152 research outputs found
Stratifying derived categories of cochains on certain spaces
In recent years, Benson, Iyengar and Krause have developed a theory of
stratification for compactly generated triangulated categories with an action
of a graded commutative Noetherian ring. Stratification implies a
classification of localizing and thick subcategories in terms of subsets of the
prime ideal spectrum of the given ring. In this paper two stratification
results are presented: one for the derived category of a commutative
ring-spectrum with polynomial homotopy and another for the derived category of
cochains on certain spaces. We also give the stratification of cochains on a
space a topological content.Comment: 27 page
Duality in algebra and topology
In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that
they can be extended to the more general rings that come up in homotopy theory.
Amongst the rings we work with are the differential graded ring of cochains on a space, the differential graded ring of chains on the loop space, and various ring spectra, e.g., the Spanier-Whitehead duals of finite spectra or chromatic localizations of the sphere spectrum.
Maybe the most important contribution of this paper is the conceptual framework, which allows us to view all of the following dualities: Poincare duality for manifolds, Gorenstein duality for commutative rings, Benson-Carlson duality for cohomology rings of finite groups, Poincare duality for groups, Gross-Hopkins duality in chromatic stable homotopy theory, as examples of a single phenomenon. Beyond setting up this framework, though, we prove some new results, both in algebra and topology, and give new proofs of a number of old results
Homotopy theory and simplicial groupoids
AbstractThe homotopy theory of simplical groups is well known [2, Ch. VI] to be equivalent to the pointed homotopy theory of reduced (i.e. only one vertex) simplicial sets (by means of a pair of adjoint functors G and W̄.The aim of this note is to show that similarly, the homotopy theory of simplical groupoids is equivalent to the (unpointed) homotopy theory of (all) simplical sets. This we do by 1.(i) showing that the category of simplicial groupoids admits a closed model catagory structure in the sense of Quillen [3], and2.(ii) extending the functors G and W̄ to pair of adjoint functorsG: (simplicial sets)↔(simplicial groupoids): W̄ which induce the desired equivalence of homotopy theories.We also show that the category of simplical groupoids admits a simplical structure which produces “function complexes” and “simplical monoids of self homotopy equivalences” of the correct homotopy types
An E2 model category structure for pointed simplicial spaces
AbstractWe find settings in which it is possible to resolve a topological space by simplicial spaces or cosimplicial spaces. We determine what such a resolution consists of, and study the sense in which any two resolutions are equivalent. As in ordinary homological algebra, these resolutions are useful for constructing spectral sequences
Topological representations of matroid maps
The Topological Representation Theorem for (oriented) matroids states that
every (oriented) matroid can be realized as the intersection lattice of an
arrangement of codimension one homotopy spheres on a homotopy sphere. In this
paper, we use a construction of Engstr\"om to show that structure-preserving
maps between matroids induce topological mappings between their
representations; a result previously known only in the oriented case.
Specifically, we show that weak maps induce continuous maps and that the
process is a functor from the category of matroids with weak maps to the
homotopy category of topological spaces. We also give a new and conceptual
proof of a result regarding the Whitney numbers of the first kind of a matroid.Comment: Final version, 21 pages, 8 figures; Journal of Algebraic
Combinatorics, 201
Class and rank of differential modules
A differential module is a module equipped with a square-zero endomorphism.
This structure underpins complexes of modules over rings, as well as
differential graded modules over graded rings. We establish lower bounds on the
class--a substitute for the length of a free complex--and on the rank of a
differential module in terms of invariants of its homology. These results
specialize to basic theorems in commutative algebra and algebraic topology. One
instance is a common generalization of the equicharacteristic case of the New
Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning
complexes over noetherian commutative rings, and of a theorem of G. Carlsson on
differential graded modules over graded polynomial rings.Comment: 27 pages. Minor changes; mainly stylistic. To appear in Inventiones
Mathematica
On the algebraic K-theory of the complex K-theory spectrum
Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and
let K(ku) be the algebraic K-theory spectrum of ku. We study the p-primary
homotopy type of the spectrum K(ku) by computing its mod (p,v_1) homotopy
groups. We show that up to a finite summand, these groups form a finitely
generated free module over a polynomial algebra F_p[b], where b is a class of
degree 2p+2 defined as a higher Bott element.Comment: Revised and expanded version, 42 pages
All-sky search for long-duration gravitational wave transients with initial LIGO
We present the results of a search for long-duration gravitational wave transients in two sets of data collected by the LIGO Hanford and LIGO Livingston detectors between November 5, 2005 and September 30, 2007, and July 7, 2009 and October 20, 2010, with a total observational time of 283.0 days and 132.9 days, respectively. The search targets gravitational wave transients of duration 10-500 s in a frequency band of 40-1000 Hz, with minimal assumptions about the signal waveform, polarization, source direction, or time of occurrence. All candidate triggers were consistent with the expected background; as a result we set 90% confidence upper limits on the rate of long-duration gravitational wave transients for different types of gravitational wave signals. For signals from black hole accretion disk instabilities, we set upper limits on the source rate density between 3.4×10-5 and 9.4×10-4 Mpc-3 yr-1 at 90% confidence. These are the first results from an all-sky search for unmodeled long-duration transient gravitational waves. © 2016 American Physical Society
All-sky search for long-duration gravitational wave transients with initial LIGO
We present the results of a search for long-duration gravitational wave transients in two sets of data collected by the LIGO Hanford and LIGO Livingston detectors between November 5, 2005 and September 30, 2007, and July 7, 2009 and October 20, 2010, with a total observational time of 283.0 days and 132.9 days, respectively. The search targets gravitational wave transients of duration 10-500 s in a frequency band of 40-1000 Hz, with minimal assumptions about the signal waveform, polarization, source direction, or time of occurrence. All candidate triggers were consistent with the expected background; as a result we set 90% confidence upper limits on the rate of long-duration gravitational wave transients for different types of gravitational wave signals. For signals from black hole accretion disk instabilities, we set upper limits on the source rate density between 3.4×10-5 and 9.4×10-4 Mpc-3 yr-1 at 90% confidence. These are the first results from an all-sky search for unmodeled long-duration transient gravitational waves. © 2016 American Physical Society
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