15,175 research outputs found

    Profinite Galois descent in K(h)-local homotopy theory

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    We investigate the category of K(h)-local spectra through the action of the Morava stabiliser group. Using condensed mathematics, we give a model for the continuous action of this profinite group on the ∞-category of K(h)-local modules over Morava E-theory, and explain how this gives rise to descent spectral sequences computing the Picard and Brauer groups of K(h)-local spectra. In the second part, we focus on the computation of these spectral sequences at height one, showing that they recover the Hopkins-Mahowald-Sadofsky computation of the Picard group, and giving a complete computation of the Brauer group relative to p-completed complex K-theory

    New formulas for cup-ii products and fast computation of Steenrod squares

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    Operations on the cohomology of spaces are important tools enhancing thedescriptive power of this computable invariant. For cohomology with mod 2coefficients, Steenrod squares are the most significant of these operations.Their effective computation relies on formulas defining a cup-ii construction,a structure on (co)chains which is important in its own right, havingconnections to lattice field theory, convex geometry and higher category theoryamong others. In this article we present new formulas defining a cup-iiconstruction, and use them to introduce a fast algorithm for the computation ofSteenrod squares on the cohomology of finite simplicial complexes. Inforthcoming work we use these formulas to axiomatically characterize thecup-ii construction they define, showing additionally that all other formulasin the literature define the same cup-ii construction up to isomorphism.<br

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Dowker Duality for Relations of Categories

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    We propose a categorification of the Dowker duality theorem for relations. Dowker's theorem states that the Dowker complex of a relation RX×YR \subseteq X \times Y of sets XX and YY is homotopy equivalent to the Dowker complex of the transpose relation RTY×XR^T \subseteq Y \times X. Given a relation RR of small categories C\mathcal{C} and D\mathcal{D}, that is, a functor of the form R ⁣:RC×DR \colon \mathcal{R} \to \mathcal{C} \times \mathcal{D}, we define the bisimplicial rectangle nerve ERER and the Dowker nerve DRDR. The diagonal d(ER)d(ER) of the bisimplicial set ERER maps to the simplicial set DRDR by a natural projection d(πR) ⁣:d(ER)DRd(\pi_R) \colon d(ER) \to DR. We introduce a criterion on relations of categories ensuring that the projection from the diagonal of the bisimplicial rectangle nerve to the Dowker nerve is a weak equivalence. Relations satisfying this criterion are called Dowker relations. If both the relation RR of categories and its transpose relation RTR^T are Dowker relations, then the Dowker nerves DRDR and DRTDR^T are weakly equivalent simplicial sets. In order to justify the abstraction introduced by our categorification we give two applications. The first application is to show that Quillen's Theorem A can be considered as an instance of Dowker duality. In the second application we consider a simplicial complex KK with vertex set VV and show that the geometric realization of KK is naturally homotopy equivalent to the geometric realization of the simplicial set with the set of nn-simplices given by functions {0,1,,n}V\{0,1,\dots,n\}\to V whose image is a simplex of KK.Comment: 16 page

    Equivariant toric geometry and Euler-Maclaurin formulae

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    We consider equivariant versions of the motivic Chern and Hirzebruch characteristic classes of a quasi-projective toric variety, and extend many known results from non-equivariant to the equivariant setting. The corresponding generalized equivariant Hirzebruch genus of a torus-invariant Cartier divisor is also calculated. Further global formulae for equivariant Hirzebruch classes are obtained in the simplicial context by using the Cox construction and the equivariant Lefschetz-Riemann-Roch theorem. Alternative proofs of all these results are given via localization at the torus fixed points in equivariant KK- and homology theories. In localized equivariant KK-theory, we prove a weighted version of a classical formula of Brion for a full-dimensional lattice polytope. We also generalize to the context of motivic Chern classes the Molien formula of Brion-Vergne. Similarly, we compute the localized Hirzebruch class, extending results of Brylinski-Zhang for the localized Todd class. We also elaborate on the relation between the equivariant toric geometry via the equivariant Hirzebruch-Riemann-Roch and Euler-Maclaurin type formulae for full-dimensional simple lattice polytopes. Our results provide generalizations to arbitrary coherent sheaf coefficients, and algebraic geometric proofs of (weighted versions of) the Euler-Maclaurin formulae of Cappell-Shaneson, Brion-Vergne, Guillemin, etc., via the equivariant Hirzebruch-Riemann-Roch formalism. Our approach, based on motivic characteristic classes, allows us to obtain such Euler-Maclaurin formulae also for (the interior of) a face, or for the polytope with several facets removed. We also prove such results in the weighted context, and for Minkovski summands of the given full-dimensional lattice polytope. Some of these results are extended to local Euler-Maclaurin formulas for the tangent cones at the vertices of the given lattice polytope.Comment: 93 pages, comments are very welcom

