Study shows equivariant Khovanov homotopy types are equivalent.
problem Understanding equivariant structures in Khovanov homotopy types.
method Investigates group actions on homotopy coherent diagrams to prove equivalence.
result Equivariant Khovanov homotopy types are equivariantly stably homotopy equivalent.
We explore homotopies in quantum field theory formalism.
problem Constructing homotopies in Batalin-Vilkovisky formalism.
method Review and construction of homotopies from renormalization group flow and gauge fixing changes.
result Constructing spans of quantum master actions with isomorphic effective actions using homotopies.
In the paper of Montgomery, D. and Yang, C.T. [5], they discuss the de-suspension of smooth free actions of S1 on (2n+1)-dimensional homotopy spheres. In this paper we discuss the de-suspension of smooth free actions of S3 on (4n + 3)-dimensional homotopy spheres.
Constructs odd Khovanov homotopy types for links, linking them to even types.
problem Understanding and constructing odd Khovanov homotopy types for links.
method Constructs stable homotopy types X^j_o(L) for links L, with cohomology matching odd Khovanov homology.
result Odd Khovanov homotopy types carry a Z/2 action whose fixed points are related to even Khovanov homotopy types.
New non-kinetic actions on four-manifolds discovered.
problem Finding non-kinetic smooth homotopy coherent actions on four-manifolds.
method Restricting a nontrivial action on a K3 surface and constructing extensions. result First example of a non-kinetic smooth homotopy coherent action of order two on a four-manifold.
This thesis extends Hamiltonian actions to multisymplectic geometry, classifying actions on spheres and constructing homotopy comomentum maps.
problem Extending Hamiltonian actions to multisymplectic geometry.
method Explicit constructions and concrete examples of homotopy comomentum maps.
result Complete classification of compact group actions on multisymplectic spheres and explicit construction of homotopy comomentum maps.
Survey on finite group actions on CW-complexes homotopy to spheres.
problem Understanding finite group actions on CW-complexes homotopy equivalent to spheres.
method Survey of extensive literature on finite G-CW-complexes homotopy equivalent to spheres. result Finite G-CW-complexes homotopy equivalent to spheres have finite group actions. Homotopy actions of Lie algebroids defined as L∞-algebra morphisms.
problem Defining and studying homotopy actions of Lie algebroids.
method Constructing homological vector fields on the semi-direct product and proving bijection.
result The construction is a bijection between homotopy actions and homological vector fields.
Study of symplectomorphisms on ruled surfaces under circle actions.
problem Homotopy type of equivariant symplectomorphisms on rational ruled surfaces.
method Analysis of action on compatible and invariant almost complex structures, use of Delzant's and Karshon's classifications.
result Equivariant symplectomorphisms are homotopy equivalent to tori or their pushout.
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. New tools prove smooth actions on exotic spheres.
problem Existence of smooth actions on exotic spheres.
method Homotopy-theoretic tools, complex and quaternionic Mahowald invariants.
result Existence of smooth U(1)- and Sp(1)-actions on exotic spheres. Smooth actions of infinite groups linked to homotopy theory.
problem Connecting infinite-dimensional smooth groups to homotopy theory.
method Two computations: diffeological homotopy groups and localization of a strict category.
result Natural constructions yield homotopically coherent group actions of G.
Study shows exotic Dehn twists on certain 3-sphere fillings.
problem Extending group actions from boundaries to interiors of 4-manifolds.
method Analyzing Dehn twists on Seifert homology spheres and their fillings.
result Dehn twists on certain fillings are infinite order exotic.
Study on group actions on spheres using multisymplectic geometry.
problem Existence of homotopy comoment maps for compact Lie group actions on spheres.
method Investigation of multisymplectic actions and comoments on spheres.
result Explicit constructions of comoments for interesting cases.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
Study modular class of Lie ∞-algebroids and their adjoint actions.
problem Understanding the modular class and adjoint actions of Lie ∞-algebroids.
method Equivalence of descriptions, homotopy invariance, explicit actions and dualities.
result Homotopy invariance of modular classes and explicit adjoint actions.
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…
Identifies LA-groups via VB-group structure and complementary actions.
problem Understanding the structure and integrability of LA-groups.
method Identifies LA-groups via VB-group structure and complementary actions up to homotopy.
result Establishes an equivalence between LA-groups and LA-matched pairs.
We show that any 3-dimensional homotopy lens space M^3 that is simple-homotopy equivalent to a lens space L(p,q) is topologically s-cobordant to the lens space. It follows that M has the same multi-signature as L(p,q) and the action of π_1(M) on the universal cover of M embeds in an orthogonal action on S^7.
Study automorphisms of pure braid groups on sphere homotopy groups.
problem Understanding automorphisms' effect on sphere homotopy groups.
method Examined Delta-group structure, proved invariance of cycle and boundary groups, computed action for few strands.
result Induced action of all automorphisms of pure braid groups on sphere homotopy groups.
Study of embedding spaces using homotopy theory and operads.
problem Understanding the stable homotopy type of embedding spaces.
method Analysis of cubes of framed configuration spaces, homotopy theory of presheaves, operadic structures.
result Induced action of the Poisson operad on the homology of configuration spaces is a homotopy invariant.
