Tsubaki's engineered solution provides secret to smooth action of innovative new rowing machine

2003 ◽  
Vol 55 (2) ◽  
Keyword(s):  
Smooth Action

The Laitinen Conjecture for finite non-solvable groups

2012 ◽  
Vol 56 (1) ◽  
pp. 303-336 ◽  
Author(s):  
Krzysztof Pawałowski ◽  
Toshio Sumi
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Solvable Group ◽  
Prime Power ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Prime Power Order ◽  
Algebraic Condition ◽  
Smooth Action

AbstractFor any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).

The Topology of a Group Action

2020 ◽  
pp. 205-212
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Point ◽  
Vector Bundle ◽  
Group Action ◽  
Lie Group ◽  
Tubular Neighborhood ◽  
Compact Lie Group ◽  
Fixed Point Set ◽  
Smooth Action ◽  
Point Set ◽  
Neighborhood Theorem

This chapter describes the topology of a group action. It proves some topological facts about the fixed point set and the stabilizers of a continuous or a smooth action. The chapter also introduces the equivariant tubular neighborhood theorem and the equivariant Mayer–Vietoris sequence. A tubular neighborhood of a submanifold S in a manifold M is a neighborhood that has the structure of a vector bundle over S. Because the total space of a vector bundle has the same homotopy type as the base space, in calculating cohomology one may replace a submanifold by a tubular neighborhood. The tubular neighborhood theorem guarantees the existence of a tubular neighborhood for a compact regular submanifold. The Mayer–Vietoris sequence is a powerful tool for calculating the cohomology of a union of two open subsets. Both the tubular neighborhood theorem and the Mayer–Vietoris sequence have equivariant counterparts for a G-manifold where G is a compact Lie group.

Rigidity of Hamiltonian Actions

2003 ◽  
Vol 46 (2) ◽  
pp. 277-290 ◽  
Author(s):  
Frédéric Rochon
Keyword(s):  
General Theory ◽  
Lie Group ◽  
Geometric Analysis ◽  
Smooth Action ◽  
Symplectic Action ◽  
Hamiltonian Actions

AbstractThis paper studies the following question: Given an ω′-symplectic action of a Lie group on a manifoldMwhich coincides, as a smooth action, with a Hamiltonian ω-action, when is this action a Hamiltonian ω′-action? Using a result of Morse-Bott theory presented in Section 2, we show in Section 3 of this paper that such an action is in fact a Hamiltonian ω′-action, provided thatM is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only.

Locally linear actions on 3-manifolds

1988 ◽  
Vol 104 (2) ◽  
pp. 253-260 ◽  
Author(s):  
SŁawomir Kwasik ◽  
Kyung Bai Lee
Keyword(s):  
Finite Group ◽  
Topological Group ◽  
Group Action ◽  
Smooth Manifold ◽  
Local Behaviour ◽  
Smooth Action ◽  
A Finite Group ◽  
Locally Linear

Let a finite group G act topologically on a closed smooth manifold Mn. One of the most natural and basic questions is whether such an action can be smoothed. More precisely, let γ:G × Mn → Mn be a topological action of G on Mn. The action γ can be smoothed if there exists a smooth action and an equivariant homeomorphism It is well known that for n ≤ 2 every finite topological group action on Mn is smoothable. However already for n = 3 there are examples of topological actions on 3-manifolds which cannot be smoothed (see [1, 2] and references there). All these actions fail to be smoothable because of bad local behaviour.

scholarly journals The Lie Group in Infinite Dimension

2011 ◽  
Vol 2011 ◽  
pp. 1-35 ◽  
Author(s):  
V. Tryhuk ◽  
V. Chrastinová ◽  
O. Dlouhý
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Vector Fields ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Symmetries Of Differential Equations ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Infinitesimal Transformations

A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local,C∞smooth) action of a Lie group on infinite-dimensional space (a manifold modelled onℝ∞) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

scholarly journals Wong’s equations in Yang-Mills theory

Open Physics ◽  
2014 ◽  
Vol 12 (4) ◽  
Author(s):  
Sergey Storchak
Keyword(s):  
Dynamical System ◽  
Compact Riemannian Manifold ◽  
Scalar Particle ◽  
Orbit Space ◽  
Yang Mills ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Gauge Fixing ◽  
Dimensional Dynamical System ◽  
Dependent Coordinates

AbstractWong’s equations for the finite-dimensional dynamical system representing the motion of a scalar particle on a compact Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are derived. The equations obtained are written in terms of dependent coordinates which are typically used in an implicit description of the local dynamics given on the orbit space of the principal fiber bundle. Using these equations, we obtain Wong’s equations in a pure Yang-Mills gauge theory with Coulomb gauge fixing. This result is based on the existing analogy between the reduction procedures performed in a finite-dimensional dynamical system and the reduction procedure in Yang-Mills gauge fields.

Fixed-point sets of smooth actions on spheres

2007 ◽  
Vol 1 (1) ◽  
pp. 95-128
Author(s):  
Masaharu Morimoto
Keyword(s):  
Fixed Point ◽  
Finite Groups ◽  
Basic Problem ◽  
Point Sets ◽  
Fixed Point Set ◽  
Surgery Theory ◽  
Smooth Action ◽  
Point Set ◽  
Current Article ◽  
Fixed Point Sets

AbstractGiven a group, it is a basic problem to determine which manifolds can occur as a fixed-point set of a smooth action of this group on a sphere. The current article answers this problem for a family of finite groups including perfect groups and nilpotent Oliver groups. We obtain the answer as an application of a new deleting and inserting theorem which is formulated to delete (or insert) fixed-point sets from (or to) disks with smooth actions of finite groups. One of the keys to the proof is an equivariant interpretation of the surgery theory of S. E. Cappell and J. L. Shaneson, for obtaining homology equivalences.

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