . \end{tikzcd}\] Here \(h\) is the Hurewicz map, \(\mathcal{R}_{N_2}\) is the linearization map, and \[\Phi(p,c)=h(p)+i(c).\] \(\Phi\) is a morphism of Mackey functors since both \(h\) and \(i\) are. It is surjective by the five lemma. If \(\Phi(j,c)=0\), then applying \(\pi\) gives \[0=\pi\Phi(j,c)=\pi h(j)=\mathcal{R}_{N_2}(j),\] so \(j\in (\underline{J}_{N_2})_2^\wedge\). By definition of \(\theta_{N_2}\), \[0=h(j)+i(c)=i(\theta_{N_2}(j)+c).\] Since \(i\) is injective, \(c=-\theta_{N_2}(j)\). Thus \[\ker(\Phi)=\{(j,-\theta_N(j)):j\in J_N\}.\] Since all maps involved are Mackey functor maps, the displayed kernel is a sub-Mackey functor and the quotient is a Mackey functor quotient. ◻
Let \(V:=C_2\times C_2\), and let \(A,B,C\subset V\) be the three subgroups of order two. Then \(J(V)\cong \mathbb{Z}\) is freely generated by \[X_V=([V/A]-1)([V/B]-1)([V/C]-1)-1.\] Following a helpful comment of Balderrama, we have the following lemma.
Lemma 9. Let \(\rho_V\) be the regular real \(V\)-representation. Then \[\theta_V(X_V)=\eta\cdot \rho_{V} \in RO(V;\mathbb{R})/2\cdot \eta.\]
Proof. Note that \(\operatorname{Res}^V_H X_V=0\) for every proper subgroup \(H \subset V\). Hence \(\theta_V(X_V)\) restricts to zero on all proper subgroups. The common kernel of the restriction maps \[RO(V;\mathbb{R})/2\longrightarrow \prod_{|H|=2} RO(H;\mathbb{R})/2\] is generated by \(\rho_V\). Thus \(\theta_V(X_V)\) is either \(0\) or \(\eta\rho_{V}\). Szymik [41] shows that \(X_V\) has nontrivial Hurewicz image, so \(\theta_V(X_V)=\eta\rho_V\). ◻
Consequently, when \(N_2\cong V\), at the top orbit one has an isomorphism of \(A(V)\)-modules \[\pi_0^V L_{KU_V/2}S_V \cong \frac{A(V)^{\wedge}_2\oplus RO(V;\mathbb{R})\{\eta\}/2}{\langle (X_V-\eta\rho_{V})\rangle}.\]
For an arbitrary finite abelian \(2\)-group \(N_2\), it follows from [43] that all Brauer relations of \(N_2\) are \(\mathbb{Z}\)-linear combinations of relations lifted from subquotients isomorphic to \(C_2\times C_2\). More precisely, for all \[L\subset K\subset N_2, \qquad K/L\cong C_2\times C_2,\] the virtual \(N_2\)-sets \[X_{K,L}=\operatorname{Tr}_K^{N_2}\operatorname{Inf}_{K/L}^{K}(X_{K/L})\in J(N_2)\] generate \(J(N_2)\). This determines the \(N_2\)-Mackey functor \(\underline{J}_{N_2}\).
Lemma 10. For every such subquotient \(K/L\cong C_2\times C_2\), \[\theta_{N_2}(X_{K,L}) = \eta\cdot \operatorname{Tr}_K^N\operatorname{Inf}_{K/L}^{K}(\rho_{K/L})= \eta\sum_{\substack{\chi:N_2\to \{\pm1\}\\ L\subseteq \ker(\chi)}}\chi \in RO(N_2;\mathbb{R})/2\cdot\eta.\]
Proof. The map \(\theta\) is defined by the Hurewicz image and is natural for transfer and for inflation. Therefore \[\begin{align} \theta_{N_2}\big(\operatorname{Tr}_K^N\operatorname{Inf}_{K/L}^{K}(X_{K/L})\big) & = \operatorname{Tr}_K^{N_2}\operatorname{Inf}_{K/L}^{K}\big(\theta_{K/L}(X_{K/L})\big)\\ & = \eta\cdot \operatorname{Tr}_K^{N_2}\operatorname{Inf}_{K/L}^{K}(\rho_{K/L}). \end{align}\] As a real \(N_2\)-representation, \(\operatorname{Tr}_K^{N_2}\operatorname{Inf}_{K/L}^{K}(\rho_{K/L})\) is isomorphic to \(\mathbb{R}\{N_2/L\}\). Its image in \(RO(N_2;\mathbb{R})/2\) is the sum of all real one-dimensional characters of \(N_2\) that are trivial on \(L\), which gives the result. ◻
Then we can determine the \(N_2\)-Mackey functor \(\underline{\pi}_0 L_{KU_{N_2}/2}S_{N_2}\).
Proposition 17. Let \(N_2\) be a finite abelian \(2\)-group. There is an isomorphism of \(N_2\)-Mackey functors \[\underline{\pi}_0 L_{KU_{N_2}/2}S_{N_2} \cong \frac{(\underline{A}_{N_2})_2^\wedge \oplus \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2}{\{j-\theta_{N_2}(j):j\in (\underline{J}_{N_2})_2^\wedge\}}.\] The map \(\theta_{N_2}\) is determined on the generators \(X_{K,L}\) by \[\theta_{N_2}(X_{K,L}) = \eta\sum_{\substack{\chi:{N_2}\to\{\pm1\}\\ L\subseteq\ker(\chi)}}\chi.\]
Proof. If a relation \(j\in J(N_2)\) is written as an integral combination \[j=\sum_{\substack{K,L\\ K/L\cong C_2\times C_2}} a_{K,L}X_{K,L},\] then \[\theta_{N_2}(j) = \eta\sum_{\substack{K,L\\ K/L\cong C_2\times C_2}} (a_{K,L}\bmod 2) \sum_{\substack{\chi:{N_2}\to\{\pm1\}\\ L\subseteq\ker(\chi)}}\chi .\] The expression of \(j\) in terms of the generators \(X_{K,L}\) is not canonical, but the resulting value of \(\theta_{N_2}(j)\) is canonical because \(\theta_{N_2}\) is defined as the Hurewicz image of \(j\).
At a general orbit \(N_2/H\), the Mackey functor map is obtained by replacing \(N_2\) with \(H\): \[\theta_{N_2}(N_2/H):J(H)\longrightarrow RO(H;\mathbb{R})/2\cdot\eta,\] and the same formula applies to subquotients \(L\subset K\subset H\) with \(K/L\cong C_2\times C_2\). ◻
When \(k=8d+1\), the short exact sequence has the form \[0\longrightarrow \underline{\mathrm{coker}}_2\{8d+2\} \xrightarrow{i} \underline{\pi}_{8d+1} L_{KU_{N_2}/2}S_{N_2}\xrightarrow{\pi} \underline{RO(-;\mathbb{R})}_{N_2}\{\eta u^d\}/2\longrightarrow 0.\] As in the non-equivariant case, after evaluating at \(N_2/K\), the sequence splits as abelian groups.
Lemma 11. For any subgroup \(K\subset N_2\), there is an isomorphism of abelian groups \(\pi_{8d+1}^K L_{KU_{N_2}/2}S_{N_2} \cong \underline{\mathrm{coker}}_2\{8d+2\}(N_2/K) \oplus \underline{\ker}_2\{8d+1\}(N_2/K)\).