Let M be a manifold homotopy equivalent to the complex projective space $\C P^m$. Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The…
The paper calculates actions of string link operations for 4- and 5-component links.
problem Classifying link-homotopy classes of string links.
method Explicit calculation of actions of partial conjugations and conjugations for 4- and 5-component string links.
result Presentations of link-homotopy classes for 4- and 5-component links.
Curved 10-manifolds with torus symmetry are spheres or complex projective spaces.
problem Characterizing positively curved manifolds with torus symmetry.
method Analyzing actions of 3-dimensional tori on closed, simply connected 10-manifolds. result Closed, simply connected, positively curved 10-manifolds with T3-symmetry are homotopy spheres or complex projective spaces. Symmetries of Poisson manifolds are in general quantized just to symmetries up to homotopy of the quantized algebra of functions. It is therefore interesting to study symmetries up to homotopy of Poisson manifolds. We notice that they are equivalent to Poisson principal bundles and describe their quantization to symmet…
If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed sets M^G and N^G are also homotopy equivalent. The replacement problem asks the converse question: If F is homotopy equivalent to the fixed set M^G, is F = N^G for a G-manifold equivariantly homotopy equivalent to M? We prove that for locally …
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and L∞-algebras. result Extension of conserved currents to spacetime vector fields.
Implemented Habegger-Lin algorithm for 4- and 5-component links.
problem Determining link-homotopy of links.
method Explicit computation of group actions and implementation of algorithm.
result Found new pairs of links not distinguishable by Milnor's invariants.
Introduces homotopy momentum sections on multisymplectic manifolds.
problem No specific problem stated; focuses on introducing a new concept.
method Introduces a new concept of homotopy momentum sections on multisymplectic manifolds.
result Shows that a gauged nonlinear sigma model with Wess-Zumino term has homotopy momentum section structure.
In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…
The paper studies Goeritz equivalence in lens spaces, describing actions and obstructions.
problem Understanding Goeritz equivalence in lens spaces L(p,1) for genus two Heegaard splittings. method Describes the Goeritz group action on the homology of the Heegaard surface and provides obstructions.
result Homology and homotopy obstructions for Goeritz equivalence of curves in the Heegaard surface.
Detects free group automorphisms using homology of covers.
problem Identifying free group automorphisms via homology.
method Analyzes the Endomorphism monoid action on homology of covers.
result Homological characterization of homotopy equivalences.
Study cyclic group actions on specific high-dimensional manifolds.
problem Classify smooth actions of cyclic groups on certain high-dimensional manifolds.
method Analyzes smooth orientation-preserving actions of Z/m on (n−1)-connected 2n-manifolds. result Classifications up to smooth conjugation for specific cases of n and m. The paper investigates exotic smooth structures on manifolds with group actions.
problem Existence of homeomorphic but not diffeomorphic smooth manifolds with shared basic spectra.
method Investigates Riemannian Laplacian eigenvalues and eigenfunctions on manifolds with compact Lie group actions.
result Establishes the existence of homeomorphic yet not diffeomorphic manifolds with shared basic spectra.
We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.
Characterizes Alexandrov spaces with Cohen-Macaulay actions.
problem Understanding actions of Lie groups on Alexandrov spaces.
method Equivariant cohomology via rational homotopy theory.
result Characterizes Alexandrov spaces with Cohen-Macaulay actions.
Decouples homotopy quotients of generalised configuration spaces on surfaces.
problem Homological stability of generalised configuration spaces on surfaces.
method Analyzes actions of diffeomorphism groups and uses homotopy quotients.
result Decouples theorem for homology of homotopy quotients on surfaces.
Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a…
New equivariant formality concepts solve the toral rank conjecture.
problem Toral rank conjecture and equivariant formality of actions.
method Rational homotopy theory, Hirsch-Brown models, A-infinity algebras.
result Actions with new properties satisfy the toral rank conjecture.
For a real oriented hyperplane arrangement, we show that the corresponding Salvetti complex is homotopy equivalent to the complement of the complexified arrangement. This result was originally proved by M. Salvetti. Our proof follows the framework of a proof given by L. Paris and relies heavily on the notation of orien…
Study homotopy sheaves on categories and their presheaves, proving descent properties.
problem Homotopy sheaves on categories and their presheaves.
method Homotopy right Kan extension, pretopologies, Yoneda embedding.
result Preserves homotopy sheaves and induces equivalence between sheaves and colimit-preserving sheaves.
We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.
A handlebody-link is a disjoint union of embeddings of handlebodies in S3 and an HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. The second author and Ryo Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numb…
The study finds infinite geodesics on manifolds with specific homotopy group properties.
problem Existence of non-contractible closed geodesics on manifolds with certain homotopy group properties.
method Analyzes the homotopy groups and their action on the fundamental group to prove the existence of infinitely many closed geodesics.
result Infinitely many geometrically distinct closed geodesics exist under specific conditions on homotopy groups.
The paper generalizes the Borsuk-Ulam theorem to surfaces and cyclic actions.
problem Generalizing the Borsuk-Ulam theorem to surfaces and cyclic actions.
method Algebraic criterion involving braid groups and homology groups.
result Determines the Borsuk-Ulam property for maps from surfaces to R^2.
We study the rational homotopy of the moduli space NX of stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g≥2. The symplectic group Aut(H1(X,Z))=Sp(2g,Z) has a natural action on the rational homotopy gr…
We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra o…
We construct an action of the free group Fn on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between homotopy classes of paths in the punctured disc and complexes of projective zigzag m…