Homotopy Groups of Transformation Groups

1969 ◽  
Vol 21 ◽  
pp. 1123-1136 ◽  
Author(s):  
F. Rhodes
Keyword(s):  
Phase Space ◽  
Continuous Flow ◽  
Homotopy Group ◽  
Transformation Group ◽  
Group Extension ◽  
Transformation Groups ◽  
Necessary Condition ◽  
Homotopy Groups ◽  
Split Extension ◽  
Discrete Flow

In a previous paper (2) I defined the fundamental group σ(X, x0, G) of a group Gof homeomorphisms of a space X, and showed that if the transformation group admits a family of preferred paths, then σ(X, x0, G) can be represented as a group extension of π1(X, x0) by G. In this paper the homotopy groups of a transformation group are defined. The nth absolute homotopy group of a transformation group which admits a family of preferred paths is shown to be representable as a split extension of the nth absolute torus homotopy group τn(X, x0) by G.In § 6 it is shown that the action of G on X induces a homomorphism of Ginto a quotient group of a subgroup of the group of automorphisms of τn(X, x0). This homomorphism is used to obtain a necessary condition for the embedding of one transformation group in another, in particular, for the embedding of a discrete flow in a continuous flow with the same phase space.

scholarly journals On locally weakly almost periodic transformation groups

1982 ◽  
Vol 25 (2) ◽  
pp. 215-219
Author(s):  
Saber Elaydi
Keyword(s):  
Phase Space ◽  
Transformation Group ◽  
Transformation Groups ◽  
Almost Periodic ◽  
Locally Compact ◽  
Weakly Almost Periodic ◽  
Phase Group ◽  
Periodic Transformation

It is shown that a transformation group with a locally compact Hausdorff phase space and an abelian phase group is locally weakly almost periodic if and only if it is P-locally weakly almost periodic for some replete semigroup P in the phase group.

Equicontinuity and -recursive transformation groups

1965 ◽  
Vol 61 (2) ◽  
pp. 333-336 ◽  
Author(s):  
Janet Allsbrook ◽  
R. W. Bagley
Keyword(s):  
Phase Space ◽  
Transformation Group ◽  
Transformation Groups ◽  
Almost Periodic ◽  
Convergence Space ◽  
Topological Transformation ◽  
Uniform Convergence Space ◽  
Left And Right ◽  
Uniformly Continuous ◽  
Special Case

In this paper we obtain results on equicontinuity and apply them to certain recursive properties of topological transformation groups (X, T, π) with uniform phase space X. For example, in the special case that each transition πt is uniformly continuous we consider the transformation group (X,Ψ,ρ), where Ψ is the closure of {πt|t∈T} in the space of all unimorphisms of X onto X with the topology of uniform convergence (space index topology) and p(x, φ) = φ(x) for (x, φ)∈ X × Ψ. (See (1), page 94, 11·18.) If π is a mapping on X × T we usually write ‘xt’ for ‘π(x, t)’ and ‘AT’ for ‘π(A × T)’ where A ⊂ X. In this case we obtain the following results:I. If (X, T, π) is almost periodic [regularly almost periodic] and πxis equicontinuous, then (X,Φ,ρ) is almost periodic [regularly almost periodic]II. Let A be a compact subset of X such that. If the left and right uniformities of T are equal and (X, T, π) is almost periodic [regularly almost periodic], then (X, T, π) is almost periodic [regularly almost periodic].

