Circle Actions Research Articles

Pointwise Periodic Flows Induced by Circle Actions

Pointwise Periodic Flows Induced by Circle Actions

Topology of Almost Complex Structures on Six-Manifolds

We study the space of (orthogonal) almost complex structures on closed six-dimensional manifolds as the space of sections of the twistor space for a given metric. For a connected six-manifold with vanishing first Betti number, we express the space of almost complex structures as a quotient of the space of sections of a seven-sphere bundle over the manifold by a circle action, and then use this description to compute the rational homotopy theoretic minimal model of the components that satisfy a certain Chern number condition. We further obtain a formula for the homological intersection number of two sections of the twistor space in terms of the Chern classes of the corresponding almost complex structures.

Constructions of exotic actions on product manifolds with an asymmetric factor

We explore transformation groups of manifolds of the form M×Sn, where M is an asymmetric manifold, that is, a manifold which does not admit any nontrivial action of a finite group. In particular, we prove that for n=2, there exists an infinite family of distinct nondiagonal effective circle actions on such products. A similar result holds for actions of cyclic groups of prime order. We also discuss free circle actions on M×S1, where M belongs to the class of “almost asymmetric” manifolds considered previously by Puppe and Kreck.

Equivariant higher Dixmier-Douady theory for circle actions on UHF-algebras

We develop an equivariant Dixmier-Douady theory for locally trivial bundles of C⁎-algebras with fibre D⊗K equipped with a fibrewise T-action, where T denotes the circle group and D=End(V)⊗∞ for a T-representation V. In particular, we show that the group of T-equivariant ⁎-automorphisms AutT(D⊗K) is an infinite loop space giving rise to a cohomology theory ED,T⁎(X). Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to ED,T1(X)≅[X,BAutT(D⊗K)]. We compute this group for tori and compare the case D=C to the equivariant Brauer group for trivial actions on the base space.

An analytic Hochschild-Kostant-Rosenberg theorem

Let R be a Banach ring. We prove that the category of chain complexes of complete bornological R-modules (and several related categories) is a derived algebraic context in the sense of Raksit. We then use the framework of derived algebra to prove a version of the Hochschild-Kostant-Rosenberg Theorem, which relates the circle action on the Hochschild algebra to the de Rham-differential-enriched-de Rham algebra of a simplicial, commutative, complete bornological algebra. This has a geometric interpretation in the language of derived analytic geometry, namely, the derived loop stack of a derived analytic stack is equivalent to the shifted tangent stack. Using this geometric interpretation we extend our results to derived schemes.

Normal real affine varieties with circle actions

We provide a complete description of normal affine algebraic varieties over the real numbers endowed with an effective action of the real circle, that is, the real form of the complex multiplicative group whose real locus consists of the unitary circle in the real plane. Our approach builds on the geometrico-combinatorial description of normal affine varieties with effective actions of split tori in terms of proper polyhedral divisors on semiprojective varieties due to Altmann and Hausen.

Almost Complex Torus Manifolds—Graphs and Hirzebruch Genera

Abstract Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As in the case of circle actions, we show that there exists a (directed labeled) multigraph that contains information on weights at the fixed points and isotropy submanifolds of $M$. This includes the notion of a GKM (Goresky-Kottwitz-MacPherson) graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition $k=n$, that is, $M$ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. We show that the Hirzebruch $\chi _y$-genus $\chi _y(M)=\sum _{i=0}^n a_i(M) \cdot (-y)^i$ of an almost complex torus manifold $M$ satisfies $a_i(M)> 0$ for $0 \leq i \leq n$. In particular, the Todd genus of $M$ is positive and there are at least $n+1$ fixed points.