Proof. The exact sequence has the form \[0\to \underline{\mathrm{coker}}_2\{8d+2\}(N_2/K) \to \pi_k^K L_{KU_{N_2}/2}S_{N_2} \to RO(K;\mathbb{R})\{\eta u^d\}/2\to 0,\] with all elements in \(\underline{\mathrm{coker}}_2\{8d+2\}(N_2/K)\) are torsion. When \(d=0\), \(\eta\in \pi_1 KO\) is the Hurewicz image of the Hopf element. The composite homomorphism \[\pi_1^K S \to \pi_1^K L_{KU_{N_2/2}}S \to \pi_1^K (KO_{N_2})_2^\wedge\] sends \(\eta\) to \(\epsilon\eta\in \pi_1^K KO_{N_2}\) and \(\operatorname{Tr}_{\ker \tau}^K(1) \eta\) to \((\epsilon+\tau)\eta \in\pi_1^K KO_{N_2}\). Then \(\tau\eta\in RO(K;\mathbb{R})\{\eta\}/2\) lifts to an element in \(\pi_1^K L_{KU_{N_2}/2}S\) of order \(2\). Thus the extension problem when \(k=1\) is trivial. For general \(d\in \mathbb{Z}\), consider the element \(\eta u^d\), a same argument shows that all the extension problems are trivial. ◻
This sequence splits after evaluating at each orbit \(N_2/K\), as a sequence of abelian groups. However, this does not imply that the sequence splits as Mackey functors. It remains to determine the \(\underline{A}\)-module structure of \(\underline{\pi}_k L_{KU_{N_2}/2}S_{N_2}\). Consider the map \[\mathcal{I}_{N_2}:\underline A_{N_2} \overset{\mathcal{R}_{N_2}}{\longrightarrow} \underline{RO}_{N_2}\longrightarrow \underline{RO(-;\mathbb{R})}_{N_2}/2.\] By the universal property of the Burnside Mackey functor, \(\mathcal{I}_{N_2}\otimes \mathbb{Z}\{\eta u^d\}\) can be lifted to \[\widetilde{h}:\underline A_{N_2}\longrightarrow \underline{\pi}_{8d+1} L_{KU_{N_2}/2}S_{N_2}.\] On the orbit \(N_2/K\), it is given by \[\widetilde{h}_K([K/H]) = \operatorname{Tr}_H^K\operatorname{Res}_H^{N_2}(\eta_{N_2}u^d)\] for \(H\subset K\), where \(\eta_{N_2}u^d\in \pi_{8d+1}^{N_2} L_{KU_{N_2}/2}S_{N_2}\) is the lift of \(\eta u^d\) as in the proof of 11. This map is determined by the action of \(\underline{A}_{N_2}\) on \(\eta_{N_2}\in \underline{\pi}_1 L_{KU_{N_2}/2}S_{N_2}\). Since \(2\eta=0\), this morphism factors through \(\underline A_{N_2}/2\). Let \[\underline{I}_{8d+1}:=\ker\big(\underline A_{N_2}/2 \to \underline{RO(-;\mathbb{R})}_{N_2}\{\eta u^d\}/2\big)\] denote its kernel, then \(\widetilde{h}\) induces \[\theta_{8d+1}: \underline{I}_{8d+1} \longrightarrow \underline{\mathrm{coker}}_2\{8d+2\}.\] With this notation, similarly to 8, the extension in degree \(8d+1\) is given by \[\underline{\pi}_{8d+1}L_{KU_{N_2}/2}S_{N_2} \cong \frac{ \underline A_{N_2}/2\oplus \underline{\mathrm{coker}}_2\{8d+2\} }{ \left\{ r-\theta_{8d+1}(r): r\in \underline{I}_{8d+1} \right\} }.\] It remains to determine \(\theta_{8d+1}\).
Lemma 12. Let \(K\subset N_2\). The group \[I_{8d+1}(N_2/K) = \ker\left( A(K)/2 \longrightarrow RO(K;\mathbb{R})\{\eta u^d\}/2 \right)\] is generated by the following two types of elements.
Let \(2K=\{2x:x\in K\}\subset K\). If \(H,H'\subset K\) satisfy \[H+2K=H'+2K,\] then \(D_{H,H'}=[K/H]+[K/H']\) lies in \(I_{8d+1}(N_2/K)\).
Let \(2K\subseteq L\subset T\subset K\) with \(T/L\cong C_2\times C_2\). Let \(M_1,M_2,M_3\) be the three intermediate subgroups between \(L\) and \(T\). Then \[B_{T,L} = [K/L]+[K/M_1]+[K/M_2]+[K/M_3]\] lies in \(I_{8d+1}(N_2/K)\).
Moreover, the elements of types \((i)\) and \((ii)\) generate \(I_{8d+1}(N_2/K)\) as an \(\mathbb{F}_2\)-vector space.
Proof. The map \[A(K)/2\longrightarrow RO(K;\mathbb{R})\{\eta u^d\}/2\] only depends on the image of a subgroup in \(K/2K\). Indeed, for \(H\subset K\), \[[K/H]\longmapsto \left( \sum_{\substack{\chi:K\to\{\pm1\}\\ H\subseteq \ker(\chi)}}\chi \right)\eta u^d,\] and the condition \(H\subseteq \ker(\chi)\) is equivalent to \(H+2K\subseteq \ker(\chi)\). Hence the elements \(D_{H,H'}\) are in the kernel, and after quotienting by these relations we may identify the source of the map with \(A(K/2K)/2\).
Put \(\overline{K}=K/2K\). It remains to determine the kernel of \[A(\overline{K})/2 \longrightarrow RO(\overline{K};\mathbb{R})/2.\] For a subgroup \(U\subset \overline{K}\), the \(\overline{K}\)-set \(\overline{K}/U\) maps to \[\sum_{\substack{\chi:\overline{K}\to\{\pm1\}\\ U\subseteq\ker(\chi)}}\chi.\] If \(U\subset W\subset \overline{K}\) and \(W/U\cong C_2\times C_2\), with intermediate subgroups \(U_1,U_2,U_3\), then \[\overline{B}_{W,U}:=[\overline{K}/U]+[\overline{K}/U_1]+[\overline{K}/U_2]+[\overline{K}/U_3]\] maps to zero. Indeed, a character trivial on \(U\) is either trivial on \(W\), in which case it is counted four times, or has kernel one of the three intermediate subgroups, in which case it is counted twice.
Let \(R\subset A(\overline{K})/2\) be the subgroup generated by the elements \(\overline{B}_{W,V}\). Then the quotient \(A(\overline{K})/(2,R)\) is generated by the class \([\overline{K}/\overline{K}]\) and the classes \([\overline{K}/H]\) with \(\overline{K}/H\cong C_2\). Their images are \(1+\chi\) for all characters \(\chi\) of \(\overline{K}\), and these elements form a basis of \(RO(\overline{K};\mathbb{R})/2\). Therefore there are no further relations. Pulling this description back along \(K\to K/2K\) gives the stated generators. ◻
Lemma 13. Let \(K\subset N_2\), and let \(H,H'\subset K\) satisfy \(H+2K=H'+2K\). For the first-type generator \(D_{H,H'}\) in 12, consider the projection onto the summand of \(\underline{\mathrm{coker}}_2\{2\}\) indexed by a subgroup \(M\cong C_{2^t}\subset K\). Let \(\rho_M\) be the irreducible rational \(K\)-representation corresponding to \(M\). Then \[\operatorname{Pr}_M\theta_{8d+1}(D_{H,H'}) = \begin{cases} 2^{t-1}, & t\geq 2 \text{ and exactly one of } H, H' \text{ is }\\ & \text{contained in } \ker \rho_M,\\ 0, & \text{otherwise.} \end{cases}\]
Proof. First we consider the case \(K\cong C_4\), the only generator is \[D_{C_2,\{e\}} = [C_4/C_2] + [C_4/\{e\}].\] Let \(\rho\) be the faithful two-dimensional roration \(C_4\)-representation. The computation of \(\underline{\pi}_1 L_{KU_{C_4}/2}S_{C_4}\) in 8 shows that \[\rho\cdot \eta \neq 0\in \pi_1^{C_4}L_{KU_{C_4}/2}S_{C_4},\] and \[\theta_{8d+1}(D_{C_2,\{e\}})=D_{C_2,\{e\}} \cdot \eta u^d = (\operatorname{Tr}_{e}^{C_4}(1)+\operatorname{Tr}_{C_2}^{C_4}(1))\eta u^d = [2\rho \beta u^d].\]
In general, let \(K\subset N_2\) and \(H,H'\subset K\) such that \(H+2K=H'+2K\), we can reduce the computation to \(D_{C_2,\{e\}}\). Let \(J = H +2 K\), then in \(A(K)/2\) we have \[[K/H] + [K/H'] = ([K/H] + [K/J]) + ([K/H'] + [K/J]).\] Hence it suffices to study the generators of the form \[[K/L] + [K/(L\langle x^2 \rangle)]\] for some subgroup \(L\subset K\) and \(x\in K\backslash L\). Let \(T = L \langle x \rangle\). Then \[[K/L] + [K/L \langle x^2 \rangle] = \operatorname{Ind}_T^K ([T/L] + [T/L \langle x^2 \rangle]).\] Now \(T/L\) is a cyclic \(2\)-group. Suppose \(T/L \cong C_{2^m}\). In \(A(C_{2^m})/2\), let \(C_{2^i}\) denote the unique subgroup of order \(2^i\). Then \[[C_{2^m}/e] + [C_{2^m}/C_{2^{m-1}}] = \sum_{i=0}^{m-2} ([C_{2^m}/C_{2^i}] + [C_{2^m}/C_{2^{i+1}}]).\] Each term \([C_{2^m}/C_{2^i}] + [C_{2^m}/C_{2^{i+1}}]\) is obtained from the basic class \[[C_4/e] + [C_4/C_2]\] on the subquotient \(C_{2^{i+2}}/C_{2^i} \cong C_4\) by inflation followed by transfer. Since \(\underline{I}_{8d+1}\) is defined by the \(\underline{A}_{N_2}\)-action on \(\eta\in \underline{\pi}_1 L_{KU_{N_2}/2}S_{N_2}\), it commutes with inflation and transfer maps, and we can compute \(\theta_{8d+1}\) of \(D_{H,H'}\) via the value of \(D_{C_2,\{e\}}\). Therefore, for every cyclic \(M\cong C_{2^t} \subset K\), we can compute \(\theta_{8d+1}\) by induction. ◻
Lemma 14. Let \(2K\subset L\subset T\subset K\) with \(T/L\cong C_2\times C_2\). Let \(M_1,M_2,M_3\) be the three intermediate subgroups between \(L\) and \(T\). For the second-type generator in 12 \[B_{T,L} = [K/L]+[K/M_1]+[K/M_2]+[K/M_3] \in I_{8d+1}(N_2/K),\] we have \[\theta_{8d+1}(B_{T,L}) = \eta^2u^d \sum_{\substack{\chi:K\to\{\pm1\}\\ L\subseteq\ker(\chi)}}\chi \in RO(K;\mathbb{R})/2\{\eta^2u^d\}.\] In particular, this value lies entirely in the real-type summand of \(\underline{\mathrm{coker}}_2\{8d+2\}(N_2/K)\).
