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.

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

Does a Predictive Minimal Model of Friction-Induced Vibration Exist?

2010 ◽  
Author(s):  
Tore Butlin ◽  
Jim Woodhouse
Keyword(s):  
Minimal Model ◽  
Mode Coupling ◽  
Test System ◽  
Minimal Models ◽  
Pin On Disc ◽  
Limited Frequency ◽  
Extensive Program ◽  
Negative Damping ◽  
Model Of Friction ◽  
Friction Induced Vibration

Highly idealised models of friction-induced vibration have been motivated by an attempt to capture what is essential to the phenomenon. This approach has resulted in a few simple mechanisms that are thought to capture common routes to instability. This paper aims to determine how well these perform as approximations to a more complex system, and whether the essential ingredients needed for a minimal model can be identified. We take a reduced-order model that exemplifies ‘mode-coupling’ and explore the extent to which it can approximate predictions based on an experimentally identified test-system. For the particular test system under study, two-mode ‘mode-coupling’ is rarely a good approximation and three modes are usually required to model a limited frequency range. We then compare predictions with results from an extensive program of sliding contact tests on a pin-on-disc rig in order to identify which ingredients are needed to explain observed squeal events. The results suggest that several minimal models would be needed to describe all observed squeal initiations, but the ‘negative-damping’ route to instability, which requires a velocity-dependent friction law, convincingly accounts for one cluster.

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

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 Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture

2019 ◽  
Vol 2019 (747) ◽  
pp. 147-174 ◽  
Author(s):  
Karol Palka
Keyword(s):  
Minimal Model ◽  
Resolution Of Singularities ◽  
Model Program ◽  
Minimal Models ◽  
Minimal Model Program ◽  
Effective Version

Abstract Let {E\subseteq\mathbb{P}^{2}} be a complex rational cuspidal curve and let {(X,D)\to(\mathbb{P}^{2},E)} be the minimal log resolution of singularities. We prove that E has at most six cusps and we establish an effective version of the Zaidenberg finiteness conjecture (1994) concerning Eisenbud–Neumann diagrams of E. This is done by analyzing the Minimal Model Program run for the pair {(X,\frac{1}{2}D)} . Namely, we show that {\mathbb{P}^{2}\setminus E} is {\mathbb{C}^{**}} -fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

On the differentials in the Adams spectral sequence

1964 ◽  
Vol 60 (3) ◽  
pp. 409-420 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Spectral Sequence ◽  
Higher Order ◽  
Adams Spectral Sequence ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Cohomology Operations ◽  
Image Position

In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This theorem will be stated precisely at the end of section 2, after a summary of the necessary information about the Adams spectral sequence and higher-order cohomology operations; the proof will follow in section 3. Finally, in section 4, we shall consider, by way of example, the Adams spectral sequence for the stable homotopy groups of spheres: we show how our theorem gives a proof of Liulevicius's result that , where the elements hn (n ≥ 0) are base elements ofcorresponding to the elements Sq2n in A, the mod 2 Steenrod algebra.

scholarly journals Polarized pairs, log minimal models, and Zariski decompositions

2014 ◽  
Vol 215 ◽  
pp. 203-224 ◽  
Author(s):  
Caucher Birkar ◽  
Zhengyu Hu
Keyword(s):  
Minimal Model ◽  
Birational Geometry ◽  
Minimal Models ◽  
Positive Part ◽  
Log Canonical ◽  
Canonical Pair ◽  
The Way

AbstractWe continue our study of the relation between log minimal models and various types of Zariski decompositions. Let (X,B) be a projective log canonical pair. We will show that (X,B) has a log minimal model if either KX + B birationally has a Nakayama–Zariski decomposition with nef positive part, or if KX +B is big and birationally has a Fujita–Zariski or Cutkosky–Kawamata–Moriwaki–Zariski decomposition. Along the way we introduce polarized pairs (X,B +P), where (X,B) is a usual projective pair and where P is nef, and we study the birational geometry of such pairs.

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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