Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
The paper identifies conditions for free circle actions on specific 7-manifolds.
problem Determining conditions for free circle actions on certain 7-manifolds.
method Analyzing specific 7-manifolds and their components.
result Identifies conditions for free circle actions on kS2imesS5#lS3imesS4#Σ. Surveying matrix group actions on manifolds.
problem Topological Zimmer's conjecture on matrix group actions on manifolds.
method Surveying existing research and literature.
result Status update on matrix group actions on manifolds.
We give an affirmative answer to the Halperin-Carlsson conjecture for the homologically injective torus actions on closed manifolds. This class contains holomorphic torus actions on compact Kahler manifolds, torus actions on compact Riemannian flat manifolds.
Study free circle actions on specific 7-manifolds with positive Ricci curvature.
problem Classify and construct metrics on manifolds with a specific cohomology ring.
method Classify manifolds, construct free circle actions, and find positive Ricci curvature metrics.
result Almost all manifolds admit infinitely many non-equivalent free circle actions with positive Ricci curvature.
Survey on finite group actions on manifolds.
problem Understanding actions of finite groups on manifolds.
method Analyzes the restriction of actions to subgroups and considers group and manifold properties.
result Provides information on subgroup actions and their properties.
Classifies real-analytic SL(n,R) actions on closed manifolds.
problem Classifying real-analytic SL(n,R) actions on closed manifolds.
method Analytic and smooth classification methods.
result Extensions and classifications of Fisher--Melnick's work.
Classifies circle actions on 6D manifolds with 4 fixed points.
problem Classifying circle actions on 6D manifolds with specific fixed points.
method Analyzes fixed point data and proves agreement with known actions.
result Agrees with actions on 6-spheres or CP3. The paper identifies manifolds with free circle actions.
problem Characterizing manifolds with free circle actions.
method Analyzing (n−1)-connected (2n+1)-manifolds with torsion-free homology. result Determining manifolds admitting free circle actions up to almost diffeomorphism.
Totally geodesic sections found in polar actions.
problem Understanding sections of polar actions on Riemannian manifolds.
method Elementary proof of a folklore result.
result Sections of polar actions are totally geodesic.
Smooth actions of cyclic groups on 3-manifolds are conjugate to smooth ones.
problem Understanding smoothness in group actions on 3-manifolds.
method Finite cyclic group actions by (1+ε)-bilipschitz homeomorphisms on closed 3-manifolds. result Finite cyclic group actions by (1+ε)-bilipschitz homeomorphisms on closed 3-manifolds are conjugate to smooth actions. Reduction principles for proper actions on smooth manifolds.
problem Proper actions on smooth manifolds and their properties.
method Exhibit constructions and prove reduction principles for proper actions.
result Reduction principles hold for proper actions, polar actions, and copolarity.
In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of class…
Classifies actions of SL(n,R) and SL(n,Z) on closed n-manifolds.
problem Classifying actions of SL(n,R) and SL(n,Z) on closed n-manifolds.
method Classification and construction of actions, using geometric structures.
result New exotic actions of SL(n,Z) on n-tori and their connected sums.
Groups can act on 3-manifolds if their Cayley complex can embed in specific types of 3-manifolds.
problem Understanding group actions on 3-manifolds.
method Proving groups admit actions on 3-manifolds if their Cayley complexes can embed in specific 3-manifolds.
result Groups with embeddable Cayley complexes can act on four specific types of 3-manifolds.
The study explores large group actions on 3-manifolds and their Heegaard splittings.
problem Understanding finite group actions on 3-manifolds and their properties.
method Investigates maximal group orders and presents specific examples of 3-manifolds with large actions.
result The existence of a unique hyperbolic 3-manifold with a specific large group action.
New submanifolds found in toric manifolds with specific actions.
problem Understanding submanifolds in toric manifolds with complex subtorus actions.
method Analyzing the closure of a complex subtorus in a toric manifold and its Hamiltonian action.
result The image of the moment map for the Hamiltonian subtorus action coincides with the image of the Delzant polytope.
In this paper we prove several related results concerning smooth Zp or $\s^1$ actions on 4-manifolds. We show that there exists an infinite sequence of smooth 4-manifolds Xn, n≥2, which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each Xn support…
Reduces proper actions to simpler core actions for analysis.
problem Understanding properties of proper actions on manifolds.
method Extending Skjelbred and Straume's construction to non-compact groups, focusing on core of actions.
result Properties of proper actions are determined by simpler core actions.
The paper classifies manifolds with free torus actions and positive Ricci curvature.
problem Classifying manifolds with specific geometric properties.
method Using circle bundles and free torus actions on connected sums of products of spheres.
result Infinitely-many manifolds with positive Ricci curvature and torus actions are classified.
Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus of (G). A torus manifold with (G)-action is defined to be a smooth connected closed oriented manifold of dimension (2\dim T) with an almost effective action of (G) such that (M^T\neq \emptyset). We show that if there is a torus manifold (M) wi…
Suspensions of manifolds by circle surgeries are key in free action constructions.
problem Understanding free S1-actions on smooth manifolds of dimension at least 3. method Circle surgeries on S1imesM yield suspensions Σ0M and Σ1M. result Suspension operations Σi are fundamental in constructing and classifying manifolds with free S1-actions. Develops a spectral sequence for Lie group actions on manifolds.
problem Understanding cohomology of manifolds with Lie group actions.
method Introduces a spectral sequence relating manifold cohomology to Lie algebra cohomology.
result Establishes a new description of de Rham cohomology for manifolds with Lie group actions.