Proof. \(\theta_{8d+1}\) is defined by the action of \(\underline{A}_{N_2}\) on \(\eta u^d\) in \(\underline{\pi}_{8d+1} L_{KU_{N_2}/2}S_{N_2}\), so we have \[\theta_{8d+1}(B_{T,L}) = (\operatorname{Tr}_{L}^{K}(1) + \operatorname{Tr}_{M_1}^{K}(1) + \operatorname{Tr}_{M_2}^{K}(1) + \operatorname{Tr}_{M_3}^{K}(1))\cdot \eta u^d.\] When \(T\cong V=V=C_2\times C_2\), \(B_{T,\{e\}}\) is the image of \(X_V\in J(V)\) under the quotient \(J(V)\to J(V)/2\), so \[\theta_{8d+1}(B_{V,\{e\}}) = X\cdot \eta u^d = \eta^2u^d \rho_{V}.\] For a genaral \(L\) and \(T\), inflating along \(T\to T/L\) and then transferring from \(T\) to \(K\) gives \[\theta_{8d+1}(B_{T,L}) = \eta^2u^d\cdot \operatorname{Tr}_{T}^{K} \operatorname{Inf}_{T/L}^{T} (\rho_{T/L}).\] Since \(2K\subseteq L\), every character of \(K\) trivial on \(L\) is a real one-dimensional character. Therefore, \[\operatorname{Tr}_{T}^{K} \operatorname{Inf}_{T/L}^{T} (\rho_{\mathrm{reg},T/L})= \sum_{\substack{\chi:K\to\{\pm1\}\\ L\subseteq\ker(\chi)}}\chi \in RO(K;\mathbb{R})/2.\] This proves the claimed formula. ◻
Proposition 18. With the notation above, there is an isomorphism of \(N_2\)-Mackey functors\[\underline{\pi}_{8d+1}L_{KU_{N_2}/2}S_{N_2} \cong \frac{ \underline A_{N_2}/2 \oplus \underline{\mathrm{coker}}_2\{8d+2\} }{ \left\{ r-\theta_{8d+1}(r): r\in I_{8d+1} \right\} }.\] The map \(\theta_{8d+1}:I_{8d+1} \to \underline{\mathrm{coker}}_2\{8d+2\}\) is completely determined by [lem:I-8d-plus-1-generators,lem:theta-8d-plus-1-type-one,lem:theta-8d-plus-1-type-two].
Proof. The proof is same as the proof of 17. ◻
When \(p=2\), we summarize the computation of \(\underline{\pi}_* L_{KU_{N_p}/p}S_{N_p}\) as following.
Proposition 19. \[\underline{\pi}_*L_{KU_{N_2}/2}S_{N_2} \cong \begin{cases} \frac{(\underline{A}_{N_2})_2^\wedge \oplus \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2}{\{j-\theta_{N_2}(j):j\in (\underline{J}_{N_2})_2^\wedge\}} & k=0\\ \frac{ \underline A_{N_2}/2 \oplus \underline{\mathrm{coker}}_2\{8d+2\} }{\left\{ r-\theta_{8d+1}(r): r\in I_{8d+1} \right\}} & k=8d+1\\ \underline{\mathrm{coker}}_2\{8d+1\} & k=8d, d\neq 0\\ \underline{\ker}_2\{8d+2\} & k=8d+2\\ \underline{\mathrm{coker}}_2\{k+1\} & k=8d+3, 8d+5, 8d+7\\ 0 & \text{otherwise}. \end{cases}\]
For the sake of self-containment, we list the computation of \(\underline{\pi}_* L_{KU_{N_p}/p}S_{N_p}\) in [1] for odd prime \(p\) as following:
Lemma 15. Define \(\underline{\mathrm{coker}}_p\{k\} := \mathrm{coker}\;( \underline{\pi}_k (KU_{N_p})_p^{\wedge} \overset{\psi^g-1}{\longrightarrow} \underline{\pi}_k (KU_{N_p})_p^{\wedge})\) and \(\underline{\ker}_p\{k\} := \ker\;(\underline{\pi}_k(KU_{N_p})_p^{\wedge} \overset{\psi^g-1}{\longrightarrow} \underline{\pi}_k (KU_{N_p})_p^{\wedge})\) for an odd prime \(p\). Then there are isomorphisms \[\underline{\pi}_{2d}L_{KU_{N_p}/p}S_{N_p}\cong \underline{\ker}_p\{2d\},\quad \underline{\pi}_{2d-1}\cong L_{KU_{N_p}/p}S_{N_p}\cong \underline{\mathrm{coker}}_p\{2d\}.\] Moreover, \[\underline{\ker}_p\{2d\}\cong\begin{cases} (\underline{R\mathbb{Q}}_p^\wedge)_{N_p}, & d=0\\ 0, & d\neq 0 \end{cases},\] \[\underline{\mathrm{coker}}_p\{2d\}(N_p/K)\cong\begin{cases} \bigoplus_{\text{cyclic }T\subset K} \mathbb{Z}_p^\wedge, & d=0\\ \mathbb{Z}/p^{\nu_p(g^d-1)} \oplus \bigoplus_{C_{p^k}\subset K, k>0} \mathbb{Z}/p^{k+\nu_p(d)}, & d\neq 0 \end{cases},\] where \(\nu_p\) is the \(p\)-adic valuation.
Proof. The computation of \(\underline{\ker}_p\{k\}\) is given in [36], [1]. The computation of \(\underline{\mathrm{coker}}_p\{k\}\) is given in [1]. The assertion about \(\underline{\pi}_*L_{KU_{N_p}/p}S_{N_p}\) follows from the fact that \(\pi_* KU_{N_p}\) is concentrated in even degrees. ◻
In this section, let \(G\) be a finite abelian group with Sylow \(p\)-subgroup \(N_p\), and let \(N\) denote the product of the Sylow \(q\)-subgroups of \(G\) for \(q\neq p\). We compute \(\underline{\pi}_*L_{KU_G}S_G\) in 21.
