15,175 research outputs found
Profinite Galois descent in K(h)-local homotopy theory
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- products and fast computation of Steenrod squares
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- 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-construction, 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- construction they define, showing additionally that all other formulasin the literature define the same cup- construction up to isomorphism.<br
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Dowker Duality for Relations of Categories
We propose a categorification of the Dowker duality theorem for relations.
Dowker's theorem states that the Dowker complex of a relation of sets and is homotopy equivalent to the Dowker complex of
the transpose relation . Given a relation of
small categories and , that is, a functor of the
form , we define the
bisimplicial rectangle nerve and the Dowker nerve . The diagonal
of the bisimplicial set maps to the simplicial set by a
natural projection .
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 of categories and its transpose
relation are Dowker relations, then the Dowker nerves and 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 with vertex set and show that the
geometric realization of is naturally homotopy equivalent to the geometric
realization of the simplicial set with the set of -simplices given by
functions whose image is a simplex of .Comment: 16 page
Equivariant toric geometry and Euler-Maclaurin formulae
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 - and homology theories. In localized equivariant
-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
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
Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology
We study smooth higher symmetry groups and moduli -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 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
-categorical quotients by the action of the higher symmetries, extract
information about the homotopy types of these moduli -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 -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 -gerbes with
-connection, comment on the relation to higher-form symmetries, and present
a new String group model. We construct smooth moduli -stacks of higher
Maxwell and Einstein-Maxwell solutions, correcting previous such considerations
in the literature, and compute the homotopy groups of several moduli
-stacks of higher - 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 -stacks of NSNS supergravity solutions.Comment: 102 pages; comments welcom
Combinatorial classification of -skew projective spaces
The noncommutative projective scheme of
a -skew polynomial algebra in variables is considered to be a
-skew projective space of dimension . In this paper, using
combinatorial methods, we give a classification theorem for -skew
projective spaces. Specifically, among other equivalences, we prove that -skew projective spaces and
are isomorphic if and only if certain
graphs associated to and are switching (or mutation) equivalent. We
also discuss invariants of -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 in the Benjamini-Schramm limit
We study eigenfunctions of the spherical Hecke algebra acting on
where with a
non-archimedean local field of characteristic zero, with the ring of integers of , and
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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