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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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References:
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