In order to compute \(\underline{\pi}_0L_{KU_G/p}S_G\), we need the following lemma.
Lemma 16. For any \(K\subset N_p\) and \(H\subset N\), \[A/J(K)\otimes A/J(H)\cong A/J(K\oplus H).\] Then there is an isomorphism of \(G\)-Mackey functors \(\underline{A/J}_{N_p}\otimes \underline{A/J}_N\cong \underline{A/J}_G\).
Proof. We first show that the map \(A(K)\otimes A(H)\to A(K\oplus H)\) is an isomorphism. For any \(X\in A(K)\) and \(Y\in A(H)\), there is a natural action of \(K\oplus H\) on \(X\times Y\); thus, \((X,Y)\mapsto X\times Y\) induces a ring homomorphism \[f:A(K)\otimes A(H)\to A(K\oplus H).\] Conversely, let \(Z\in A(K\oplus H)\), \(Z\) can be uniquely expressed as a direct sum of \(K \oplus H\)-orbits \[Z \cong \bigsqcup_{i=1}^r Z_i,\] where for each \(Z_i\), there exists \(S \subset K \oplus H\) such that \(Z_i = (K \oplus H)/S\). Let \(S_1\) be the image of \(S\) under the projection to \(K\), and \(S_2\) be the image of \(S\) under the projection to \(H\). Since \(|K|\) and \(|H|\) are coprime, \(|S| = |S_1| \times |S_2|\), which implies \(S \cong S_1 \oplus S_2\) and \(Z_i \cong K/S_1 \times H/S_2\). It follows that any \(Z\in A(K\oplus H)\) can be expressed uniquely as \[Z\cong \bigsqcup_{i=1}^r X_i\times Y_i,\] where \(X_i\), \(Y_i\) are \(K\)-orbits and \(H\)-orbits, respectively. Therefore, there is a ring homomorphism \[g:A(K\oplus H)\to A(K)\otimes A(H),\quad Z\mapsto \sum_{i=1}^r (X_i,Y_i),\] which is the inverse of \(f\). Since any subgroup \(S \subset K \oplus H\) satisfies \(S \cong (S\cap K) \oplus (S\cap H)\), \(f\) commutes with \(\operatorname{Res}_{S}^{K \oplus H}\) and \(\operatorname{Tr}_{S}^{K \oplus H}\), \(f\) induces an isomorphism of \(G\)-Green functors \(\underline{A}_{N_p} \otimes \underline{A}_N \to \underline{A}_G\).
Consider the following commutative diagram with exact rows \[\begin{tikzcd} 0 \ar[r] & \underline{\ker}(\mathcal{R}_{N_p}\otimes \mathcal{R}_N) \ar[r] \ar[d,dashed] & \underline{A}_{N_p}\otimes\underline{A}_N \ar[r,"\mathcal{R}_{N_p}\otimes \mathcal{R}_N"]\ar[d,"\cong"] & \underline{RU}_{N_p}\otimes \underline{RU}_N \ar[d,"\cong"]\\ 0 \ar[r] & \underline{J}_G \ar[r] & \underline{A}_G \ar[r,"\mathcal{R}_G"] &\underline{RU}_G \end{tikzcd},\] there exists an isomorphism \(\underline{\ker}(\mathcal{R}_{N_p}\otimes \mathcal{R}_N)\cong \underline{J}_G\) indicated by the dashed arrow. Thus \[\underline{A/J}_{N_p}\otimes \underline{A/J}_N\cong \mathrm{Im}(\mathcal{R}_{N_p}\otimes \mathcal{R}_N)\cong \mathrm{Im}\mathcal{R}_G\cong \underline{A/J}_G.\] Here the first isomorphism follows from the fact that \(RU(K)\) is a free abelian group for every subgroup \(K\subset G\), and the second follows from the Five Lemma. ◻
Combining the above calculations, we give \(\underline{\pi}_* L_{KU_G/p}S\) as following:
Proposition 20. Let \(G = N_p \oplus N\), where \(N_p\) is the Sylow \(p\)-subgroup. For \(p=2\), \[\underline{\pi}_n(L_{KU_G/2}S_G) \cong \begin{cases} \frac{(\underline{A}_{N_2})_2^\wedge \oplus \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2}{\{j-\theta_{N_2}(j):j\in (\underline{J}_{N_2})_2^\wedge\}} \otimes \underline{A/J}_N, & n=0\\ \underline{\mathrm{coker}}_2\{0\}\otimes \underline{A/J}_N & n=-1\\ \underline{\mathrm{coker}}_2\{8d+1\}\otimes \underline{A/J}_N & n=8d, d\neq 0\\ \underline{\pi}_{8d+1} L_{KU_{N_2}/2}S_{N_2}\otimes \underline{A/J}_N & n=8d+1\\ \underline{\ker}_2\{8d+2\}\otimes \underline{A/J}_N & n=8d+2\\ \underline{\mathrm{coker}}_2\{n+1\}\otimes \underline{A/J}_N & n=8d+3, 8d+5,\\&\qquad 8d+7,n\neq -1\\ 0 & \text{otherwise}. \end{cases}\] For odd prime \(p\), \[\underline{\pi}_n(L_{KU_G/p}S_G) \cong \begin{cases} (\underline{A/J}_G)_p^\wedge, & n=0\\ \underline{\mathrm{coker}}_p\{0\}\otimes \underline{A/J}_N & n=-1\\ \underline{\mathrm{coker}}_p\{2d\}\otimes \underline{A/J}_N & n=2d-1, d\neq 0\\ 0 & \text{otherwise} \end{cases}.\] Here \(\underline{\ker}_p\{k\}\) and \(\underline{\mathrm{coker}}_p\{k\}\) are listed in [lem:psi-1,lem:psi-1 for odd p], \(\theta_{N_2}\) is given in 17, and \(\underline{\pi}_{8d+1} L_{KU_{N_2}/2}S_{N_2}\) is listed in 18. In particular, \(\underline{\pi}_n(L_{KU_G/p}S)\) is torsion-free when \(n = -1\). For \(n \neq 0, -1\), \(\underline{\pi}_n(L_{KU_G/p}S)\) are all torsion groups.
Proof. The computation follows from 14 and 19. Note that by 16, \[(\underline{R\mathbb{Q}}_{N_p}\otimes \underline{A/J}_N)_p^\wedge\cong (\underline{A/J}_{N_p}\otimes \underline{A/J}_N)_p^\wedge\cong (\underline{A/J}_G)_p^\wedge.\] Then \(\underline{\pi}_0(L_{KU_G/p}S_G) \cong (\underline{A/J}_G)_p^\wedge\). ◻
Finally, we compute \(\underline{\pi}_*L_{KU_G}S_G\) via the Arithmetic fracture square \[\begin{tikzcd}[ampersand replacement=\&] L_{KU_G}S_G \ar[r] \ar[d] \& \prod_{p} L_{KU_G/p} S_G \ar[d] \\ L_{KU_G\otimes\mathbb{Q}} S_G \ar[r] \& (\prod_{p} L_{KU_G/p}S_G)_{\mathbb{Q}} . \end{tikzcd}\]
Theorem 21. Let \(G\) be a finite abelian group. For any prime \(p\), let \(N_p\) be the Sylow \(p\)-subgroup of \(G\), and let \(G/N_p\) denote the product of the Sylow \(q\)-subgroups of \(G\) for \(q\neq p\). \[\underline{\pi}_kL_{KU_G}S_G\cong \begin{cases} \underline{A/J}_{G/N_2} \otimes \frac{\underline{A}_{N_2} \oplus \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2}{\{j-\theta_{N_2}(j):j\in \underline{J}_{N_2}\}} & \quad k=0\\ 0 & \quad k=-1\\ \mathbb{Q}/\mathbb{Z}\otimes (\prod_p \underline{\operatorname{coker}}_p\{0\}\otimes \underline{A/J}_{G/N_p}) & \quad k=-2\\ \prod_p \underline{\pi}_kL_{KU_G/p}S_G & \quad \text{otherwise} \end{cases}\] Here \(\theta_{N_2}:\underline{J}_{N_2} \to \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2\) is induced by the Hurewicz map \(\underline{A}_{N_2}\cong \underline{\pi}_0 S_{N_2}\to \underline{\pi}_0 L_{KU_{N_2}/2}S_{N_2}\). The computation of \(\theta_{N_2}\) is carried out in 17.
