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We show a number of Toda brackets in the homotopy of the motivic bordism spectrum MGL and of the Real bordism spectrum MUR. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height n, the bracket itself will be of chromatic height (n+1). Using these, we deduce a family of exotic multiplications in the π_{**}MGL-module structure of the motivic Morava K-theories, including non-trivial multiplications by 2. These in turn imply the analogous family of exotic multiplications in the π_{\star}MUR-module structure on the Real Morava K-theories.
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In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin ( 2 ) \operatorname {Pin}(2) -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the Pin ( 2 ) \operatorname {Pin}(2) -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from B Pin ( 2 ) B\operatorname {Pin}(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j j -based Atiyah–Hirzebruch spectral sequence.more » « less
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We study modules over the commutative ring spectrum 𝐻𝔽₂∧𝐻𝔽₂, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator ξ_{k} in the category of associative algebras freely kills the higher generators ξ_{k+n}. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative 𝐻𝔽₂∧𝐻𝔽₂-algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.more » « less
