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The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum 𝐸𝑛, whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that 𝐸𝑛 is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group 𝔾𝑛 of the formal group in question. In this paper we find that the 𝔾𝑛-equivariant dual of 𝐸𝑛 is in fact 𝐸𝑛 twisted by a sphere with a non-trivial (when 𝑛>1) action by 𝔾𝑛. This sphere is a dualizing module for the group 𝔾𝑛, and we construct and study such an object 𝐼𝒢 for any compact p-adic analytic group 𝒢. If we restrict the action of 𝒢 on 𝐼𝒢 to certain type of small subgroups, we identify 𝐼𝒢 with a specific representation sphere coming from the Lie algebra of 𝒢. This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of 𝐸ℎ𝐻𝑛 for select choices of p and n and finite subgroups H of 𝔾𝑛. 

Abstract For a subring $$R$$ of the rational numbers, we study $$R$$ -localization functors in the local homotopy theory of simplicial presheaves on a small site and then in $${\mathbb {A}}^1$$ -homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in $${\mathbb {A}}^1$$ -homotopy theory, paying attention to future applications for vector bundles. We show that $$R$$ -localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $$BGL_n$$ is $${\mathbb {A}}^1$$ -nilpotent when $$n$$ is odd, and analyze the (more complicated) situation where $$n$$ is even as well. We establish analogs of various classical results about rationalization in the context of $${\mathbb {A}}^1$$ -homotopy theory: if $-1$ is a sum of squares in the base field, $${\mathbb {A}}^n \,{\setminus}\, 0$$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres. 

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. 

Abstract We prove that there is no parity anomaly in M-theory in the low-energy field theory approximation. Our approach is computational. We determine the generators for the 12-dimensional bordism group of pin manifolds with a w1-twisted integer lift of w4; these are the manifolds on which Wick-rotated M-theory exists. The anomaly cancellation comes down to computing a specific η-invariant and cubic form on these manifolds. Of interest beyond this specific problem are our expositions of computational techniques for η-invariants, the algebraic theory of cubic forms, Adams spectral sequence techniques and anomalies for spinor fields and Rarita–Schwinger fields.