Proof. For \(KU_G\otimes \mathbb{Q}\)-localization, it follows from [36] that that \(L_{KU_G \otimes \mathbb{Q}} S_G \simeq H(\mathbb{Q} \otimes \underline{A/J})\), which is a \(G\)-equivariant Eilenberg-MacLane spectrum. So \[\underline{\pi}_* L_{KU_G \otimes \mathbb{Q}} S_G \cong \mathbb{Q} \otimes \underline{A/J}\] concentrated in degree \(0\). Furthermore, since \(\underline{\pi}_k \prod_{p} L_{KU_G/p} S_G\) are torsion groups for \(k \neq 0, -1\), \((\prod_{p} L_{KU_G/p} S_G)_{\mathbb{Q}}\) has non-trivial homotopy groups only in degrees \(0\) and \(-1\). Consequently, the long exact sequence on homotopy groups induced by the arithmetic fracture square takes the following form: \[\begin{align} 0\to & \underline{\pi}_0 L_{KU_G}S_G \to \underline{\pi}_0(\prod_{p} L_{KU_G/p} S_G)\bigoplus (\mathbb{Q}\otimes \underline{A/J}) \overset{f}{\longrightarrow} \underline{\pi}_0(\prod_{p} L_{KU_G/p} S_G)\otimes \mathbb{Q} \\ & \to \underline{\pi}_{-1}L_{KU_G}S_G \to \underline{\pi}_{-1}(\prod_{p} L_{KU_G/p} S_G) \overset{g}{\longrightarrow} \underline{\pi}_{-1}(\prod_{p} L_{KU_G/p} S_G)\otimes \mathbb{Q} \\ & \to \underline{\pi}_{-2}L_{KU_G}S_G \to 0, \end{align}\] and for \(k\neq 0.-1,-2\), \[0\to \underline{\pi}_{k}L_{KU_G}S_G \to \underline{\pi}_{k}(\prod_{p} L_{KU_G/p} S_G)\to 0.\] Then for \(k\neq 0.-1,-2\), \(\underline{\pi}_kL_{KU_G}S_G\cong\prod_p \underline{\pi}_kL_{KU_G/p}S_G\).
When \(k=0\), \(\underline{\pi}_0 L_{KU_G}S_G\cong \ker f\). Let \[M=\frac{\underline{A}_{N_2} \oplus \underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2}{\{j-\theta_{N_2}(j):j\in \underline{J}_{N_2}\}},\] we have \(M_2^\wedge \cong \underline{\pi}_0 L_{KU_{N_2}/2}S_{N_2}\). Note that all elements of \(\underline{RO(-;\mathbb{R})}_{N_2}\{\eta\}/2\) are \(2\)-torsion, after \(p\)-completion or rationalization, \(M_p^\wedge \cong (\underline{A/J}_{N_2})_p^\wedge\), and \(M\otimes \mathbb{Q}\cong \underline{A/J}_{N_2}\otimes \mathbb{Q}\). Therefore, \[\underline{A/J}_p^\wedge \cong \underline{A/J}_{G/N_2} \otimes M_p^\wedge, \quad \underline{A/J}\otimes \mathbb{Q} \cong \underline{A/J}_{G/N_2} \otimes M\otimes \mathbb{Q}.\] By 20, \(f\) is given by \[f : (\prod_p M_p^\wedge \bigoplus M\otimes \mathbb{Q})\otimes \underline{A/J}_{G/N_2} \to (\prod_p M_p^\wedge )\otimes \mathbb{Q} \otimes \underline{A/J}_{G/N_2}.\] which is exactly the arithmetic pullback of \(M\otimes \underline{A/J}_{G/N_2}\), then \[\underline{\pi}_0 L_{KU_G}S_G\cong \ker f \cong M\otimes \underline{A/J}_{G/N_2}.\]
When \(k=-1\), it follows from [1] that \(\mathrm{coker} \;f=0\), so \(\underline{\pi}_{-1} L_{KU_G}S_G\cong \ker g\). Here \[g : \underline{\pi}_{-1}(\prod_{p} L_{KU_G/p} S_G) \to \underline{\pi}_{-1}(\prod_{p} L_{KU_G/p} S_G)\otimes \mathbb{Q}\] is an inclusion induced by \(\mathbb{Z}\to \mathbb{Q}\) since \(\underline{\pi}_{-1}(\prod_{p} L_{KU_G/p} S_G)\) is torsion free, thus \(\underline{\pi}_{-1} L_{KU_G}S_G\cong \ker g=0\).
When \(k=-2\), \(\underline{\pi}_{-2} L_{KU_G}S_G\cong \mathrm{coker}\;g\). Therefore, \[\underline{\pi}_{-2} L_{KU_G}S_G\cong \underline{\pi}_{-1}(\prod_{p} L_{KU_G/p} S_G)\otimes \mathbb{Q}/\mathbb{Z}\cong \mathbb{Q}/\mathbb{Z} \otimes (\prod_p \underline{\operatorname{coker}}_p\{0\}\otimes \underline{A/J}_{G/N_p}).\] ◻
Remark 22 (A remark for \(\underline{\pi}_0\)). If \(N_2\) is cyclic, then \(J(N_2)=\{0\}\), and the extension problem for \(k=0\) in 17 is trivial. In this case, \[\underline{\pi}_0 L_{KU_G}S_G\cong \underline{A/J}\bigoplus \underline{A/J}_{G/N_2}\otimes \underline{RO(-;\mathbb{R})}/2.\]
Remark 23. It follows from the proof that 21 applies to every finite nilpotent group \(G\) whose Sylow \(2\)-subgroup \(N_2\) is abelian. When \(N_2\) is non-abelian, the classification of the Brauer relations in [43] implies that it remains to resolve the extension problem for \(N_2\cong D_{2^n}\), in a manner analogous to 9.
In this section, let \(G\) be a finite abelian group, and let \(\mathrm{Cyc}\) denote the family of cyclic subgroups of \(G\). For a fixed prime \(p\), we study the role of \(KU_G/p\) in equivariant chromatic homotopy theory, which yields an alternative description of \(L_{KU_G/p}S_G\) in 24. We then compute \(\underline{\pi}_V L_{KU_G/p}S_G\) for \(V\in RO(G)\).
For \(H\subset G\), let \(T(H)=G/H_+\wedge S[\mathcal{F}_{H\not\subset}^{-1}]\in Sp^G\). It follows from [10] that for all \(G\)-spectrum \(X\), \[L_{T(H)}X \simeq L_{\widetilde{E\mathcal{F}_{H\not\subset}}} L_{G/H_+} X \simeq F(EG/H_+, X)[\mathcal{F}_{H\not\subset}^{-1}],\] and \(L_{T(H_1)}L_{T(H_2)}X\simeq \ast\) if \(H_2\not\subset H_1\). In particular, if \(X\) is a non-equivariant spectrum, we can describe \(L_{T(H)}\operatorname{Inf}_e^G(X)\) as following.
Lemma 17. For any \(X\in Sp\), we have \[L_{T(H)} \operatorname{Inf}_e^G X = \operatorname{Inf}_{G/H}^G F(EG/H_+, X)[\mathcal{F}_{H\not\subset}^{-1}].\]
Proof. For any \(T(H)\) acyclic spectrum \(M\), \(\Phi^H(M\wedge T(H))\simeq \Phi^H M\simeq \ast\). By 9, there are isomorphisms \[[M,\operatorname{Inf}_{G/H}^G F(EG/H_+, X)[\mathcal{F}_{H\not\subset}^{-1}]]^G\cong [\Phi^H M, F(EG/H_+, X)]^{G/H}\cong 0.\] Therefore, \(\operatorname{Inf}_{G/H}^G F(EG/H_+, X)[\mathcal{F}_{H\not\subset}^{-1}]\) is \(T(H)\)-local.
