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.
Homotopy operators help describe structures in equivariant deformation problems.
problem Equivariant deformation problems in algebraic structures.
method Use homotopy operators for an L∞-algebra associated with the problem. result Smooth parametrization of the space of structures around a given one.
Generalizes Floer homotopy via Morse-Bott theory.
problem Constructing equivariant models in Floer theory.
method Morse-Bott theory, flow categories, stable homotopy types.
result Equivalence of Borel equivariant spectra for certain Lagrangians.
Stable approach solves equivariant Hopf theorem for G-manifolds.
problem Describe homotopy classes of G-equivariant maps into a G-sphere.
method Equivariant stable homotopy theory with semi-free G-universe.
result Degrees of maps are characterized by congruences.
Unified classification of equivariant principal bundles using higher homotopy theory.
problem Unified classification of equivariant principal bundles.
method Smooth Oka principle, singular-cohesive homotopy theory, internally describing principal bundles.
result Unified classification results for equivariant principal bundles.
Added examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.
problem Constructing examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.
method Explicit equivariant surgeries to construct examples.
result Construction of new examples with finite 2nd homotopy group and non-zero A-genus.
Study compares weak and homotopy moment maps in multisymplectic geometry.
problem Existence and equivariance of moment maps in multisymplectic geometry.
method Comparison of weak and homotopy moment maps.
result Analysis of existence and equivariance phenomena.
A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic Z/2-equivariant unitary bordism ring, in…
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.
Novikov theorem extended to rational Pontryagin classes for cyclic group C4.
problem Classifying stable Cp-smoothings of high-dimensional manifolds. method Computing equivariant homotopy groups and applying to C4. result Novikov's theorem extended to rational Pontryagin classes for C4. Refines quantum annular homology using stable homotopy methods.
problem Quantum annular homology lacks a stable homotopy refinement.
method Equivariant Burnside category approach, cyclic group action.
result Stable homotopy refinement of quantum annular homology constructed.
New infinite family of 4-manifolds with same stable properties but not homotopy equivalent.
problem Finding infinite homotopy stable classes of 4-manifolds with boundary.
method Construction of an infinite family of topological 4-manifolds with specific properties.
result Infinite family of 4-manifolds that are stably homeomorphic but not homotopy equivalent.
Analyzes string topology operations using Chen's integrals and homotopy transfer.
problem Relating string topology to perturbative Chern-Simons theory.
method Develops integrals over configuration spaces and applies homotopy transfer.
result Intertwines involutive Lie bialgebra structures on homology.
Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.
problem Finding exotic diffeomorphisms on contractible 4-manifolds.
method Developed a Pin(2) × Z_2-equivariant refinement for computing Seiberg-Witten Floer homotopy types.
result Constructed an exotic diffeomorphism that survives two stabilizations.
By results of Loeffler and Comezana, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G = S^1 x ... x S^1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of D. Sinha. …
The first author's geometric Hopf invariant of a stable map F:Σ∞X→Σ∞Y is a stable Z2-equivariant map h(F):Σ∞X→Σ∞(Y∧Y) constructed by an explicit difference construction applied to (F∧F)ΔX−ΔYF. The stable Z2-equivariant homotopy c…
Constructs a functor for equivariant smooth h-cobordisms.
problem Defines a functor for equivariant smooth h-cobordisms.
method Constructs an (∞,1)-functor mapping smooth G-manifolds to spaces of equivariant h-cobordisms. result The functor structure is subtle and relies on new ideas.
In this paper we classify maps from a torus phase space X to Hn∗, the space of n×n, non-singular hermitian operators up to equivariant homotopy. The equivariance is with respect to a time-reversal involution on X and an involution on Hn∗ defining a certain symmetry class. Furthe…
New rigidity results for complex and quaternionic moment-angle manifolds.
problem Equivariant topological rigidity of complex and quaternionic moment-angle manifolds.
method Reduction to equivariant rigidity of quasitoric (or quoric) quotients and principal bundles.
result Full equivariant rigidity for manifolds with four-dimensional quoric quotients and primary rigidity for higher dimensions.
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.
Homotopy equivalent boundaries of cube complexes are studied.
problem The equivalence of different boundaries of cube complexes.
method Using a partial order on a quotient of the Roller boundary, we obtain the simplicial Roller boundary and show homotopy equivalence among the Tits, simplicial, and simplicial Roller boundaries.
result The Tits, simplicial, and simplicial Roller boundaries are homotopy equivalent.
For a finite group G, we define an equivariant cobordism category CdG. Objects of the category are (d−1)-dimensional closed smooth G-manifolds and morphisms are smooth d-dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (i.e. geometric realization of its si…
Abstract: Homotopy Poisson algebra models for reduced spaces derived from Poisson structures.
problem Homotopy Poisson algebra models for reduced spaces.
method Cattaneo-Zambon compatibility and regularity conditions, equivariant map, homotopy Poisson algebra.
result Derivation of homotopy Poisson algebra generalizing classical BFV algebra.