We show that every continuous action of a finite group on a smooth three-manifold is a uniform limit of smooth actions.
Study circle actions on 4-manifolds, deriving formulas and graphs.
problem Understanding circle actions on 4-dimensional manifolds.
method Derive the Atiyah-Hirzebruch formula and associate graphs to fixed point data.
result Show existence of 4D oriented S^1-manifolds from satisfying graphs.
The paper classifies actions of a specific group on certain manifolds.
problem Classifying analytic actions of a specific semi-orthogonal group on manifolds.
method Adapting Uchida's construction, the paper explicitly constructs actions on specific manifolds and demonstrates that any action is covered by these.
result Any analytic action of the semi-orthogonal group on a closed, connected manifold is covered by the constructed actions.
Resolves singular fibers of a 5-manifold with S1 action.
problem Discrete singular fibers of a closed 5-manifold with S1 action. method Construction of resolution compatible with cyclic surface singularities.
result Compatibility proved between resolution and cyclic surface singularities.
In this paper, we study the action of Homeo0(M), the identity component of the group of homeomorphisms of an n-dimensional manifold M with an Fp-free action, on another manifold N of dimension n+k<2n. We prove that if M is not an Fp-homology sphere, then N≅M×K fo…
We explore transformation groups of manifolds of the form M×Sn, where M is an asymmetric manifold, i.e. a manifold which does not admit any non-trivial action of a finite group. In particular, we prove that for n=2 there exists an infinite family of distinct non-diagonal effective circle actions on such pr…
The main result of this paper is that a polar action on a compact irreducible homogeneous Kaehler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.
We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants, which are invariants of the topology of the manifold, of th…
Extends Cheeger's method to Lie groupoid actions on manifolds.
problem Smooth Lie group actions on manifolds with singularities.
method Extension of Cheeger's deformation techniques to Lie groupoid actions.
result Explicit sectional curvature description of the deformation.
We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows tha…
Study circle actions on unitary manifolds with discrete fixed points.
problem Understanding circle actions on compact unitary manifolds with discrete fixed points.
method Prove relationships between weights at fixed points and derive results regarding the first equivariant Chern class and Hirzebruch χy-genus. result Derive a multigraph encoding fixed point data, leading to new insights into unitary S1-manifolds. We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds t…
Study polar actions on Damek-Ricci spaces, proving existence and finding examples.
problem Characterize polar actions on Damek-Ricci spaces.
method Prove criteria for isometric actions to be polar, find examples, and classify actions.
result Non-trivial polar actions exist on all Damek-Ricci spaces.
A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. Such group actions have been studied extensively on symmetric spaces. We show how to construct a manifold admitting a polar group action by prescribing their isotropy groups along a fundamental d…
Geodesics on Kähler manifold potentials are paths of least action.
problem Understanding geodesics on the space of Kähler potentials.
method Study Lagrangians and geodesics on the Fréchet manifold of Kähler potentials, showing geodesics are paths of least action.
result Geodesics on the space of Kähler potentials are paths of least action, and conversely under suitable conditions.
Morse inequalities for noncompact manifolds with group action.
problem Establishing inequalities for noncompact manifolds with group action.
method Using L2-Betti numbers and functions describing critical points. result Morse inequalities given in terms of L2-Betti numbers and group functions. Conditions for equivariant bundles on 4-manifolds with cyclic actions.
problem Existence of equivariant bundles on 4-manifolds with cyclic actions.
method Conditions derived from the twisted signature formula and congruence relations between fixed point data and isotropy representations.
result Necessary and sufficient conditions for the existence of equivariant bundles.
Classifies symplectic torus actions up to equivariant symplectomorphism.
problem Classifying symplectic torus actions up to equivariant symplectomorphism.
method Classification theorems based on Duistermaat and Pelayo's work on symplectic torus actions with coisotropic orbits.
result Every almost isotropy-maximal symplectic torus action is equivariantly diffeomorphic to a product of a symplectic toric manifold and a torus.
We show that the simply-connected four-manifolds which admit locally linear, homologically trivial actions by rank two finite abelian groups are homeomorphic to connected sums of CP^2, -CP^2, and S^2 x S^2 (with one exception: pseudofree Z_3 x Z_3 actions on the Chern manifold), and also establish an equivariant decomp…
Example of group action on surface that can't extend to 3-manifold.
problem Finding group actions on surfaces that can't be extended to 3-manifolds.
method Provided a first known example and sufficient conditions for infinitely many such actions.
result Finite group actions on surfaces that are free and orientation-preserving do not extend to arbitrary actions on 3-manifolds.
A survey of finite group actions on symplectic 4-manifolds is given with a special emphasis on results and questions concerning smooth or symplectic classification of group actions, group actions and exotic smooth structures, and homological rigidity and boundedness of group actions. We also take this opportunity to in…
We give a congruence for the quantum invariant of a Z_p-quotient of a 3$-manifold with a Z_{p^2} action. We show the congruence does not hold for quotients of 3--manifolds with a Z_{5}xZ_{5} action.
It is known that the hard Lefschetz action, together with Kähler identities for Kähler (resp. hyperkähler) manifolds, determines a su(1,1)sup (resp. sp(1,1)sup) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also gener…
In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected 5-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension 4 are connected sums of S3…
Study a 10D symplectic manifold with 6 fixed points, linking to G2 orbit.
problem Understanding fixed points and Chern classes in Hamiltonian S1 actions. method Analyzing manifold data, comparing to G2 orbit. result Certain data uniquely determine others, showing similarities to G2 orbit.