The map \(i:X\to \operatorname{Inf}_{G/H}^G F(EG/H_+, X)[\mathcal{F}_{H\not\subset}^{-1}]\) induced by \(EG/H_+\to S^0\) is a \(T(H)\)-equivalence. Indeed, \(\Phi^N(i \wedge T(H))\) is a map between trivial spectra for all \(N\neq H\), and \(\Phi^H(i\wedge T(H))\) is the equivalence of non-equivariant spectra \(X\wedge G/H_+\to F(EG/H_+, X)\wedge G/H_+\) induced by \(EG/H_+\to S^0\). ◻
Comparing 17 with [10], for any non-equivariant spectrum \(X\), there is an equivalence of \(G\)-spectra \[\operatorname{Inf}_{G/H}^G F(EG/H_+, X)[\mathcal{F}_{H\not\subset}^{-1}]\simeq F(EG/H_+, \operatorname{Inf}_{e}^G X)[\mathcal{F}_{H\not\subset}^{-1}].\] So we can omit the notation of inflation functor in those cases.
Lemma 18. For \(H\subset G\), let \[K(H,n):=G/H_+\wedge K(n) [\mathcal{F}_{H\not\subset}^{-1}]\in Sp_{(p)}^G.\] Then \(KU_G/p\) is Bousfield equivalent to \(\bigvee_{H\in Cyc, H\cap N_p=\emptyset} K(H,1)\).
Proof. This follows from the fact that \[\Phi^N K(H,n)\simeq\begin{cases} G/H_+\wedge K(n) & N=H,\\ \ast & \text{otherwise.} \end{cases}\] ◻
Lemma 19. Let \(G\) be a finite abelian group, and let \(H\subset G\) be a subgroup. For any prime \(p\), any positive integer \(n\), and any \(G\)-spectrum \(X\), we have \[\begin{align} L_{K(H,n)}X & \simeq \operatorname{Inf}_{G/H}^G F(EG/H_+,L_{K(n)}\Phi^H X)[\mathcal{F}_{H\not\subset}^{-1}]\\ & \simeq L_{T(H)}L_{K(n)}\Phi^H X, \end{align}\] When \(n=1\), for any odd prime \(p\), let \(g=(\zeta_{p-1},p+1)\) be a topological generator of \(\mathbb{Z}_p^{\times}\) and \(B=KU_p^\wedge\), there is a fiber sequence \[L_{K(H,1)}S_G\rightarrow \operatorname{Inf}_{G/H}^G F(EG/H_+,B)[\mathcal{F}_{H\not\subset}^{-1}]\overset{\psi^g-1}{\longrightarrow}\operatorname{Inf}_{G/H}^G F(EG/H_+,B)[\mathcal{F}_{H\not\subset}^{-1}].\] When \(p=2\), this is a fiber sequence with \(g\) a generator of \((\mathbb{Z}_2^\wedge)^{\times}/{\pm 1}\) and \(B=KO_2^\wedge\).
Proof. Let \(M=\operatorname{Inf}_{G/H}^G F(EG/H_+,L_{K(n)}\Phi^H X)[\mathcal{F}_{H\not\subset}^{-1}]\), we need to show that \(M\) is \(T(H)\) and \(K(n)\)-local, and there exists a \(K(H,n)\)-equivalence \(X\to M\).
By 17, \(M\simeq L_{T(H)}L_{K(n)}\Phi^H X\) is \(T(H)\)-local. For any \(K(n)\)-acyclic \(G\)-spectrum \(W\), \[\begin{align} [W, M]^G & = F(EG/H_+,L_{K(n)}\Phi^H X)^0(\Phi^H W)\\ & \cong [\Phi^H W, F(EG/H_+, L_{K(n)}\Phi^H X)]^{G/H}\\ & \cong [BG/H_+ \wedge \Phi^H W, L_{K(n)}\Phi^H X]. \end{align}\] The first isomorphism follows from 9, while the last follows from the fact that Borel \(G/H\)-equivariant cohomology depends only on the underlying non-equivariant equivalence type. Since \(W\wedge \operatorname{Inf}_{e}^G K(n)\simeq \ast\), the spectrum \(\Phi^H W\) is \(K(n)\)-acyclic. Hence \(BG/H+\wedge \Phi^H W\) is also \(K(n)\)-acyclic. It follows that \([W,M]^G\cong 0\), so \(M\) is \(K(n)\)-local.
There is a map \(i: X\to M\) induced by the \(K(n)\)-localization \(\Phi^H X\to L_{K(n)} \Phi^H X\) via the isomorphism \[[X,M]^G\cong [\Phi^H X, F(EG/H_+, L_{K(n)}\Phi^H X)]^{G/H}.\] For any \(N\subset G\), \(\Phi^N K(H,n)\simeq \ast\) if \(N\neq H\). When \(N=H\), \(\Phi^H(i\wedge K(H,n))\) is the map \[\Phi^H X\wedge G/H_+ \wedge K(n) \longrightarrow L_{K(n)}\Phi^H X \wedge G/H_+ \wedge K(n)\] induced by the \(K(n)\)-localization \(\Phi^H X\to L_{K(n)} \Phi^H X\), which is an equivalence. Therefore, \(\Phi^N(i\wedge K(H,n))\) is an equivalence for all \(N\subset G\). Hence \(i\) is a \(K(H,n)\)-equivalence.
When \(n=1\) and \(X=S_G\), we have \(\Phi^H S_G\simeq S\) for all \(H\subset G\). The fiber sequences in the lemma are obtained by applying the functor \(\operatorname{Inf}_{G/H}^G F(EG/H_+,-)[\mathcal{F}_{H\not\subset}^{-1}]\) to the fiber sequence \(L_{K(1)}S\to B\overset{\psi^g-1}{\longrightarrow} B\). ◻
Note that \(K(H,n)\)-local objects are automatically \(p\)-complete, which implies that \(\operatorname{Inf}_{G/H}^G F(EG/H_+,L_{K(n)}\Phi^H X)[\mathcal{F}_{H\not\subset}^{-1}]\) is \(p\)-complete.
Lemma 20. Let \(I\) be a set of subgroups of \(G\), and let \(H\) be a subgroup of \(G\) such that \(H\not\subset T\) for any \(T\in I\). Then for any \(X\in Sp^G\), there is a pullback square \[\begin{tikzcd} L_{\bigvee_{T\in I\cup\{H\}}K(T,n)}X \arrow[r]\arrow[d] & L_{K(H,n)} X\arrow[d]\\ L_{\bigvee_{T\in I}K(T,n)}X \arrow[r] & L_{K(H,n)}L_{\bigvee_{T\in I}K(T,n)}X. \end{tikzcd}\]
Proof. By 7, it suffices to show that \[(\bigvee_{T\in I}K(T,n)) \wedge L_{K(H,n)} X\simeq \ast.\] Indeed, \(\Phi^L K(T,1)\not\simeq \ast\) if and only if \(L=T\). On the other hand, by 19, \(\Phi^L L_{K(H,n)}X\not\simeq \ast\) only if \(H\subset L\). Since \(H\not\subset T\) for all \(T\in I\), we have \[\Phi^L((\bigvee_{T\in I}K(T,n)) \wedge L_{K(H,n)} X)\simeq \ast\] for all \(L\subset G\). ◻
Theorem 24. Let \(G\) be a finite abelian group, let \(Cyc\) be a family of cyclic subgroups of \(G\), and let \(N_p\) be the Sylow \(p\)-subgroup of \(G\). For any prime \(p\) and any \(G\)-spectrum \(X\), there is an equivalence of \(G\)-equivariant ring spectrum \[L_{KU_G/p}X \simeq L_{\bigvee_{H\in Cyc, H\cap N_p=e}K(H,1)}X \simeq \bigvee_{H\in Cyc, H\cap N_p=e} L_{K(H,1)}X.\]
Proof. Let \(I=\{H\in Cyc:H\cap N_p=e\}\), we have \(L_{KU_G/p}S_G\simeq L_{\bigvee_{H\in I}K(H,1)}S_G\) since \(\langle KU_G\rangle=\langle \bigvee_{H\in I}K(H,1)\rangle\).
