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

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

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.

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.

Sullivan's Minimal Models and Higher Order Whitehead Products

1978 ◽  
Vol 30 (5) ◽  
pp. 961-982 ◽  
Author(s):  
Peter Andrews ◽  
Martin Arkowitz
Keyword(s):  
Minimal Model ◽  
Additive Group ◽  
Higher Order ◽  
Homotopy Groups ◽  
Rational Homotopy ◽  
Minimal Models

The theory of minimal models, as developed by Sullivan [6; 8; 16] gives a method of computing the rational homotopy groups of a space X (that is, the homotopy groups of X tensored with the additive group of rationals Q). One associates to X a free, differential, graded-commutative lgebra , over Q, called the minimal model of X, from which one can read off the rational homotopy groups of X.

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