The Relative Burnside Module and the Stable Maps Between Classifying Spaces of Compact Lie Groups

Minami, Norihiko

doi:10.2307/2154897

Cited by 3 publications

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“…We note that applying π C2 0,0 (−) recovers a case of Norihiko Minami's work on relative Burnside modules [56]. With this theorem in hand, we define the C 2 -equivariant Mahowald invariant in Section 4 following the classical and motivic definitions.…”

Section: Introductionmentioning

confidence: 98%

Real motivic and $C_2$-equivariant Mahowald invariants

Quigley

2019

Preprint

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The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. In [59], we studied an analog of this construction in the setting of motivic stable homotopy theory over Spec(C). In this paper, we define analogs in the settings of motivic stable homotopy theory over Spec(R) and C 2 -equivariant stable homotopy theory. To do so in the C 2 -equivariant setting, we prove a C 2 -equivariant analog of Lin's Theorem by adapting the motivic Singer construction developed by Gregersen [24] to the C 2 -equivariant setting. In both the real motivic and C 2 -equivariant settings, we prove an analog of the theorem of Mahowald and Ravenel in the classical setting [46] and the author in the complex motivic setting that the Mahowald invariant of (2 + ρη) i , i ≡ 2, 3 mod 4, is v 1 -periodic. Up to a conjecture about the R-motivic stable stems, we also prove that the Mahowald invariant of η i , i ≥ 1, is w 1 -periodic. Finally, we study the behavior of the Mahowald invariant under various functors between the motivic, equivariant, and classical stable homotopy categories. ContentsAbstract 1 1. Introduction 1 2. The parametrized Tate construction 7 3. C 2 -equivariant Lin's Theorem 16 4. Real motivic and C 2 -equivariant Mahowald invariants 23 5. Some real motivic and parametrized Σ 2 -Tate constructions 27 6. v 1 -periodicity over R and generalized Mahowald invariants of (2 + ρη) i 36 7. Real motivic and C 2 -equivariant Mahowald invariants of η i 41 Appendix A. Atiyah-Hirzebruch spectral sequence charts 50 References 55

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Section: Introductionmentioning

confidence: 98%