Loops in the fundamental group of which are not represented by circle actions

Abstract We study generators of the fundamental group of the group of symplectomorphisms $\operatorname {\mathrm{Symp}} (\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2, \omega )$ for some particular symplectic forms. It was observed by Kȩdra (2009, Archivum Mathematicum 45) that there are many symplectic 4-manifolds $(M, \omega )$ , where M is neither rational nor ruled, that admit no circle action and $\pi _1 (\operatorname {\mathrm {Ham}} (M,\omega ))$ is nontrivial. On the other hand, it follows from Abreu and McDuff (2000, Journal of the American Mathematical Society 13, 971–1009), Anjos and Eden (2019, Michigan Mathematical Journal 68, 71–126), Anjos and Pinsonnault (2013, Mathematische Zeitschrift 275, 245–292), and Pinsonnault (2008, Compositio Mathematica 144, 787–810) that the fundamental group of the group $ \operatorname {\mathrm{Symp}}_h(\mathbb C\mathbb P^2\#\,k\overline { \mathbb C\mathbb P}\,\!^2,\omega )$ , of symplectomorphisms that act trivially on homology, with $k \leq 4$ , is generated by circle actions on the manifold. We show that, for some particular symplectic forms $\omega $ , the set of all Hamiltonian circle actions generates a proper subgroup in $\pi _1(\operatorname {\mathrm{Symp}}_{h}(\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2,\omega )).$ Our work depends on Delzant classification of toric symplectic manifolds, Karshon’s classification of Hamiltonian $S^1$ -spaces, and the computation of Seidel elements of some circle actions.

Higher systolic inequalities for 3-dimensional contact manifolds

A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.

Non-Hamiltonian Actions With Fewer Isolated Fixed Points

Abstract In an earlier paper, the second author resolved a question of McDuff by constructing a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points. In this paper, we improve on this example by reducing the number of fixed points. More concretely, we construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly $2k$ fixed points for any $k \geq 5$.

Hamiltonian circle actions on complete intersections

We study the problem of determining which diffeomorphism classes of K\"{a}hler manifolds admit a Hamiltonian circle action. Our main result is the following: Let $M$ be a closed symplectic manifold, diffeomorphic to a complete intersection with complex dimension $4k$, having a Hamiltonian circle action such that each component of the fixed point set is an isolated fixed point or has dimension $2 \mod 4$. Then $M$ is diffeomorphic to $\mathbb{CP}^{4k}$, a quadric $Q \subset \mathbb{CP}^{4k+1}$ or an intersection of two quadrics $Q_1 \cap Q_2 \subset \mathbb{CP}^{4k+2}$.

Equivariant fixed point formulae and Toeplitz operators under Hamiltonian torus actions and remarks on equivariant asymptotic expansions

Suppose given a holomorphic and Hamiltonian action of a compact torus on a polarized Hodge manifold. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of the torus on the associated Hardy space which decomposes into isotypes. The main result of this paper is the description of asymptotics along rays in weight space of traces of equivariant Toeplitz operators composed with quantomorphisms for the torus action. The main ingredient in the proof is the micro-local analysis of the equivariant Szegő kernels. As a particular case we obtain a simple approach for asymptotics of the Lefschetz fixed point formula and traces of Toeplitz operators in the setting of ladder representations. We also consider equivariant asymptotic when the decomposition given by the standard circle action is taken into account, in this case one can recall previous results of X. Ma and W. Zhang or of R. Paoletti. We address some explicit computations for the action of the special unitary group of dimension two.

Symmetry and isoperimetry for Riemannian surfaces

For a domain \(\Omega \) in a geodesically convex surface, we introduce a scattering energy \(\mathcal {E}(\Omega )\), which measures the asymmetry of \(\Omega \) by quantifying its incompatibility with an isometric circle action. We prove several sharp quantitative isoperimetric inequalities involving \(\mathcal {E}(\Omega )\) and characterize the domains with vanishing scattering energy by their convexity and rotational symmetry. We also give a new of the sharp Sobolev inequality for Riemannian surfaces.

Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori

We show that every parabolic orbit of a two-degree of freedom integrable system admits a $C^\infty$-smooth Hamiltonian circle action, which is persistent under small integrable $C^\infty$ perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the integrals of motion is Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets.

Equivariant KK-Theory and the Continuous Rokhlin Property

Abstract We introduce and study the continuous Rokhlin property for actions of compact groups on $C^*$-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong $KK$-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the Universal Coefficient Theorem (UCT) is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. Our analysis of the $KK$-theory of the crossed product allows us to prove a ${{\mathbb {T}}}$-equivariant version of Kirchberg–Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are $KK^{{{\mathbb {T}}}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant $K$-theory. We moreover characterize the $KK^{{{\mathbb {T}}}}$-theoretical invariants that arise in this way. Finally, we identify a $KK^{{{\mathbb {T}}}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.

