When looking at Bott's original proof of his periodicity theorem for the stable homotopy groups of the orthogonal and unitary groups, one sees in the background a differential geometric periodicity phenomenon. We show that this geometric phenomenon extends to the standard inclusion of the orthogonal group into the unit…
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48 results for “Bott periodicity”
We use Bott periodicity to relate previously defined quantum classes to certain "exotic Chern classes" on . This provides an interesting computational and theoretical framework for some Gromov-Witten invariants connected with cohomological field theories. This framework has applications to study of higher dimension…
Geometrically proves a theorem linking Clifford modules to vector bundles over spheres.
problem Relating Clifford modules to vector bundles over spheres.
method Direct geometric proof based on explicit deformations, avoiding Bott periodicity.
result Establishes a correspondence between Clifford modules and stable vector bundles over spheres modulo certain conditions.
For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e. symplectic) magnetic field has periodic orbits on a sequence o…
Paper reviews spheres with almost complex structures.
problem Identifying spheres with almost complex structures.
method Characteristic classes and Bott periodicity theorem.
result Only S^2 and S^6 admit almost complex structures.
In this paper we produce a lower bound for the number of periodic orbits of certain Hamiltonian vector fields near Bott-nondegenerate symplectic critical submanifolds. This result is then related to the problem of finding closed orbits of the motion of a charged low energy particle on a Riemannian manifold under the in…
Floer theory constructs filtrations on quantum cohomology for symplectic manifolds.
problem Quantum cohomology of symplectic manifolds with -actions.
method Floer theory applied to -actions on symplectic manifolds.
result Constructs a family of filtrations on quantum cohomology for Conical Symplectic Resolutions.
In the framework of fibred cusp operators on a manifold associated to a boundary fibration $Φ: \pa X\to Y$, the homotopy groups of the space of invertible smoothing perturbations of the identity are computed in terms of the K-theory of . It is shown that there is a periodicity, namely the odd and the even h…
We give here some extensions of Gromov's and Polterovich's theorems on $\karea$ of , particularly in the symplectic and Hamiltonian context. Our main methods involve Gromov-Witten theory, and some connections with Bott periodicity, and loop groups. The argument is closely connected with study of jump…
The Hamiltonian flow of the standard metric Hamiltonian with respect to the twisted symplectic structure on the cotangent bundle describes the motion of a charged particle on the base. We prove that under certain natural hypotheses the number of periodic orbits on low energy levels for this flow is at least the sum of …
We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,Ω) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all suffici…
Proposes a new Hodge conjecture in Bott-Chern cohomology.
problem Hodge conjecture in Bott-Chern cohomology.
method Characterization of real holomorphic chains, atomic section theory, refined Bott-Chern classes.
result Proof of a new Hodge conjecture in Bott-Chern cohomology.
The infinite matrix `Schwartz' group is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on We show that while the higher (even, Schwartz) loop groups of $G^{-\infty…
Deformation K-theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G. In all known examples, this spectrum is 2-periodic above the rational cohomological dimension of G (minus 2), in the sense that T. Lawson's Bott map is an isomorphism on homotopy in …
Discrete Morse-Bott theory on CW complexes generalizes Forman's theory.
problem No specific problem stated; focuses on theory development.
method Derived a discrete Morse-Bott theory on CW complexes.
result Discrete Morse-Bott theory is a generalization of Forman's theory.
Defines Morse-Bott functions on manifolds with boundary and proves inequalities.
problem No specific problem stated; focuses on generalizing Morse theory.
method Defines Morse-Bott functions and proves inequalities for manifolds with boundary.
result Proves Morse-Bott inequalities for manifolds with boundary.
Formula derived for Bott-Chern classes in complex blow-ups.
problem Calculating Bott-Chern classes in blow-ups of complex manifolds.
method Proved blow-up formula for Bott-Chern classes, established Riemann-Roch without denominators.
result Formula for Bott-Chern classes in blow-ups.
Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.
problem Characterize Bott-Chern cohomology of Vaisman manifolds.
method Explicit description via basic cohomology, infer relationships between cohomology groups, show invariants are unbounded, cohomological characterization of formality.
result Bott-Chern and Dolbeault numbers determine each other for Vaisman manifolds, and cohomological invariants are unbounded.
