Foliations by planes and Lie group actions

J. A. Álvarez López; J. L. Arraut; C. Biasi

Displaying similar documents to “Foliations by planes and Lie group actions”

Correspondence between diffeomorphism groups and singular foliations

Tomasz Rybicki (2012)

Annales Polonici Mathematici

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It is well-known that any isotopically connected diffeomorphism group G of a manifold determines a unique singular foliation $ℱ_{G}$ . A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup [G,G] of an isotopically connected, factorizable and non-fixing $C^{r}$ diffeomorphism group G is simple iff the foliation $ℱ_{[G, G]}$ defined by [G,G] admits no proper...

Warped compact foliations

Szymon M. Walczak (2008)

Annales Polonici Mathematici

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The notion of the Hausdorffized leaf space $\tilde{}$ of a foliation is introduced. A sufficient condition for warped compact foliations to converge to $\tilde{}$ is given. Moreover, a necessary condition for warped compact Hausdorff foliations to converge to $\tilde{}$ is shown. Finally, some examples are examined.

Finite actions on the Klein four-orbifold and prism manifolds

John Kalliongis, Ryo Ohashi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We describe the finite group actions, up to equivalence, which can act on the orbifold $Σ (2, 2, 2)$ , and their quotient types. This is then used to consider actions on prism manifolds $M (b, d)$ which preserve a longitudinal fibering, but do not leave any Heegaard Klein bottle invariant. If $ϕ : G \to Homeo (M (b, d))$ is such an action, we show that $M (b, d) = M (b, 2)$ and $M (b, 2) / ϕ$ fibers over a certain collection of 2-orbifolds with positive Euler characteristic which are covered by $Σ (2, 2, 2)$ . For the standard actions, we compute the fundamental group of $M (b, 2) / ϕ$ and...

Real algebraic actions on projective spaces - A survey

Ted Petrie (1973)

Annales de l'institut Fourier

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Let $G$ be a compact lie group. We introduce the set $S_{G} (Y)$ for every smooth $G$ manifold $Y$ . It consists of equivalence classes of pair $(X, f)$ where $f : X \to Y$ is a $G$ map which defines a homotopy equivalence from $X$ to $Y$ . Two pairs $(X_{i}, f_{i})$ , for $i = 0, 1$ , are equivalent if there is a $G$ homotopy equivalence $φ : X_{0} \to X_{1}$ such that $f_{0}$ is $G$ homotopic to $f_{1} \circ φ$ . Properties of the set $S_{G} (Y)$ and related to the representation of $G$ on the tangent spaces of $X$ and $Y$ at the fixed points. For the case $G = S^{1}$ and $Y$ is the $S^{1}$ manifold defined by a “linear”...

Correction for the paper “ $S^{3}$ -bundles and exotic actions”

T. E. Barros (2001)

Bulletin de la Société Mathématique de France

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In [R] explicit representatives for $S^{3}$ -principal bundles over $S^{7}$ are constructed, based on these constructions explicit free $S^{3}$ -actions on the total spaces are described, with quotients exotic $7$ -spheres. To describe these actions a classification formula for the bundles is used. This formula is not correct. In Theorem 1 below, we correct the classification formula and in Theorem 2 we exhibit the correct indices of the exotic $7$ -spheres that occur as quotients of the free $S^{3}$ -actions described...

Groups of $C^{r, s}$ -diffeomorphisms related to a foliation

Jacek Lech, Tomasz Rybicki (2007)

Banach Center Publications

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The notion of a $C^{r, s}$ -diffeomorphism related to a foliation is introduced. A perfectness theorem for the group of $C^{r, s}$ -diffeomorphisms is proved. A remark on $C^{n + 1}$ -diffeomorphisms is given.

Minimal, rigid foliations by curves on $ℂ ℙ^{n}$

Frank Loray, Julio C. Rebelo (2003)

Journal of the European Mathematical Society

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We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space $ℂ ℙ^{n}$ for every dimension $n \geq 2$ and every degree $d \geq 2$ . Precisely, we construct a foliation $ℱ$ which is induced by a homogeneous vector field of degree $d$ , has a finite singular set and all the regular leaves are dense in the whole of $ℂ ℙ^{n}$ . Moreover, $ℱ$ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if $ℱ$ is conjugate to another holomorphic...

On finite groups acting on a connected sum of 3-manifolds S² × S¹

Bruno P. Zimmermann (2014)

Fundamenta Mathematicae

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Let $H_{g}$ denote the closed 3-manifold obtained as the connected sum of g copies of S² × S¹, with free fundamental group of rank g. We prove that, for a finite group G acting on $H_{g}$ which induces a faithful action on the fundamental group, there is an upper bound for the order of G which is quadratic in g, but there does not exist a linear bound in g. This implies then a Jordan-type bound for arbitrary finite group actions on $H_{g}$ which is quadratic in g. For the proofs we develop a calculus...

Topological friction in aperiodic minimal $ℝ^{m}$ -actions

Jarosław Kwapisz (2010)

Fundamenta Mathematicae

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For a continuous map f preserving orbits of an aperiodic $ℝ^{m}$ -action on a compact space, its displacement function assigns to x the “time” $t \in ℝ^{m}$ it takes to move x to f(x). We show that this function is continuous if the action is minimal. In particular, f is homotopic to the identity along the orbits of the action.

