scholarly journals The homotopy fixed point set of Lie group actions on elliptic spaces

2015 ◽  
Vol 110 (5) ◽  
pp. 1135-1156 ◽  
Author(s):  
Urtzi Buijs ◽  
Yves Félix ◽  
Sergio Huerta ◽  
Aniceto Murillo
Keyword(s):  
Fixed Point ◽  
Lie Group ◽  
Group Actions ◽  
Fixed Point Set ◽  
Lie Group Actions ◽  
Point Set ◽  
Elliptic Spaces

scholarly journals Equivariant basic cohomology of Riemannian foliations

2018 ◽  
Vol 2018 (745) ◽  
pp. 1-40 ◽  
Author(s):  
Oliver Goertsches ◽  
Dirk Töben
Keyword(s):  
Fixed Point ◽  
Lie Group ◽  
Betti Number ◽  
Group Actions ◽  
Fixed Point Set ◽  
Foliated Manifold ◽  
Riemannian Foliations ◽  
Basic Cohomology ◽  
Lie Group Actions ◽  
Point Set

Abstract The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino’s observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.

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.

scholarly journals Almost free actions on manifolds

1974 ◽  
Vol 10 (2) ◽  
pp. 177-196 ◽  
Author(s):  
Philip T. Church ◽  
Klaus Lamotke
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lie Group ◽  
Orbit Space ◽  
Fixed Point Set ◽  
Point Set ◽  
Free Actions ◽  
Connected Lie Group

Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and(2) the (even) number of fixed points.Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.

scholarly journals A non-fixed point theorem for Hamiltonian lie group actions

2002 ◽  
Vol 354 (7) ◽  
pp. 2971-2982 ◽  
Author(s):  
Christopher Allday ◽  
Volker Hauschild ◽  
Volker Puppe
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Lie Group ◽  
Group Actions ◽  
Lie Group Actions

scholarly journals Non-existence of odd periodic maps on certain spaces without fixed points

1985 ◽  
Vol 32 (3) ◽  
pp. 389-397
Author(s):  
Tej Bahadur Singh
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Group Actions ◽  
Fixed Point Set ◽  
Circle Group ◽  
Finitistic Space ◽  
Point Set

In this paper, we show that the fixed point set of Zp-actions, p an odd prime, on a finitistic space X of type (a, b) is non-empty, whenever b ≡ 0 (mod p). We also prove a similar result for circle group actions of finitistic spaces of (a, 0) type.

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

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