A notion of geometric formality in the context of Bott-Chern and Aeppli cohomologies on a complex manifold is discussed. In particular, by using Aeppli-Bott-Chern-Massey triple products, it is proved that geometric Aeppli-Bott-Chern formality is not stable under small deformations of the complex structure.
Paper computes Stiefel-Whitney classes on real Bott manifolds.
problem Computing Stiefel-Whitney classes on real Bott manifolds.
method Analyzes real Bott manifolds with holonomy group and diagonal type.
result Extends results to compute even and odd Stiefel-Whitney classes.
New findings on complex manifold properties under deformations.
problem Properties of Dolbeault and Bott-Chern formalities are not preserved under holomorphic deformations.
method Construction of a complex manifold to demonstrate non-preservation of properties.
result Existence of a manifold satisfying -lemma but with non-vanishing Aeppli-Bott-Chern-Massey product.
In studying the Bott-Chern and Aeppli cohomologies for q-complete manifolds, we introduce the class of cohomologically Bott-Chern q-complete manifolds.
The paper generalizes the Bott-Virasoro group and derives new Euler equations.
problem Understanding the generalized Bott-Virasoro group and its dynamics.
method Generalizing the Bott-Virasoro group using connection cochain and deriving Euler equations.
result New Euler equations derived from the generalized Bott-Virasoro group.
Study of deformed Bott-Chern cohomology on complex manifolds.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that these assumptions are insufficient with an explicit counterexample…
Study on Kähler manifolds proves weak decompositions and relates harmonic forms.
problem Analyzing harmonic forms on Kähler manifolds.
method Proves weak Bott-Chern and Dolbeault decompositions.
result Strict relation between Bott-Chern harmonic forms and the Bott-Chern decomposition.
The study describes Lefschetz-Bott fibrations on symplectic manifolds and their applications.
problem Understanding Lefschetz-Bott fibrations on symplectic manifolds.
method Explicit construction and analysis of Lefschetz-Bott fibrations over line bundles.
result Construction of strong symplectic fillings of symplectic manifolds.
Spin-structures on real Bott manifolds with Kähler structure are characterized.
problem Existence of spin-structures on real Bott manifolds with Kähler structure.
method Ishida characterization and techniques from \cite{PS16} using characteristic classes.
result Necessary and sufficient condition for the existence of spin-structures on .
3D foliation study finds Poisson structure with specific singularities.
problem Characterizing Poisson structures on Bott-Morse foliations in 3D.
method Analyzes Bott-Morse foliations, computes Poisson bivectors and symplectic forms.
result Linear, singular Poisson structure of rank 2 with Bott-Morse singularities found.
Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.
problem Well-definedness of Morse-Bott-Smale chain complex.
method Unified five degeneracy relations into a single condition.
result Quasi-isomorphic to Morse-Smale-Witten chain complex, alternative proof of Morse Homology Theorem.
Condition found for spinc structures on a specific type of manifold.
problem Existence of spinc structures on real Bott manifolds.
method Provided a necessary and sufficient condition.
result Condition for spinc structures on real Bott manifolds established.
Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.
problem Proving inequalities for four-manifolds of Seiberg-Witten simple type.
method Uses virtual Morse-Bott indices and Hirzebruch-Riemann-Roch Theorem.
result Proves positivity of virtual Morse-Bott indices, leading to inequalities.
The Alexander polynomial is linked to Bott-Cattaneo-Rossi invariants via Chern-Simons theory.
problem Expressing Alexander polynomial of long knots in terms of invariants.
method Using a previously established formula relating Bott-Cattaneo-Rossi invariants to the Alexander polynomial and Chern-Simons theory.
result Relating Bott-Cattaneo-Rossi invariants to the Alexander polynomial and Chern-Simons theory.
Affine Ricci Solitons associated to the Bott connection on three-dimensional Lorentzian Lie groupsmath.DG
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on Lorentzian Lie groups.
method Computed Bott connections and their curvature; classified Ricci solitons.
result Classification of Ricci solitons on three-dimensional Lorentzian Lie groups.
