Hurewicz images of real bordism theory and real Johnson–Wilson theories

Abstract

We show that the Hopf elements, the Kervaire classes, and theκfamily in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the C 2 -fixed points of the Real bordism spectrum. A subset of these families is detected by the C 2 -fixed points of Real Johnson-Wilson theory ER(n), depending on n. In the proof, we establish an isomorphism between the slice spectral sequence and the C 2 -equivariant May spectral sequence of BP R .

On equivariant and motivic slices

“…Definition 2. 19 We say that F : (C, C q ) → (D, D q ) is compatible with the slice filtration at E if α q and β q are equivalences for all q ∈ Z. If the slices filtrations on C and D are understood, then we will simply say that F : C → D is compatible with the slice filtration at E .…”

“…The differentials in the equivariant slice spectral sequence for BPR/2 can be determined by the differentials in the slice spectral sequence for BPR itself. The differentials in the latter can be determined by the work of Hu and Kriz [13] or as a consequence of Hill, Hopkins, Ravenel [10, Theorem 9.9], and are fully described in [19,Proposition 3.4]. To wit, the E 1 -term is given in positive degrees by Z[a σ , u 2σ , v i ].…”

Models of Lubin-Tate spectra via Real bordism theory

“…Periodic spectra such as these higher real K-theories play a central role in modern detection theorems. These are results about families in the stable homotopy groups of spheres obtained by studying the Hurewicz homomorphisms of such periodic spectra [Rav78,HHR16,LSWX19]. 1.2.…”

Symmetric Powers and Eilenberg--Maclane Spectra

“…They then proved that π C 2 ⋆ (HF 2 ∧ HF 2 ) ≃ (HF 2 ) ⋆ [ξ i , τ i ]/ τ 0 a σ = u σ + η R (u σ ), τ 2 i = τ i+1 a σ + ξ i+1 η R (u σ ) An exposition of this computation is given in [14]. It is curious that this result bears a similarity to the classical odd primary dual Steenrod algebra computed by Milnor.…”