It follows from 19 that for any subgroups \(H_1, H_2\in I\), \[L_{K(H_1,1)}L_{K(H_2,1)}X = L_{T(H_1)}L_{K(1)}\Phi^{H_1}(L_{T(H_2)}L_{K(1)}\Phi^{H_2}X).\] Here \(H_1\cap N_p=e\), by 6 we have \[\begin{align} \Phi^{H_1}(L_{T(H_2)}L_{K(1)}\Phi^{H_2}X) & \simeq \Phi^{H_1}(L_{T(H_2)/p}L_{K(1)}\Phi^{H_2}X)\\ &\simeq L_{\Phi^{H_1}T(H_2)/p}(\Phi^{H_1}L_{K(1)}\Phi^{H_2}X) \\ &\simeq \begin{cases} L_\ast(\Phi^{H_1}L_{K(1)}\Phi^{H_2}X)\simeq \ast & H_1\neq H_2\\ L_{K(H_1,1)}X & H_1=H_2. \end{cases} \end{align}\] Therefore, \(L_{K(H_1,1)}L_{K(H_2,1)}X\simeq \ast\) for all \(H_1\neq H_2\) with \(H_1\cap N_p=e\), and the pullback square in 20 implies that \[L_{K(H_1,1)\vee K(H_2,1)}X\simeq L_{K(H_1,1)}X \vee L_{K(H_2,1)}X.\]
We can order the elements of \(I\) by the order of the corresponding subgroups. For subgroups of the same order, we assign an arbitrary order, since they cannot contain one another. Based on this order, \(I\) is totally ordered. We write \[I=\{H_1,H_2,\ldots,H_n\},\] where \(|H_i|\leq |H_{i+1}|\). Let \(I_1=\{H_1\}\) and \(I_k=I_{k-1} \cup \{H_k\}\).
We study \(L_{\bigvee_{H\in I}K(H,1)}S_G\) by induction on \(I_k\), using the pullback diagrams in 20. Assume that for any \(k<n\), \(L_{\bigvee_{H\in I_k}K(H,1)}X \simeq \bigvee_{H\in I_k} L_{K(H,1)}X\). Then \[L_{K(H_n,1)}L_{\bigvee_{H\in I_{n-1}}K(H,1)}X\simeq \bigvee_{H\in I_{n-1}}L_{K(H_n,1)}L_{K(H,1)}X\simeq \ast,\] thus \(L_{\bigvee_{H\in I_n}K(H,1)}X \simeq \bigvee_{H\in I_n} L_{K(H,1)}X\). ◻
This theorem allows us to use 9 to compute \(\underline{\pi}_V L_{KU_G/p}S_G\) for \(V\in RO(G)\).
Corollary 2. Let \(G\) be a finite abelian group with Sylow \(p\)-subgroup \(N_p\), and let \(N\) denote the product of the Sylow \(q\)-subgroups of \(G\) for \(q\neq p\). For any \(V\in RO(G)\), \[\underline{\pi}_VL_{KU_G/p}S_G\cong \bigoplus_{H\in Cyc, p\nmid |H|}\underline{\pi}_{n_{V,H}} L_{KU_{N_p}/p}S_{N_p},\] where \(n_{V,H}\) is the dimension of \(V^H\).
For any \(P_1\subset P_2\subset N_p\), the restriction \(\operatorname{Res}_{P_1}^{P_2}\) and the transfer \(\operatorname{Tr}_{P_1}^{P_2}\) are induced by those in \(\underline{\pi}_{n_{V,H}} L_{KU_{N_p}/p}S_{N_p}\). For \(L_1\subset L_2\subset N\), the restriction and transfer maps are the natural projection and inclusion, respectively.
Proof. For any cyclic subgroup \(H\subset G\) such that \(p\nmid |H|\), it follows from [prop:pushout type,lem:morava k local] that for any \(V\in RO(G)\), \[\begin{align} \pi_V^G L_{K(H,1)}S_G & \cong \pi^G_V\operatorname{Inf}_{G/H}^G F(EG/H_+,L_{K(1)}S)[\mathcal{F}_{H\not\subset}^{-1}] \\ &\cong \pi^G_{V^H}F(EG/H_+, L_{K(1)}S)\cong L_{K(1)}S^{-n_{V,H}}(BG/H). \end{align}\] Since \(p\nmid |H|\), \(H\subset N\) and \(G/H\cong N_p\oplus N/H\). Stably, there is a transfer map \(BG/H \to BN_p\) such that the composite \[BG/H \longrightarrow BN_p \longrightarrow BG/H\] induces multiplication by the constant \([G/H : N_p] = |N/H|\) on homology. After \(p\)-completion, this composite is an equivalence of spectra since \(p\nmid |N|\). Then \[L_{K(1)}S^*(BG/H)\cong L_{K(1)}S^*(BN_p).\] Note that for a finite \(p\)-group \(N_p\), \(KU_{N_p}/p\) Bousfield equivalent to \((N_p)_+\wedge KU/p\), we have \(L_{KU_{N_p}/p}S_{N_p}\simeq F((EN_p)_+, L_{K(1)}S)\), and \[\pi_V^G L_{K(H,1)}S_G \cong L_{K(1)}S^{-n_{V,H}}(BN_p)\cong \pi_{n_{V,H}}^G L_{KU_{N_p}/p}S_{N_p}.\] It follows from 24 that \[\pi_V L_{KU_G/p}S_G \cong \bigoplus_{\substack{H\in \mathrm{Cyc}\\ p\nmid |H|}} \pi_{n_{V,H}}\, L_{KU_{N_p}/p}S_{N_p}.\]
Since \(L_{KU_G/p}S_G \simeq \bigvee_{\substack{H\in \mathrm{Cyc}\\ p\nmid |H|}} L_{K(H,1)}S_G\), the restriction and transfer maps of \(\underline{\pi}_V L_{KU_G/p}S_G\) act independently on each direct-summand \(\underline{\pi}_V L_{K(H,1)}S_G\). Therefore, for any \(H\in Cyc\) such that \(p\nmid |H|\), and for any \(T\subset G\), it suffices to compute \[\begin{align} \operatorname{Res}_T^G(H): &\pi_V L_{K(H,1)}S_G \to \pi_{\operatorname{Res}_T^G V}L_{\operatorname{Res}_T^G K(H,1)} S_T,\\ \operatorname{Tr}_T^G(H): &\pi_{\operatorname{Res}_T^G V}L_{\operatorname{Res}_T^G K(H,1)} S_T\to \pi_V L_{K(H,1)}S_G. \end{align}\] Let \(P=T\cap N_p\) and \(L=T\cap N\). If \(H\subset L\), \(\langle \operatorname{Res}_T^G K(H,1)\rangle = \langle K(H,1) \rangle\). Thus \[\operatorname{Res}_T^G(H): L_{K(1)}S^{-n_{V,H}}(BN_p)\to L_{K(1)}S^{-n_{V,H}}(BP)\] is induced by \(BP\to BN_p\), and \(\operatorname{Tr}_T^{G}(H)\) is induced by the transfer map \(BN_p\to BP\).
If \(H\not\subset L\), \(\operatorname{Res}_T^G K(H,1)\) is a trivial \(T\)-spectrum, which implies that \[\operatorname{Res}_T^G(H):\pi_V L_{K(H,1)}S_G\to 0, \quad \operatorname{Tr}_T^G(H):0\to \pi_V L_{K(H,1)}S_G\] are the natural projection and inclusion, respectively. Thus in \(\underline{\pi}_VL_{KU_G/p}S_G\), for any \(P\subset N_p\), \(\operatorname{Res}_P^{N_p}\) and \(\operatorname{Tr}_P^{N_p}\) is determined by those in \(\underline{\pi}_{n_{V,H}} L_{KU_{N_p}/p}S_{N_p}\); For any \(L_1\subset L_2\subset N\), the restriction and transfer maps are the natural projection and inclusion, respectively. ◻
In particular, if \(n_{V,H}=n\) is constant for all \(H\in \mathrm{Cyc}\), then the computation above shows that \[\underline{\pi}_V L_{KU_G/p}S_G \cong \underline{A/J}_N \otimes \underline{\pi}_n L_{KU_{N_p}/p}S_{N_p}.\]
Remark 25. Compared with the approach given by 11, the computation in 2 does not make the generators of the homotopy groups as explicit. Nevertheless, we hope that 24 will help us study the case \(G=S^1\).