Develops a new method for equivariant Lagrangian Floer homology using symplectic homotopy quotients.
problem Constructing equivariant Lagrangian Floer homology for symplectic manifolds with group actions.
method Using symplectic homotopy quotients involving cotangent bundles of an approximation of EG, and Wehrheim and Woodward's theory of quilts. result Shows that the constructed groups are independent of auxiliary choices and are H∗(BG)-bimodules. A key open problem in M-theory is the identification of the degrees of freedom that are expected to be hidden at ADE-singularities in spacetime. Comparison with the classification of D-branes by K-theory suggests that the answer must come from the right choice of generalized cohomology theory for M-branes. Here we show…
Cyclification of orbifolds explained in cohesive higher topos theory.
problem Understanding cyclification of orbifolds in geometric and algebraic contexts.
method Cohesive higher topos theory and transgression of cohomological charges.
result Cyclification of orbifolds is a fundamental base-change construction.
Constructs 2-vector bundles and 2K-theory for Lie groupoids and 2-equivariant settings.
problem Developing a theory of 2-vector bundles and 2K-theory for Lie groupoids and their equivariant versions.
method Defines 2-vector bundles over Lie groupoids, constructs 2K-theory as Grothendieck completion, and proves classification theorems.
result Establishes an equivalence between homotopy categories of 2-vector bundles and simplicial maps, and computes 2-equivariant 2K-theories for specific Lie groups.
The study confirms that certain symmetric spaces are formal.
problem Understanding the equivariant cohomology of symmetric spaces.
method Analyzing the isotropy action and using Rational Homotopy Theory.
result Formality of Z2×Zk-symmetric spaces. Authors prove existence of exotic surfaces and invariants not detecting self-diffeomorphisms.
problem Existence and properties of exotic surfaces and diffeomorphisms.
method Vanishing theorem of family Bauer--Furuta invariant for diffeomorphisms on spin 4-manifolds.
result Family Bauer--Furuta invariants do not detect exotic self-diffeomorphisms on S4 or S2imesS2. We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vecto…
Homotopy types of 4-manifolds tied to their fundamental groups.
problem Determining the homotopy type of 4-manifolds based on their fundamental groups.
method Uses the fundamental group, second homotopy group, first Stiefel-Whitney class, and equivariant intersection pairing.
result Homotopy type of 4-manifolds is determined by given group properties.
Global group laws connect equivariant bordism rings to formal group laws.
problem Establishing connections between equivariant bordism rings and formal group laws.
method Global homotopy theory framework; proving isomorphisms and universal properties.
result Equivariant bordism rings are isomorphic to Lazard rings for abelian Lie groups.
In this paper, a classification of free involutions on 3-dimensional homotopy complex projective spaces is given. By the Z2-equivariant Montgomery-Yang correspondence, we obtain all smooth involutions on S6 with fixed-point set an embedded S3.
Authors classify 3D locally standard T-pseudomanifolds under weaker conditions.
problem Classifying equivariant homeomorphism types of 3D locally standard T-pseudomanifolds.
method Introduced and classified by characteristic data under homotopy equivalence condition.
result Condition for classification can be removed when dimension is at most three.
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. Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper w…
We compute the homotopy type of the space of T^n-equivariant symplectic embeddings from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic-toric manifold M, and use this computation to define a Z-valued step function on the positive real line which is an invariant of the symplectic-t…
The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "N-F-amenability" or "amenability dimension", and "d-BLR condition") and its generalisation, transfer reducibility, which are versions of asymptotic dimension invented for the proofs of the Farrell--Jones and Bo…
We give an analytical proof of the Poincare-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space.
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…
The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in t…
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant G-Poincaré spaces and gap conditions. result Space of isovariant structures on semifree G-Poincaré spaces is highly connected. New invariant connects symplectic fillings and contact structures.
problem Understanding symplectic fillings and contact structures.
method Floer homotopy theory and KO-cohomology.
result Constraint on symplectic fillings using KO-cohomology.
Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.
problem Finite group actions on rational homology 3-spheres.
method Equivariant version of Seiberg-Witten-Floer stable homotopy type.
result Definition of d-invariants with Froyshov-type inequality. In this paper, we introduce the classification of equivariant principal bundles over the 2-sphere. Isotropy representations provide tools for understanding the classification of equivariant principal bundles. We consider a Γ-equivariant principal G-bundle over S2 with structural group G a compact connected Lie…
Proves a specific knot is not smoothly slice using real invariants.
problem Determining the smooth sliceness of (2n,1)-cables of the figure-eight knot. method Used real Seiberg-Witten Frøyshov invariant and developed an equivariant lattice homotopy type.
result Proves the (2n,1)-cable of the figure-eight knot is not smoothly slice when n is odd. This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this paper, we compute this cohomology for spheres and show that for any simple connec…
Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the topological rigidity of compact aspherical manifolds. Our goal is to strip the basi…