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Abstract Working at the prime 2 and chromatic height 2, we construct a finite resolution of the homotopy fixed points of MoravaE-theory with respect to the subgroup$$\mathbb {G}_2^1$$ $G_2^1$of the Morava stabilizer group. This is an upgrade of the finite resolution of the homotopy fixed points ofE-theory with respect to the subgroup$$\mathbb {S}_2^1$$ $S_2^1$constructed in work of Goerss–Henn–Mahowald–Rezk, Beaudry and Bobkova–Goerss. 

Abstract In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm by permutation summands. These quotients are of interest because of their close relationship with higher real ‐theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories . These spectra serve as natural equivariant generalizations of connective integral Morava ‐theories. We provide a complete computation of the ‐localized slice spectral sequence of , where is the real sign representation of . To achieve this computation, we establish a correspondence between this localized slice spectral sequence and the ‐based Adams spectral sequence in the category of ‐modules. Furthermore, we provide a full computation of the ‐localized slice spectral sequence of the height‐4 theory . The ‐slice spectral sequence can be entirely recovered from this computation. 

In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite-dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite-dimensional Hilbert space [Formula: see text] to the pure state space of the quasi-local algebra of the quantum spin system with Hilbert space [Formula: see text] at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to [Formula: see text]-spaces for an operad we call the “multiplicative” linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step toward understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected. 

This paper is concerned with the physics of parametrized gapped quantum many-body systems, which can be viewed as a generalization of conventional topological phases of matter. In such systems, rather than considering a single Hamiltonian, one considers a family of Hamiltonians that depend continuously on some parameters. After discussing the notion of phases of parametrized systems, we formulate a bulk-boundary correspondence for an important bulk quantity, the Kapustin-Spodyneiko higher Berry curvature, first in one spatial dimension and then in arbitrary dimension. This clarifies the physical interpretation of the higher Berry curvature, which in one spatial dimension is a flow of (ordinary) Berry curvature. In 𝑑 dimensions, the higher Berry curvature is a flow of (𝑑−1)-dimensional higher Berry curvature. Based on this, we discuss one-dimensional systems that pump Chern number to/from spatial boundaries, resulting in anomalous boundary modes featuring isolated Weyl points. In higher dimensions, there are pumps of the analogous quantized invariants obtained by integrating the higher Berry curvature. We also discuss the consequences for parametrized systems of Kitaev's proposal that invertible phases are classified by a generalized cohomology theory, and emphasize the role of the suspension isomorphism in generating new examples of parametrized systems from known invertible phases. Finally, we present a pair of general quantum pumping constructions, based on physical pictures introduced by Kitaev, which take as input a 𝑑-dimensional parametrized system, and produce new (𝑑+1)-dimensional parametrized systems. These constructions are useful for generating examples, and we conjecture that one of the constructions realizes the suspension isomorphism in a generalized cohomology theory of invertible phases. 

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. 

We consider how the outputs of the Kadison transitivity theorem and Gelfand–Naimark–Segal (GNS) construction may be obtained in families when the initial data are varied. More precisely, for the Kadison transitivity theorem, we prove that for any nonzero irreducible representation [Formula: see text] of a [Formula: see text]-algebra [Formula: see text] and [Formula: see text], there exists a continuous function [Formula: see text] such that [Formula: see text] for all [Formula: see text], where [Formula: see text] is the set of pairs of [Formula: see text]-tuples [Formula: see text] such that the components of [Formula: see text] are linearly independent. Versions of this result where [Formula: see text] maps into the self-adjoint or unitary elements of [Formula: see text] are also presented. Regarding the GNS construction, we prove that given a topological [Formula: see text]-algebra fiber bundle [Formula: see text], one may construct a topological fiber bundle [Formula: see text] whose fiber over [Formula: see text] is the space of pure states of [Formula: see text] (with the norm topology), as well as bundles [Formula: see text] and [Formula: see text] whose fibers [Formula: see text] and [Formula: see text] over [Formula: see text] are the GNS Hilbert space and closed left ideal, respectively, corresponding to [Formula: see text]. When [Formula: see text] is a smooth fiber bundle, we show that [Formula: see text] and [Formula: see text] are also smooth fiber bundles; this involves proving that the group of ∗-automorphisms of a [Formula: see text]-algebra is a Banach Lie group. In service of these results, we review the topology and geometry of the pure state space. A simple non-interacting quantum spin system is provided as an example illustrating the physical meaning of some of these results. 

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.