    A convenient category of parametrized spectra

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    We describe a point-set category of parametrized orthogonal spectra, a model structure on this category, and a separate, more geometric class of cofibrant-and-fibrant objects. The structures we describe are "convenient" in that they are preserved by the most common operations. They allow us to reduce sophisticated statements about the homotopy category to straightforward claims at the point-set level. We use this framework to give a construction of the bicategory of parametrized spectra, one that is far more direct than earlier approaches. This gives a clean bridge between the concrete index theory pioneered by Dold, and the formal bicategorical theory developed by May and Sigurdsson, Ponto, and Shulman.Comment: 97 pages with references. This article is a condensation of arXiv:1906.0477

    Sparse polynomial prediction

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    Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology

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    We study smooth higher symmetry groups and moduli \infty-stacks of generic higher geometric structures on manifolds. Symmetries are automorphisms which cover non-trivial diffeomorphisms of the base manifold. We construct the smooth higher symmetry group of any geometric structure on MM and show that this completely classifies, via a universal property, equivariant structures on the higher geometry. We construct moduli stacks of higher geometric data as \infty-categorical quotients by the action of the higher symmetries, extract information about the homotopy types of these moduli \infty-stacks, and prove a helpful sufficient criterion for when two such higher moduli stacks are equivalent. In the second part of the paper we study higher U(1)\mathrm{U}(1)-connections. First, we observe that higher connections come organised into higher groupoids, which further carry affine actions by Baez-Crans-type higher vector spaces. We compute a presentation of the higher gauge actions for nn-gerbes with kk-connection, comment on the relation to higher-form symmetries, and present a new String group model. We construct smooth moduli \infty-stacks of higher Maxwell and Einstein-Maxwell solutions, correcting previous such considerations in the literature, and compute the homotopy groups of several moduli \infty-stacks of higher U(1)\mathrm{U}(1)- connections. Finally, we show that a discrepancy between two approaches to the differential geometry of NSNS supergravity (via generalised and higher geometry, respectively) vanishes at the level of moduli \infty-stacks of NSNS supergravity solutions.Comment: 102 pages; comments welcom

    Combinatorial classification of (±1)(\pm 1)-skew projective spaces

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    The noncommutative projective scheme ProjncS\operatorname{\mathsf{Proj_{nc}}} S of a (±1)(\pm 1)-skew polynomial algebra SS in nn variables is considered to be a (±1)(\pm 1)-skew projective space of dimension n1n-1. In this paper, using combinatorial methods, we give a classification theorem for (±1)(\pm 1)-skew projective spaces. Specifically, among other equivalences, we prove that (±1)(\pm 1)-skew projective spaces ProjncS\operatorname{\mathsf{Proj_{nc}}} S and ProjncS\operatorname{\mathsf{Proj_{nc}}} S' are isomorphic if and only if certain graphs associated to SS and SS' are switching (or mutation) equivalent. We also discuss invariants of (±1)(\pm 1)-skew projective spaces from a combinatorial point of view.Comment: 14 pages, v2: minor modifications, v3: removed an exampl

    Quantum ergodicity on the Bruhat-Tits building for PGL(3,F)\text{PGL}(3, F) in the Benjamini-Schramm limit

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    We study eigenfunctions of the spherical Hecke algebra acting on L2(Γn\G/K)L^2(\Gamma_n \backslash G / K) where G=PGL(3,F)G = \text{PGL}(3, F) with FF a non-archimedean local field of characteristic zero, K=PGL(3,O)K = \text{PGL}(3, \mathcal{O}) with O\mathcal{O} the ring of integers of FF, and (Γn)(\Gamma_n) is a sequence of cocompact torsionfree lattices. We prove a form of equidistribution on average for eigenfunctions whose spectral parameters lie in the tempered spectrum when the associated sequence of quotients of the Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table
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