Let \(G=C_4=\langle \gamma\rangle\), \(C_2=\langle \gamma^2\rangle\), and set \[J:=L_{KU_{C_4}/2}S_{C_4}.\] For \(H\leq C_4\), we write \[M:=\underline{\pi}_1J, \qquad M_H:=M(G/H)=\pi_1^H J.\] We can compute \(M\) via the fiber sequence \[X\longrightarrow (KO_{C_4})_2^{\wedge} \xrightarrow{\psi^5-1} (KO_{C_4})_2^{\wedge}.\] For every subgroup \(H\leq C_4\), the associated long exact sequence gives a short exact sequence \[\begin{align} 0\longrightarrow \operatorname{coker}&\left(\psi^5-1:\pi_2^H (KO_{C_4})_2^{\wedge} \to \pi_2^H (KO_{C_4})_2^{\wedge} \right) \longrightarrow M_H \\ &\longrightarrow \ker\left(\psi^5-1:\pi_1^H (KO_{C_4})_2^{\wedge} \to \pi_1^H (KO_{C_4})_2^{\wedge} \right) \longrightarrow 0. \end{align}\]
Let \(\epsilon\) denote the non-trivial real one-dimensional representation of \(C_2\), and let \[\sigma:C_4\longrightarrow \{\pm 1\}, \qquad \sigma(\gamma)=-1,\] be the sign representation of \(C_4\). Thus \(\sigma|_{C_2}=1\). Let \(L\) denote the faithful complex one-dimensional representation of \(C_4\), that is, \(L(\gamma)=i\). Its underlying real representation is the faithful two-dimensional rotation representation, which we denote by \(\lambda\).
11 implies that this short exact sequence is pointwise split as a sequence of abelian groups. Therefore, the values of \(M\) are \[M_e\cong \mathbb{F}_2\{a,b\}, \quad M_{C_2}\cong \mathbb{F}_2\{a_1,a_\epsilon,b_1,b_\epsilon\},\] and \[M_{C_4}\cong \mathbb{F}_2\{A_1,A_\sigma,B_1,B_\sigma\} \oplus \mathbb{Z}/4\{c\}.\] The classes \(a\), \(a_1\), \(a_\epsilon\), \(A_1\), and \(A_\sigma\) are lifts of the classes associated to \(\eta\) in \(\ker(\psi^5-1)\). The classes \(b\), \(b_1\), \(b_\epsilon\), \(B_1\), and \(B_\sigma\) come from the classes associated to \(\eta^2\) in \(\mathrm{coker}(\psi^5-1)\). The class \[c\in M_{C_4}\] is the class of the cokernel associated to \(\beta L\); more explicitly, if \[r_{\mathbb{R}}:KU\longrightarrow KO\] denotes realification, then \(c\) is represented by \(r_{\mathbb{R}}(\beta L)\), where \(\beta\in \pi_2KU\) is the complex Bott class.
The restriction maps are given by \[\operatorname{Res}_e^{C_2}(a_1)=a, \qquad \operatorname{Res}_e^{C_2}(a_\epsilon)=a,\] \[\operatorname{Res}_e^{C_2}(b_1)=b, \qquad \operatorname{Res}_e^{C_2}(b_\epsilon)=b,\] and \[\operatorname{Res}_{C_2}^{C_4}(A_1)=a_1, \qquad \operatorname{Res}_{C_2}^{C_4}(A_\sigma)=a_1,\] \[\operatorname{Res}_{C_2}^{C_4}(B_1)=b_1, \qquad \operatorname{Res}_{C_2}^{C_4}(B_\sigma)=b_1,\] \[\operatorname{Res}_{C_2}^{C_4}(c)=b_\epsilon.\] The transfer maps are given by \[\operatorname{Tr}_e^{C_2}(a)=a_1+a_\epsilon, \qquad \operatorname{Tr}_e^{C_2}(b)=b_1+b_\epsilon,\] \[\operatorname{Tr}_{C_2}^{C_4}(a_1)=A_1+A_\sigma, \quad \operatorname{Tr}_{C_2}^{C_4}(b_1)=B_1+B_\sigma, \qquad \operatorname{Tr}_{C_2}^{C_4}(b_\epsilon)=0.\] All these results are determined by the restriction and transfer maps in \(RO(C_4)\), so it remains to determine \(\operatorname{Tr}_{C_2}^{C_4}(a_\epsilon)\).
Proposition 26. \(\operatorname{Tr}_{C_2}^{C_4}(a_\epsilon)=2c\).
Proof. Note that \(a_\epsilon\) is a lifting of \(\eta\epsilon\in \pi_1^{C_4}KO_{C_4}\) and as the construction in the proof of 11, \(a_\epsilon\) is the Hurewicz image of \(\operatorname{Tr}_e^{C_2}(\eta)-\eta\) for the Hopf element \(\eta\in \pi_1 S\). Let \[J_{\mathbb{C}} := \operatorname{hofib} \left( (KU_{C_4})^\wedge_2\xrightarrow{\psi^5-1}(KU_{C_4})^\wedge_2 \right),\] the complexification map \(KO_{C_4}\to KU_{C_4}\) induces a \(C_4\)-map \(f:J\to J_{\mathbb{C}}\). Here \[\pi_1^{C_2} J_{\mathbb{C}}\cong \mathbb{Z}/4\{\beta, \epsilon\beta\}, \qquad \pi_1^{C_4} J_{\mathbb{C}}\cong \mathbb{Z}/4\{L^i\beta:1\leq i\leq 3\}.\] The homomorphism \(f_*: \underline{\pi}_1 J \to \underline{\pi}_1 J_{\mathbb{C}}\) satisfies that \[f_*(a_\epsilon)= (\operatorname{Tr}_e^{C_2}(1)-1) f_*(\eta) = \epsilon f_*(\eta) =\epsilon \cdot 2\beta,\] and \[f_*(\operatorname{Tr}_{C_2}^{C_4}(a_\epsilon))=\operatorname{Tr}_{C_2}^{C_4} 2\epsilon \beta = 2 \beta (L+L^3) \neq 0.\] Therefore \(\operatorname{Tr}_{C_2}^{C_4}(a_\epsilon) = 2c \neq 0\). ◻
Finally, We record the nonequivariant \(f_*(\eta)=2\beta\) used above. Restrict to the trivial group \(\{e\}\), \[\pi_1J_{\mathbb{C}} \cong \mathbb{Z}/4\{\beta\}.\] Since \(\eta\in \pi_1 L_{KU/2}S\) is the Hurewicz image of the Hopf element \(\eta\in \pi_1S\), \(f_*\eta\) is the element represented by \[S^1\xrightarrow{\eta} S^0 \to J \xrightarrow{f} J_{\mathbb{C}}.\] Consider the cofiber sequence \(S^1\xrightarrow{\eta} S^0 \to C\eta\simeq \Sigma^{-2}\mathbb{C}P^2\). Let \(u=[\mathcal{O}(1)]-1\) be the generator of \(\widetilde{KU}^*(\mathbb{C}P^2)\) such that \(\widetilde{KU}(\mathbb{C}P^2)\cong \mathbb{Z}[u]/u^3\), we have \(\psi^5(u)=5u+10u^2\). Stably \(f_*\eta\) is the obstruction of the extension of \(S^0\to J\) via \(S^0\to C\eta\). Consider the diagram \[\begin{tikzcd} S^1 \arrow[r,"\eta"] & S^0 \arrow[r]\arrow[d] & C\eta\arrow[ldd,"\beta^{-1}u"]\\ & J_{\mathbb{C}} \arrow[d] &\\ & KU_2^\wedge &, \end{tikzcd}\] \(\beta^{-1}u\) gives an extension of \(S^0\to KU_2^\wedge\). Since \((\psi^5-1)(\beta^{-1}u)=2\beta^{-1}u^2\), \(\beta^{-1}u\) can’t lifts to \(J_{\mathbb{C}}\), which implies that \(f^*\eta\neq 0\in \pi_1 J_{\mathbb{C}}\), thus \(f_*\eta=2\beta\).