IMBEDDING OF TRANSFORMATION GROUPS IN AFFINE ONES AND RENORMALIZATION OF TWO-DIMENSIONAL HOMOGENEOUS σ-MODELS

1993 ◽  
Vol 08 (31) ◽  
pp. 2937-2942
Author(s):  
A. V. BRATCHIKOV
Keyword(s):  
Homogeneous Space ◽  
Transformation Group ◽  
Affine Group ◽  
Coupling Constants ◽  
Transformation Groups ◽  
Two Dimensional

The BLZ method for the analysis of renormalizability of the O(N)/O(N − 1) model is extended to the σ-model built on an arbitrary homogeneous space G/H and in arbitrary coordinates. For deriving Ward-Takahashi (WT) identities an imbedding of the transformation group G in an affine group is used. The structure of the renormalized action is found. All the infinities can be absorbed in a coupling constants renormalization and in a renormalization of auxiliary constants which are related to the imbedding.

scholarly journals QUASI-PERIODIC SOLUTIONS OF THE RELATIVISTIC TODA HIERARCHY

2012 ◽  
Vol 19 (04) ◽  
pp. 1250030 ◽  
Author(s):  
DONG GONG ◽  
XIANGUO GENG
Keyword(s):  
Meromorphic Function ◽  
Periodic Solutions ◽  
Continuous Flow ◽  
Hyperelliptic Curve ◽  
Algebraic Curves ◽  
Asymptotic Properties ◽  
Toda Hierarchy ◽  
Discrete Flow

On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the relativistic Toda hierarchy are straightened out using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the relativistic Toda hierarchy are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the hyperelliptic curve.

Symmetric Powers of Motives

2019 ◽  
pp. 232-252
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Homotopy Group ◽  
Orbit Space ◽  
Symmetric Power ◽  
Homotopy Groups ◽  
Integral Homology ◽  
Symmetric Powers ◽  
Mac Lane ◽  
Abelian Monoid

This chapter develops the basic theory of symmetric powers of smooth varieties. The constructions in this chapter are based on an analogy with the corresponding symmetric power constructions in topology. If 𝐾 is a set (or even a topological space) then the symmetric power 𝑆𝑚𝐾 is defined to be the orbit space 𝐾𝑚/Σ‎𝑚, where Σ‎𝑚 is the symmetric group. If 𝐾 is pointed, there is an inclusion 𝑆𝑚𝐾 ⊂ 𝑆𝑚+1𝐾 and 𝑆∞𝐾 = ∪𝑆𝑚𝐾 is the free abelian monoid on 𝐾 − {*}. When 𝐾 is a connected topological space, the Dold–Thom theorem says that ̃𝐻*(𝐾, ℤ) agrees with the homotopy groups π‎ *(𝑆∞𝐾). In particular, the spaces 𝑆∞(𝑆 𝑛) have only one homotopy group (𝑛 ≥ 1) and hence are the Eilenberg–Mac Lane spaces 𝐾(ℤ, 𝑛) which classify integral homology.

A Mapping Problem and Jp-Index. I

1970 ◽  
Vol 22 (4) ◽  
pp. 705-712 ◽  
Author(s):  
Masami Wakae ◽  
Oma Hamara
Keyword(s):  
Cyclic Group ◽  
Prime Order ◽  
Transformation Group ◽  
Classical Group ◽  
Cohomology Ring ◽  
Transformation Groups ◽  
Normal Space ◽  
Mapping Problem ◽  
Countable Basis ◽  
Kakutani Theorem

Indices of normal spaces with countable basis for equivariant mappings have been investigated by Bourgin [4; 6] and by Wu [11; 12] in the case where the transformation groups are of prime order p. One of us has extended the concept to the case where the transformation group is a cyclic group of order pt and discussed its applications to the Kakutani Theorem (see [10]). In this paper we will define the Jp-index of a normal space with countable basis in the case where the transformation group is a cyclic group of order n, where n is divisible by p. We will decide, by means of the spectral sequence technique of Borel [1; 2], the Jp-index of SO(n) where n is an odd integer divisible by p. The method used in this paper can be applied to find the Jp-index of a classical group G whose cohomology ring over Jp has a system of universally transgressive generators of odd degrees.

