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Hurewicz images of real bordism theory and real Johnson-Wilson theories. (English) Zbl 1417.55019
Let \(\mathbb{S}\) be the sphere spectrum and let \(MU_{\mathbb{R}}\) be the real cobordism spectrum in the sense of P. S. Landweber [Bull. Am. Math. Soc. 74, 271–274 (1968; Zbl 0181.26801)]. In [Ann. Math. (2) 184, No. 1, 1-262 (2016; Zbl 1366.55007)], M. A. Hill et al. showed that the Kervaire invariant elements \(\theta_j \in \pi_{2^{j+1}-2} \mathbb {S}\) do not exist for \(j \geq 7\). It is well known that the spectrum \(MU_{\mathbb{R}}\) splits as a wedge of suspensions of \(BP_{\mathbb {R}}\), a real analogue of the Brown-Peterson spectrum, in [S. Araki, Jpn. J. Math., New Ser. 5, 403–430 (1979; Zbl 0443.55003)]; that is, \[ MU_{\mathbb{R}} = \bigvee_{m_i} \Sigma^{m_i(1+\rho)} BP_{\mathbb {R}}, \] where \(\rho\) is the non-trivial one dimensional real representation of \(C_2\). It is also well known that the Hopf elements are represented by the elements \(h_i \in \text{Ext}_{\mathcal A_*}^{1,2^i}(\mathbb F_2, \mathbb F_2 )\) on the \(E_2\)-page of the classical Adams spectral sequence at the prime \(2\). W. Browder [Ann. Math. (2) 90, 157–186 (1969; Zbl 0198.28501)] showed that the Kervaire classes \(\theta_j \in \pi_{2^{j+1}-2}\mathbb {S}\) are represented by the elements \(h_j^2 \in \text{Ext}_{\mathcal A_*}^{2,2^{j+1}}(\mathbb F_2, \mathbb F_2 )\) on the \(E_2\)-page if they exist. W.-H. Lin [Topology Appl. 155, No. 5, 459–496 (2008; Zbl 1140.55012)] proved that there is a family \(\{g_k \mid k \geq 1 \}\) of indecomposable elements with \(g_k \in \text{Ext}_{\mathcal A_*}^{4,2^{k+2} + 2^{k+3}}(\mathbb F_2, \mathbb F_2 )\). The elements \(g_1\) and \(g_2\) of the family survive the Adams spectral sequence to become \(\bar \kappa \in \pi_{20}\mathbb {S}\) and \(\bar \kappa_2 \in \pi_{44}\mathbb {S}\), respectively. The \(\bar \kappa\)-family consists of the homotopy classes detected by the surviving \(g_k\)-family.
In this paper, the authors show that the Hopf elements, the Kervaire classes and the \(\bar \kappa\)-family are detected by the Hurewicz maps \(\pi_* \mathbb{S} \rightarrow \pi_* MU_{\mathbb{R}}^{C_2}\) and \(\pi_* \mathbb{S} \rightarrow \pi_* BP_{\mathbb{R}}^{C_2}\), and that the \(G\)-fixed point of \(MU^{((G))}\) detects the Hopf elements, the Kervaire classes and the \(\bar \kappa\)-family for any finite group \(G\) containing \(C_2\). Moreover, the authors show that the images of the elements \(\{ h_i \mid i \geq 1\}\), \(\{ h_j^2 \mid j \geq 1\}\) and \(\{ g_k \mid k \geq 1\}\) on the \(E_2\)-page of the classical Adams spectral sequence of the sphere spectrum are nonzero on the \(E_2\)-page of the \(C_2\)-equivariant Adams spectral sequence of \(BP_{\mathbb {R}}\). They also prove that the integer-graded \(C_2\)-equivariant May spectral sequence of \(BP_{\mathbb {R}}\) is isomorphic to the associated-graded slice spectral sequence of \(BP_{\mathbb {R}}\), and that a subset of those families is detected by the \(C_2\)-fixed points of Real Johnson-Wilson theory \(E\mathbb R (n)\) depending on \(n\). More precisely, (1) if the element \(h_i \in \text{Ext}_{\mathcal A_*}^{1,2^i}(\mathbb F_2, \mathbb F_2 )\) or \(h_j^2 \in \text{Ext}_{\mathcal A_*}^{2,2^{j+1}}(\mathbb F_2, \mathbb F_2 )\) survives to the \(E_\infty\)-page of the Adams spectral sequence, then its image under the Hurewicz map \(\pi_* \mathbb{S} \rightarrow \pi_* E\mathbb R (n)^{C_2}\) is nonzero for \(1 \leq i,j \leq n\), and (2) if the element \(g_k \in \text{Ext}_{\mathcal A_*}^{4,2^{k+2} + 2^{k+3}}(\mathbb F_2, \mathbb F_2 )\) survives to the \(E_\infty\)-page of the Adams spectral sequence, then its image under the Hurewicz map \(\pi_* \mathbb{S} \rightarrow \pi_* E\mathbb R (n)^{C_2}\) is nonzero for \(1 \leq k \leq n-1\).

MSC:
55Q45 Stable homotopy of spheres
55T20 Eilenberg-Moore spectral sequences
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