The Space of Traces in Symmetric Monoidal Infinity Categories

Abstract We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,\mathtt{C}}:\hom_\mathtt{C}(x, x) \to \hom_\mathtt{C}(𝟙,𝟙)$ for all dualizable objects x in any symmetric monoidal $\infty$-category $\mathtt{C}$. This generalizes the trace from linear algebra that assigns a scalar $\operatorname{Tr}(\,f\,) \in k$ to any endomorphism f : V → V of a finite-dimensional k-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence, we show that the trace $\operatorname{Tr}$ can be uniquely extended to a tracelike transformation up to a contractible space of choices. This allows us to give several model-independent characterizations of the $\infty$-categorical trace. By restricting the aforementioned notion of tracelike transformations from endomorphisms to automorphisms one can in particular recover a theorem of Toën and Vezzosi. Other examples of tracelike transformations are for instance given by $f \mapsto \operatorname{Tr}(\,f^{\,n})$. Unlike for $\operatorname{Tr}$, the relevant connected component of the moduli space is not contractible, but rather equivalent to $B\mathbb{Z}/n\mathbb{Z}$ or BS1 for n = 0. As a result, we obtain a $\mathbb{Z}/n\mathbb{Z}$-action on $\operatorname{Tr}(\,f^{\,n})$ as well as a circle action on $\operatorname{Tr}(\operatorname{id}_x)$.

Interactive System Based on Leap Motion for 3D Medical Model

An interactive visualization of the patients’ 3D medical anatomical model as guide is often helpful for doctors during complex surgery. However, there are certain limitations according to the actual requirements of building sterile operating environment. Traditional human–computer interaction tools (mouse and keyboard) must be disinfected regularly and cannot be used in the process. A noncontact gesture control medical model based on Leap Motion is proposed in this study. The gesture is recognized and localized without using mouse and keyboards through a binocular camera assembled on Leap Motion. Hence, the model is directly controlled by the gesture to complete the operation of rotation, zoom, and other functions. In this study, a 3D heart model is combined with pseudo-color processing technology to enhance the observability of its 3D structure. Gesture recognition technology is then utilized to control the rendered model as rotation and zoom. Experimental results show that our system has an absolute accuracy in recognizing circle, swipe, and other actions. Thus, rotation is proposed as a new motion that can be identified steadily. Rotation plays an essential role in usability, intuition, and interactive efficiency of future system design. The system is applicable to sterile operating environments due to its stable recognition process and small space occupation.

Exotic symplectomorphisms and contact circle actions

Using Floer-theoretic methods, we prove that the non-existence of an exotic symplectomorphism on the standard symplectic ball, [Formula: see text] implies a rather strict topological condition on the free contact circle actions on the standard contact sphere, [Formula: see text] We also prove an analogue for a Liouville domain and contact circle actions on its boundary. Applications include results concerning the symplectic mapping class group and the fundamental group of the group of contactomorphisms.

Circle Actions on Oriented Manifolds with Few Fixed Points

Circle Actions on Oriented Manifolds with Few Fixed Points

Peningkatan Kinerja Guru dalam Pembelajaran Melalui Pola Kepemimpinan Transformasional Kepala Sekolah di SMP Santu Fransiskus Xaverius Ruteng, Kabupaten Manggarai

The aim of this study is to improve the science of school based management interest and outcomes using a Transformational Leadership model to manage the performance of teachers on school. This study was done at a secondary school SMP St. FransiskusXaveriusRuteng in Manggarai regency. The kind ofthis study is an action research in school and were done in two circle of acts. This study involved the principal, the teachers, and some students as informant and participant. The data collecting techniques were interview, observation, questioners, and document study.The results of the study have shown that before the transformational leadership wasapplied, there were 52,65% low performance teachers in pedagogic, professional, personality, and social ability. Daily routine activity with no creativity and innovationwere found as the roots main problems and also the weakness of principal leadership. On the first circle actions there were four actions as the solutions to overcome the problem; 1) teacher self-evaluation, 2) application of transformational leadership, 3) strengths capacity and capability of teaching and learning, and 4) peer evaluation for teachers.On the second circle the actions were focused on communication and human relationships among the teacher and their principal. The solutions were conducted by the principal through a transformational leadership model; 1) idealized influence, 2) intellectual stimulation, 3) inspirational motivation, 4) charisma, and 5) individual consideration. The results of applying transformational leadership model were the increase of teacher performance on pedagogic; on the first circle 78,90% and on the second circle 84,20%. Performance on professional on the first circle 77,60% and on the second circle 83,30%. Performance on personality; on the first circle 77,70% and on the second circle 83,20%. Performance on social; on the first circle 77,80% and on the second circle 83,70%.The results recommend the principal of secondary school SMP St. Fransiskus Xaverius Ruteng toapply the transformational leadership model to conduct some activities on the school life and would improve the teacher performance.