On Localized Unstable K1-groups and Applications to Self-homotopy Groups

2014 ◽  
Vol 57 (2) ◽  
pp. 344-356
Author(s):  
Daisuke Kishimoto ◽  
Akira Kono ◽  
Mitsunobu Tsutaya
Keyword(s):  
Homotopy Group ◽  
The Self ◽  
Explicit Description ◽  
Homotopy Groups

AbstractThe method for computing the p-localization of the group [X, U(n)], by Hamanaka in 2004, is revised. As an application, an explicit description of the self-homotopy group of Sp(3) localized at p ≥ 5 is given and the homotopy nilpotency of Sp(3) localized at p ≥ 5 is determined.

Some properties of wild orbits of infinite-dimensional transformation groups

1982 ◽  
Vol 92 (3) ◽  
pp. 429-435
Author(s):  
R. J. Magnus
Keyword(s):  
Transformation Group ◽  
Transformation Groups ◽  
Infinite Dimensional

A wild orbit is an orbit of a transformation group which has the property that its dimension appears to increase in a paradoxical way. The existence of such orbits was noted in a previous paper (1).

scholarly journals A Simple Derivation of Tropical Cyclone Ventilation Theory and Its Application to Capped Surface Entropy Fluxes

2017 ◽  
Vol 74 (9) ◽  
pp. 2989-2996 ◽  
Author(s):  
Daniel R. Chavas
Keyword(s):  
Phase Space ◽  
Wind Speed ◽  
Tropical Cyclone ◽  
Analytical Solutions ◽  
Heat Engine ◽  
Vertical Wind Shear ◽  
Intensity Change ◽  
Necessary Condition ◽  
Surface Entropy ◽  
Entropy Flux

Abstract In a recent study, a theory was presented for the dependence of tropical cyclone intensity on the ventilation of dry air by environmental vertical wind shear. This theory was found to successfully capture the statistics of intensity dynamics in the historical record. This theory is rederived here from a simple three-term power budget and extended to analytical solutions for the complete phase space, including the change in storm intensity itself. The derivation is then generalized to the case of a capped surface entropy flux wind speed, including analytical solutions defined relative to both the traditional potential intensity and the capped-flux potential intensity. The results demonstrate that a cap on the surface entropy flux wind speed reduces the potential intensity of the system and effectively amplifies the detrimental effect of ventilation on the tropical cyclone heat engine. However, such a cap does not alter the qualitative structure of the phase-space solution for intensity change phrased relative to the capped-flux potential intensity. Thus, the wind speed dependence of surface entropy fluxes is important for intensity change in real-world storms, though it is not a necessary condition for intensification in general. Indeed, a residual power surplus may remain available to intensify a storm even in the presence of a cap, though intensification may be fully suppressed for sufficiently strong ventilation. This work complements a recent numerical simulation study and provides further evidence that there is no disconnect between extant tropical cyclone theory and the finding in numerical simulations that a storm may intensify in the presence of capped surface entropy fluxes.

The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions II

1992 ◽  
Vol 122 (1-2) ◽  
pp. 127-135 ◽  
Author(s):  
John W. Rutter
Keyword(s):  
Group Extension ◽  
Equivalence Classes ◽  
Homotopy Groups ◽  
Geometric Proof ◽  
Homotopy Equivalence ◽  
Connected Space ◽  
Abelian Kernel ◽  
Simply Connected ◽  
Extension Sequence ◽  
Connected Spaces

SynopsisWe give here an abelian kernel (central) group extension sequence for calculating, for a non-simply-connected space X, the group of pointed self-homotopy-equivalence classes . This group extension sequence gives in terms of , where Xn is the nth stage of a Postnikov decomposition, and, in particular, determines up to extension for non-simplyconnected spaces X having at most two non-trivial homotopy groups in dimensions 1 and n. We give a simple geometric proof that the sequence splits in the case where is the generalised Eilenberg–McLane space corresponding to the action ϕ: π1 → aut πn, and give some information about the class of the extension in the general case.

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