scholarly journals Representation stability for homotopy groups of configuration spaces

2018 ◽  
Vol 2018 (737) ◽  
pp. 217-253 ◽  
Author(s):  
Alexander Kupers ◽  
Jeremy Miller
Keyword(s):  
General Theorem ◽  
Configuration Spaces ◽  
Homotopy Groups ◽  
Rational Homotopy ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Connected Manifold ◽  
Representation Stability ◽  
Gerstenhaber Algebras

AbstractWe prove that the dual rational homotopy groups of the configuration spaces of a 1-connected manifold of dimension at least 3 are uniformly representation stable in the sense of [6], and that their derived dual integral homotopy groups are finitely generated as{{\mathsf{FI}}}-modules in the sense of [4]. This is a consequence of a more general theorem relating properties of the cohomology groups of a 1-connected co-{{\mathsf{FI}}}-space to properties of its dual homotopy groups. We also discuss several other applications, including free Lie and Gerstenhaber algebras.

An arithmetic characterization of the rational homotopy groups of certain spaces

1979 ◽  
Vol 53 (2) ◽  
pp. 117-133 ◽  
Author(s):  
John B. Friedlander ◽  
Stephen Halperin
Keyword(s):  
Homotopy Groups ◽  
Rational Homotopy

scholarly journals Coformality and rational homotopy groups of spaces of long knots

2008 ◽  
Vol 15 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Greg Arone ◽  
Pascal Lambrechts ◽  
Victor Turchin ◽  
Ismar Volić
Keyword(s):  
Homotopy Groups ◽  
Rational Homotopy

Ideals with Trivial Conormal Bundle

1980 ◽  
Vol 32 (1) ◽  
pp. 210-218 ◽  
Author(s):  
A. V. Geramita ◽  
C. A. Weibel
Keyword(s):  
Polynomial Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Question Answering ◽  
General Theorem ◽  
Closed Field ◽  
Natural Question ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Conormal Bundle

Throughout this paper all rings considered will be commutative, noetherian with identity. If R is such a ring and M is a finitely generated R-module, we shall use v(M) to denote that non-negative integer with the property that M can be generated by v(M) elements but not by fewer.Since every ideal in a noetherian ring is finitely generated, it is a natural question to ask what v(I) is for a given ideal I. Hilbert's Nullstellensatz may be viewed as the first general theorem dealing with this question, answering it when I is a maximal ideal in a polynomial ring over an algebraically closed field.More recently, it has been noticed that the properties of an R-ideal I are intertwined with those of the R-module I/I2.

Rational Equivalence of Fibrations with Fibre G/K

1982 ◽  
Vol 34 (1) ◽  
pp. 31-43 ◽  
Author(s):  
Stephen Halperin ◽  
Jean Claude Thomas
Keyword(s):  
Finite Type ◽  
Homotopy Type ◽  
Lie Group ◽  
The Other ◽  
Homotopy Groups ◽  
Homotopy Equivalence ◽  
Rational Homotopy ◽  
Algebraic Invariants ◽  
Associated Bundles ◽  
Connected Lie Group

Let be two Serre fibrations with same base and fibre in which all the spaces have the homotopy type of simple CW complexes of finite type. We say they are rationally homotopically equivalent if there is a homotopy equivalence between the localizations at Q which covers the identity map of BQ.Such an equivalence implies, of course, an isomorphism of cohomology algebras (over Q) and of rational homotopy groups; on the other hand isomorphisms of these classical algebraic invariants are usually (by far) insufficient to establish the existence of a rational homotopy equivalence.Nonetheless, as we shall show in this note, for certain fibrations rational homotopy equivalence is in fact implied by the existence of an isomorphism of cohomology algebras. While these fibrations are rare inside the class of all fibrations, they do include principal bundles with structure groups a connected Lie group G as well as many associated bundles with fibre G/K.

scholarly journals CRITICAL POINTS OF PAIRS OF VARIETIES OF ALGEBRAS

2009 ◽  
Vol 19 (01) ◽  
pp. 1-40 ◽  
Author(s):  
PIERRE GILLIBERT
Keyword(s):  
Natural Number ◽  
Critical Points ◽  
Modular Lattice ◽  
General Theorem ◽  
Finitely Generated ◽  
Congruence Distributive

For a class [Formula: see text] of algebras, denote by Conc[Formula: see text] the class of all (∨, 0)-semilattices isomorphic to the semilattice ConcA of all compact congruences of A, for some A in [Formula: see text]. For classes [Formula: see text] and [Formula: see text] of algebras, we denote by [Formula: see text] the smallest cardinality of a (∨, 0)-semilattices in Conc[Formula: see text] which is not in Conc[Formula: see text] if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties [Formula: see text] and [Formula: see text], [Formula: see text] is either finite, or ℵn for some natural number n, or ∞. We also find two finitely generated modular lattice varieties [Formula: see text] and [Formula: see text] such that [Formula: see text], thus answering a question by J. Tůma and F. Wehrung.

CELLULAR AUTOMATA OVER SEMI-DIRECT PRODUCT GROUPS: REDUCTION AND INVERTIBILITY RESULTS

2006 ◽  
Vol 16 (06) ◽  
pp. 1071-1085
Author(s):  
SILVIO CAPOBIANCO
Keyword(s):  
Cellular Automata ◽  
Direct Product ◽  
Configuration Spaces ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Current State ◽  
Global Properties ◽  
Finite Factor ◽  
Finiteness Properties

Cellular automata are transformations of configuration spaces over finitely generated groups, such that the next state in a point only depends on the current state of a finite neighborhood of the point itself. Many questions arise about retrieving global properties from such local descriptions, and finding algorithms to perform these tasks. We consider the case when the group is a semi-direct product of two finitely generated groups, and show that a finite factor (whatever it is) can be thought of as part of the alphabet instead of the group, preserving both the dynamics and some "finiteness" properties. We also show that, under reasonable hypotheses, this reduction is computable: this leads to some reduction theorems related to the invertibility problem.

Rational homotopy groups and Koszul algebras

2002 ◽  
Vol 335 (1) ◽  
pp. 53-58 ◽  
Author(s):  
Stefan Papadima ◽  
Alexander I. Suciu
Keyword(s):  
Homotopy Groups ◽  
Koszul Algebras ◽  
Rational Homotopy

scholarly journals Goodwillie Approximations to Higher Categories

2021 ◽  
Vol 272 (1333) ◽  
Author(s):  
Gijs Heuts
Keyword(s):  
Homotopy Theory ◽  
Symmetric Groups ◽  
Homotopy Groups ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Content Type ◽  
Simply Connected ◽  
Derivatives Of ◽  
Connected Spaces

We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of ∞ \infty -categories C \mathcal {C} and classify such Goodwillie towers in terms of the derivatives of the identity functor of C \mathcal {C} . As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras in spectra with Tate diagonals. Our classification of Goodwillie towers simplifies considerably in settings where the Tate cohomology of the symmetric groups vanishes. As an example we apply our methods to rational homotopy theory. Another application identifies the homotopy theory of p p -local spaces with homotopy groups in a certain finite range with the homotopy theory of certain algebras over Ching’s spectral version of the Lie operad. This is a close analogue of Quillen’s results on rational homotopy.

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