Research resolves sign conventions in Floer theory for Morse-Bott case.
problem Sign conventions in filtered -operations for Lagrangian Floer theory.
method Defined filtered -operations and verified formulae using de Rham model.
result Resolved sign issues in Bott-Morse setting.
Study bigraded formality and Aeppli-Bott-Chern-Massey products on complex manifolds.
problem Formality and higher Aeppli-Bott-Chern-Massey products on complex manifolds.
method Introduce and study bigraded formality and Aeppli-Bott-Chern-Massey products, showing non-trivial pullbacks on blow-ups.
result Aeppli-Bott-Chern-Massey products on complex manifolds pull back non-trivially to blow-ups under certain conditions.
Proof that critical knots of Morse-Bott functions are graph knots.
problem Characterizing critical knots in Morse-Bott functions.
method Inductive proof on the number of index-1 critical knots.
result Critical knots of Morse-Bott functions are graph knots.
We study Kähler geometry of Bott manifolds and find extremal metrics.
problem Understanding Kähler geometry on Bott manifolds.
method Simple induction and generalized Calabi construction.
result Any stage n Bott manifold admits an extremal Kähler metric.
Proves a Baum--Bott formula for foliations by curves with logarithmic terms.
problem Analyzing singularities and smoothness in foliations by curves.
method Logarithmic Baum--Bott residues for foliated triples , relating to Poincaré's Problem and GSV indices.
result Logarithmic Baum--Bott residues generalize Aleksandrov logarithmic index for vector fields on hypersurfaces.
The paper extends Bott-Chern Laplacian definition and explores its properties on almost Hermitian manifolds.
problem Exploring the properties of Bott-Chern Laplacian on almost Hermitian manifolds.
method Extending the definition of Bott-Chern Laplacian, proving ellipticity, and analyzing kernels on different types of manifolds.
result The dimensions of Bott-Chern and Dolbeault harmonic forms differ on almost complex 4-manifolds with specific metrics.
Spin-structures on real Bott manifolds with Kähler structures are characterized.
problem Existence of Spin-structures on real Bott manifolds with Kähler structures.
method Ishida characterization and Popko-Szczepański cohomological rigidity technique.
result Necessary and sufficient condition for the existence of Spin-structures on M.
This paper extends Bott-Chern cohomology to coherent sheaves using superconnections.
problem Extending Bott-Chern cohomology to coherent sheaves.
method Using superconnections and the theory of superconnections in the sense of Quillen.
result Characteristic classes are independent of Hermitian metrics and depend on the homotopy class of the coherent module.
A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a …
The paper calculates cohomology for complex 3-folds using new cohomology types.
problem Calculating cohomology for compact complex 3-folds.
method Defined new cohomology types and used them to calculate cohomology.
result Complete Bott-Chern-Aeppli cohomology for various complex 3-folds.
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
The paper proves a blow-up formula for Bott-Chern cohomology and shows the bimeromorphic invariance of non-Kählerness degrees on threefolds.
problem The bimeromorphic invariance of the -Lemma on threefolds.
method Proves a blow-up formula for Bott-Chern cohomology and applies it to show bimeromorphic invariance of non-Kählerness degrees.
result The non-Kählerness degrees are bimeromorphic invariants on compact complex threefolds, implying the bimeromorphic invariance of the -Lemma.
Study Hilbert complexes on complex manifolds.
problem Analyse Hilbert complexes on complex manifolds.
method Define and study Aeppli-Bott-Chern Hilbert complex; examine properties on various manifolds; use self-adjoint extensions of differential operators.
result Kernels of operators on compact Hermitian manifolds are isomorphic to Aeppli or Bott-Chern cohomology.
Study new invariants in complex geometry using Bott-Chern hypercohomology.
problem Understanding geometry through Bott-Chern hypercohomology and bimeromorphic invariants.
method Construct new invariants involving sheaf cohomology, establish blow-up formula and canonical morphism.
result Compute invariants for specific complex threefolds like Iwasawa manifolds and quintic threefolds.