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

Grzegorz Dylawerski; Kazimierz Gęba; Jerzy Jodel; Wacław Marzantowicz

Displaying similar documents to “An $S^{1}$ -equivariant degree and the Fuller index”

Degree of T-equivariant maps in ℝⁿ

Joanna Janczewska, Marcin Styborski (2007)

Banach Center Publications

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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

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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...

Equivariant measurable liftings

Nicolas Monod (2015)

Fundamenta Mathematicae

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We discuss equivariance for linear liftings of measurable functions. Existence is established when a transformation group acts amenably, as e.g. the Möbius group of the projective line. Since the general proof is very simple but not explicit, we also provide a much more explicit lifting for semisimple Lie groups acting on their Furstenberg boundary, using unrestricted Fatou convergence. This setting is relevant to $L^{\infty}$ -cocycles for characteristic classes.

$Z ₂^{k}$ -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,...

The equivariant universality and couniversality of the Cantor cube

Michael G. Megrelishvili, Tzvi Scarr (2001)

Fundamenta Mathematicae

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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

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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.

Equivariant algebraic topology

Sören Illman (1973)

Annales de l'institut Fourier

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Let $G$ be a topological group. We give the existence of an equivariant homology and cohomology theory, defined on the category of all $G$ -pairs and $G$ -maps, which both satisfy all seven equivariant Eilenberg-Steenrod axioms and have a given covariant and contravariant, respectively, coefficient system as coefficients. In the case that $G$ is a compact Lie group we also define equivariant $C W$ -complexes and mention some of their basic properties. The paper is a short abstract...

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

Aleksander Misiak, Eugeniusz Stasiak (2001)

Mathematica Bohemica

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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 $\underset{1}{\to} u, \underset{2}{\to} u, \dots, \underset{s}{\to} u$ .

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

Michał Sadowski (1991)

Annales Polonici Mathematici

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Equivariant deformation quantization for the cotangent bundle of a flag manifold

Ranee Brylinski (2002)

Annales de l’institut Fourier

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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...

Classification of homotopy classes of equivariant gradient maps

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

Fundamenta Mathematicae

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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).

Equivariant K-theory of flag varieties revisited and related results

V. Uma (2013)

Colloquium Mathematicae

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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...

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

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

Colloquium Mathematicae

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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.

Noncommutative Borsuk-Ulam-type conjectures

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

Banach Center Publications

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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...

Displaying similar documents to “An S 1 -equivariant degree and the Fuller index”

Displaying similar documents to “An $S^{1}$ -equivariant degree and the Fuller index”