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.
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.
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.
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. We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each r≥2 we associate to an annular link L a naive Z/rZ-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of L as …
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…
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…
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. 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.
Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the …
Study cohomotopy classes for 4-manifolds using complex spin structures.
problem Understanding cohomotopy classes for families of 4-manifolds with complex spin structures.
method Using Bauer--Furuta invariants in parametrised stable homotopy theory.
result Definition of characteristic cohomotopy classes on Thom spectra.
In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariant…
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.
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…
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. We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Flo…
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.
Study homotopy groups in GIT quotients using transversality methods.
problem Homotopy groups of stable loci in affine GIT.
method Infinite-dimensional transversality framework extended to GIT setting.
result Generic homotopies avoid unstable and strictly semistable strata under certain conditions.
New mathematical framework connects M-theory charges to stable homotopy groups.
problem Quantization of fluxes in M-theory and their mathematical representation.
method Establishing a correspondence between M-theory phenomena and stable homotopy theory concepts.
result Found a direct link between M-theory charges and stable homotopy groups.
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.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
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.
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.
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.
Novel construction of Bauer--Furuta invariant using sheaves of spectra.
problem Constructing the Bauer--Furuta invariant without finite-dimensional approximations.
method Using sheaves of spectra and Borel--Moore homology, avoiding approximations.
result Defines the shriek functors and Thom spectra for index calculations.
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. …
Study stable equivalence relations on 4-manifolds, proving homotopy equivalent manifolds with abelian fundamental group are stably diffeomorphic.
problem Classifying stable equivalence relations on 4-manifolds.
method Combination of modified and classical surgery, focusing on homotopy equivalence up to stabilisation.
result Closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic.
The paper develops methods for calculating equivariant homology from Morse functions.
problem Calculating equivariant homology from equivariant Morse functions.
method Alter equivariant Morse functions to stable ones, use generic equivariant metrics, and analyze the Morse spectral sequence.
result Equivariant Morse functions induce a filtration that computes equivariant homology.
New homotopy theory reveals the structure of stable curves.
problem Understanding the structure of the moduli stack of stable curves.
method Using stratified homotopy theory, the category of stable curves captures the stratified homotopy type of the moduli stack.
result The category of stable curves classifies constructible sheaves via an exodromy equivalence.
We set up foundations of representation theory over S, the sphere spectrum, which is the `initial ring' of stable homotopy theory. In particular, we treat S-Lie algebras and their representations, characters, gln(S)-Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. As an application, w…
Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.
problem Understanding the homotopy types of free racks and quandles.
method Proved analogs of Milnor's theorem for racks and quandles and their pointed variants.
result Identified the homotopy types of free racks and quandles on spaces of generators.
We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the …
Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.
problem Studying Steenrod squares for virtual links.
method Defines a second Steenrod square for virtual links.
result First meaningful nontrivial example of the second Steenrod square on Khovanov homology.
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…
The paper studies equivariant sheaves on toric varieties and their quotients.
problem Understanding stability of sheaves on toric GIT quotients.
method Defining equivariant sheaves and showing stability preservation under certain conditions.
result Stability of sheaves on toric GIT quotients is related to combinatorial criteria.
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.
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.
Refines Khovanov homology using signed Burnside categories.
problem Stable homotopy refinement of Khovanov homology.
method Signed Burnside category approach to compare Blanchet and Khovanov chain complexes.
result Stable homotopy type construction for link diagrams.
In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except 15,31,63,127 is obtained. It is proved that for n>127 in the stable homotopy group o…
Revisits Pontryagin's proof of stable stems 0, 1, and 2.
problem Computing stable stems in dimensions 0, 1, and 2.
method Introduction of homotopy theory concepts, framed cobordism, and Pontryagin-Thom construction.
result Corrects the result of \(\pi_2^S\) from \cite{Pont2} and computes stable stems in dimensions 0, 1, and 2.
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.
Researchers prove a method to upgrade Morse-Bott homology to stable homotopy invariants.
problem Proving a method to upgrade Morse-Bott homology to stable homotopy invariants rigorously.
method Rigorous construction of stable normal framings and proof of stable homotopy type recovery.
result The stable homotopy type recovers Σ∞+M and Thom spectra for all reduced KO-theory classes.
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…
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.
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.