Equivariant measurable liftings

Nicolas Monod

Displaying similar documents to “Equivariant measurable liftings”

Degree of T-equivariant maps in ℝⁿ

Joanna Janczewska, Marcin Styborski (2007)

Banach Center Publications

Similarity:

A special case of G-equivariant degree is defined, where G = ℤ₂, and the action is determined by an involution $T : ℝ^{p} \oplus ℝ^{q} \to ℝ^{p} \oplus ℝ^{q}$ given by T(u,v) = (u,-v). The presented construction is self-contained. It is also shown that two T-equivariant gradient maps $f, g : (ℝ ⁿ, S^{n - 1}) \to (ℝ ⁿ, ℝ ⁿ ∖ 0)$ are T-homotopic iff they are gradient T-homotopic. This is an equivariant generalization of the result due to Parusiński.

Positivity and Kleiman transversality in equivariant $K$ -theory of homogeneous spaces

Dave Anderson, Stephen Griffeth, Ezra Miller (2011)

Journal of the European Mathematical Society

Similarity:

We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth–Ram [GrRa04] concerning the alternation of signs in the structure constants for torus-equivariant $K$ -theory of generalized ﬂag varieties $G / P$ . These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with ﬁnitely many orbits. The computation of the coefficients in the expansion...

An $S^{1}$ -equivariant degree and the Fuller index

Grzegorz Dylawerski, Kazimierz Gęba, Jerzy Jodel, Wacław Marzantowicz (1991)

Annales Polonici Mathematici

Similarity:

$Z ₂^{k}$ -actions fixing point ∪ Vⁿ

Pedro L. Q. Pergher (2002)

Fundamenta Mathematicae

Similarity:

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,...

On isomorphisms between fibrations over $T^{k}$ associated with a $T^{k}$ action

Michał Sadowski (1991)

Annales Polonici Mathematici

Similarity:

The equivariant universality and couniversality of the Cantor cube

Michael G. Megrelishvili, Tzvi Scarr (2001)

Fundamenta Mathematicae

Similarity:

Let ⟨G,X,α⟩ be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let $⟨ H ({0, 1}^{ℵ ₀}), {0, 1}^{ℵ ₀}, τ ⟩$ be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding $φ : G ↪ H ({0, 1}^{ℵ ₀})$ ; (2) there exists an embedding $ψ : X ↪ {0, 1}^{ℵ ₀}$ , equivariant with respect to φ, such that ψ(X) is an equivariant retract of ${0, 1}^{ℵ ₀}$ with respect...

Stable soliton resolution for equivariant wave maps exterior to a ball

Andrew Lawrie (2014-2015)

Séminaire Laurent Schwartz — EDP et applications

Similarity:

In this report we review the proof of the stable soliton resolution conjecture for equivariant wave maps exterior to a ball in $ℝ^{3}$ and taking values in the $3$ -sphere. This is joint work with Carlos Kenig, Baoping Liu, and Wilhelm Schlag.

Commuting involutions whose fixed point set consists of two special components

Pedro L. Q. Pergher, Rogério de Oliveira (2008)

Fundamenta Mathematicae

Similarity:

Let Fⁿ be a connected, smooth and closed n-dimensional manifold. We call Fⁿ a manifold with property when it has the following property: if $N^{m}$ is any smooth 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. Examples of manifolds with this property are: the real, complex and quaternionic even-dimensional projective spaces $R P^{2 n}$ , $C P^{2 n}$ and $H P^{2 n}$ , and the connected sum of $R P^{2 n}$ and any number of copies of Sⁿ × Sⁿ, where Sⁿ is the n-sphere...

Equivariant deformation quantization for the cotangent bundle of a flag manifold

Ranee Brylinski (2002)

Annales de l’institut Fourier

Similarity:

Let $X$ be a (generalized) flag manifold of a complex semisimple Lie group $G$ . We investigate the problem of constructing a graded star product on $ℛ = R (T^{☆} X)$ which corresponds to a $G$ -equivariant quantization of symbols into twisted differential operators acting on half-forms on $X$ . We construct, when $ℛ$ is generated by the momentum functions $μ^{x}$ for $G$ , a preferred choice of $☆$ where $μ^{x} ☆ φ$ has the form $μ^{x} φ + \frac{1}{2} {μ^{x}, φ} t + Λ^{x} (φ) t^{2}$ . Here $Λ^{x}$ are operators on $ℛ$ . In the known examples, $Λ^{x}$ ( $x \neq 0$ ) is not a differential operator, and so the star...

A note on the theorems of Lusternik-Schnirelmann and Borsuk-Ulam

T. E. Barros, C. Biasi (2008)

Colloquium Mathematicae

Similarity:

Let p be a prime number and X a simply connected Hausdorff space equipped with a free $ℤ_{p}$ -action generated by $f_{p} : X \to X$ . Let $α : S^{2 n - 1} \to S^{2 n - 1}$ be a homeomorphism generating a free $ℤ_{p}$ -action on the (2n-1)-sphere, whose orbit space is some lens space. We prove that, under some homotopy conditions on X, there exists an equivariant map $F : (S^{2 n - 1}, α) \to (X, f_{p})$ . As applications, we derive new versions of generalized Lusternik-Schnirelmann and Borsuk-Ulam theorems.