Shadowing in multi-dimensional shift spaces

Piotr Oprocha (2008)

Colloquium Mathematicae

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We show that the class of expansive $ℤ^{d}$ actions with P.O.T.P. is wider than the class of actions topologically hyperbolic in some direction $ν \in ℤ^{d}$ . Our main tool is an extension of a result by Walters to the multi-dimensional symbolic dynamics case.

Non-orbit equivalent actions of $𝔽_{n}$

Adrian Ioana (2009)

Annales scientifiques de l'École Normale Supérieure

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For any $2 \leq n \leq \infty$ , we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group $𝔽_{n}$ . These actions arise as diagonal products between a generalized Bernoulli action and the action $𝔽_{n} ↷ (𝕋^{2}, λ^{2})$ , where $𝔽_{n}$ is seen as a subgroup of ${SL}_{2} (ℤ)$ .

$Z ₂^{k}$ -actions fixing point ∪ Vⁿ

Pedro L. Q. Pergher (2002)

Fundamenta Mathematicae

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We describe the equivariant cobordism classification of smooth actions $(M^{m}, Φ)$ of the group $G = Z ₂^{k}$ on closed smooth m-dimensional manifolds $M^{m}$ for which the fixed point set of the action is the union F = p ∪ Vⁿ, where p is a point and Vⁿ is a connected manifold of dimension n with n > 0. The description is given in terms of the set of equivariant cobordism classes of involutions fixing p ∪ Vⁿ. This generalizes a lot of previously obtained particular cases of the above question; additionally,...

Shadowing in actions of some Abelian groups

Sergei Yu. Pilyugin, Sergei B. Tikhomirov (2003)

Fundamenta Mathematicae

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We study shadowing properties of continuous actions of the groups $ℤ^{p}$ and $ℤ^{p} \times ℝ^{p}$ . Necessary and sufficient conditions are given under which a linear action of $ℤ^{p}$ on $ℂ^{m}$ has a Lipschitz shadowing property.

Homogenization of codimension 1 actions of $ℝ^{n}$ near a compact orbit

Marcos Craizer (1994)

Annales de l'institut Fourier

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Let $Φ$ be a $C^{\infty} ℝ^{n}$ -action on an orientable $(n + 1)$ -dimensional manifold. Assume $Φ$ has an isolated compact orbit $T$ and let $W$ be a small tubular neighborhood of it. By a $C^{\infty}$ change of variables, we can write $W = ℝ^{n} / ℤ^{n} \times I$ and $T = 𝕋^{n} \times [0]$ , where $I$ is some interval containing 0. In this work, we show that by a $C^{0}$ change of variables, $C^{\infty}$ outside $T$ , we can make $Φ_{| W}$ invariant by transformations of the type $(x, z) \to (x + a, z), a \in ℝ^{n}$ , where $x \in ℝ^{n} / ℤ^{n}$ and $z \in I$ . As a corollary one cas describe completely the dynamics of $Φ$ in $W$ .

Homotopy decompositions of orbit spaces and the Webb conjecture

Jolanta Słomińska (2001)

Fundamenta Mathematicae

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Let p be a prime number. We prove that if G is a compact Lie group with a non-trivial p-subgroup, then the orbit space $(B_{p} (G)) / G$ of the classifying space of the category associated to the G-poset $_{p} (G)$ of all non-trivial elementary abelian p-subgroups of G is contractible. This gives, for every G-CW-complex X each of whose isotropy groups contains a non-trivial p-subgroup, a decomposition of X/G as a homotopy colimit of the functor $X^{E ₙ} / (N E ₀ \cap . . . \cap N E ₙ)$ defined over the poset $(s d_{p} (G)) / G$ , where sd is the barycentric subdivision....

On the group of real analytic diffeomorphisms

Takashi Tsuboi (2009)

Annales scientifiques de l'École Normale Supérieure

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The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the $n$ -dimensional torus, its identity component is a simple group. For $U (1)$ fibered manifolds, for manifolds admitting special semi-free $U (1)$ actions and for 2- or 3-dimensional manifolds with nontrivial $U (1)$ actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

Torus embeddings, polyhedra, k*-actions and homology

Jerzy Jurkiewicz

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CONTENTSIntroduction..............................................................................................51. General torus embeddings...................................................................7 1.1. Sets of subrings..............................................................................7 1.2. Complex of cones and torus embeddings. Basic properties and notation...............8 1.3. Jets of 1-p.s. at 0...........................................................................12 1.4....

Foliations by complex manifolds involving the complex Hessian

Julian Ławrynowicz, Jerzy Kalina, Masami Okada

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SummaryIn 1979 the second named author proved, in a joint paper with J. Ławrynowicz, the existence of a foliation of a bounded domain in $ℂ^{n}$ by complex submanifolds of codimension k+p-1, connected in some sense with a real (1,1) C³-form of rank k and the pth power of the complex Hessian of a C³-function u with im u plurisubharmonic and the property that for every leaf of this foliation the restricted functions im u, re u and $(\partial / \partial z_{j}) i m u$ , $(\partial / \partial z_{j}) r e u$ are pluriharmonic and holomorphic, respectively.Now the...

$Z ₂^{k}$ -actions with a special fixed point set

Pedro L. Q. Pergher, Rogério de Oliveira (2005)

Fundamenta Mathematicae

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Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^{m}$ is any smooth and closed m-dimensional manifold with m > n and $T : N^{m} \to N^{m}$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^{m}; Φ)$ of the group $G = Z ₂^{k}$ on closed smooth m-dimensional manifolds $M^{m}$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....