Projectively equivariant quantization and symbol on supercircle $S^{1 | 3}$

Taher Bichr (2021)

Czechoslovak Mathematical Journal

Similarity:

Let $𝒟_{λ, μ}$ be the space of linear differential operators on weighted densities from $ℱ_{λ}$ to $ℱ_{μ}$ as module over the orthosymplectic Lie superalgebra $𝔬𝔰𝔭 (3 | 2)$ , where $ℱ_{λ}$ , $ł \in ℂ$ is the space of tensor densities of degree $λ$ on the supercircle $S^{1 | 3}$ . We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.

Equivariant K-theory of flag varieties revisited and related results

V. Uma (2013)

Colloquium Mathematicae

Similarity:

We obtain several several results on the multiplicative structure constants of the T-equivariant Grothendieck ring $K_{T} (G / B)$ of the flag variety G/B. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T} (G / B)$ to R(T) ⊗ R(T), where R(T) denotes the representation ring of the torus T. We further apply our results to describe the multiplicative structure constants of $K {(X)}_{ℚ}$ where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure...

Noncommutative Borsuk-Ulam-type conjectures

Paul F. Baum, Ludwik Dąbrowski, Piotr M. Hajac (2015)

Banach Center Publications

Similarity:

Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if $δ : A \to A ⨂_{m i n} H$ is a free coaction of the C*-algebra H of a non-trivial compact quantum group on a unital C*-algebra A, then there is no H-equivariant *-homomorphism from A to the equivariant join C*-algebra $A ⊛_{δ} H$ . For A being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of functions on ℤ/2ℤ, we recover the celebrated...

On the stratification of the orbit space for the action of automorphisms on connections

Witold Kondracki, Jan Rogulski

Similarity:

CONTENTSIntroduction..................................................................................................................................................5§1. Basic notions and notation.....................................................................................................................7 1.1. Automorphisms of principal bundles....................................................................................................7 1.2. Connections and parallel...

Equivariant maps between certain $G$ -spaces with $G = O (n - 1, 1)$ .

Aleksander Misiak, Eugeniusz Stasiak (2001)

Mathematica Bohemica

Similarity:

In this note, there are determined all biscalars of a system of $s \leq n$ linearly independent contravariant vectors in $n$ -dimensional pseudo-Euclidean geometry of index one. The problem is resolved by finding a general solution of the functional equation $F (A \underset{1}{\to} u, A \underset{2}{\to} u, \dots, A \underset{s}{\to} u) = (sign (det A)) F (\underset{1}{\to} u, \underset{2}{\to} u, \dots, \underset{s}{\to} u)$ for an arbitrary pseudo-orthogonal matrix $A$ of index one and the given vectors $\underset{1}{\to} u, \underset{2}{\to} u, \dots, \underset{s}{\to} u$ .

The local index density of the perturbed de Rham complex

Jesús Álvarez López, Peter B. Gilkey (2021)

Czechoslovak Mathematical Journal

Similarity:

A perturbation of the de Rham complex was introduced by Witten for an exact 1-form $Θ$ and later extended by Novikov for a closed 1-form on a Riemannian manifold $M$ . We use invariance theory to show that the perturbed index density is independent of $Θ$ ; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence...

A complete analogue of Hardy's theorem on semisimple Lie groups

Rudra P. Sarkar (2002)

Colloquium Mathematicae

Similarity:

A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are $O (| x |^{m} e^{- α x ²})$ and $O (| x | ⁿ e^{- x ² / (4 α)})$ respectively for some m,n ≥ 0 and α > 0, then f and f̂ are $P (x) e^{- α x ²}$ and $P^{'} (x) e^{- x ² / (4 α)}$ respectively for some polynomials P and P’. If in particular f is as above, but f̂ is $o (e^{- x ² / (4 α)})$ , then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.

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

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

Fundamenta Mathematicae

Similarity:

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....

Classification of homotopy classes of equivariant gradient maps

E. N. Dancer, K. Gęba, S. M. Rybicki (2005)

Fundamenta Mathematicae

Similarity:

Let V be an orthogonal representation of a compact Lie group G and let S(V),D(V) be the unit sphere and disc of V, respectively. If F: V → ℝ is a G-invariant C¹-map then the G-equivariant gradient C⁰-map ∇F: V → V is said to be admissible provided that ${(\nabla F)}^{- 1} (0) \cap S (V) = \emptyset$ . We classify the homotopy classes of admissible G-equivariant gradient maps ∇F: (D(V),S(V)) → (V,V∖0).

Fourier analysis, linear programming, and densities of distance avoiding sets in $ℝ^{n}$

Fernando Mário de Oliveira Filho, Frank Vallentin (2010)

Journal of the European Mathematical Society

Similarity:

We derive new upper bounds for the densities of measurable sets in $ℝ^{n}$ which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions $2, \dots, 24$ . This gives new lower bounds for the measurable chromatic number in dimensions $3, \dots, 24$ . We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems...

Shadowing in actions of some Abelian groups

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

Fundamenta Mathematicae

Similarity:

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.