The paper proves a generalized inverse function theorem for curved L∞ spaces.
problem Proving a generalized inverse function theorem for curved L∞ spaces. method Obstruction theory for L∞ homomorphisms and homotopy transfer theorem for curved L∞ algebras. result A morphism of curved L∞ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, A∞ spaces, E∞ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends…
New theory of Kuranishi manifolds derived from homotopy L∞ spaces.
problem Defining and structuring homotopy L∞ spaces. method Developed a new theory of Kuranishi manifolds, proving they form a 2-category. result Kuranishi manifolds form a 2-category with invertible 2-morphisms. We show that conically smooth stratified spaces embed fully faithfully into ∞-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each ∞-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…
Let $\g\_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g\_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any G_∞-structure ({\rm i.e.} Gerstenhaber algebra up to homotopy structure) on $\g\_2$, and any C_∞-morphism φ ({\rm i.e.} morphism of co…
Study rational homotopy types of embedding spaces of manifolds.
problem Understanding the rational homotopy types of embedding spaces of manifolds.
method Express rational homotopy types through combinatorially defined L-infinity algebras of diagrams.
result Expressed the rational homotopy type of connected components of embedding spaces.
Curved L∞ spaces form a category of fibrant objects.
problem Understanding the structure of curved L∞ spaces. method Proving L∞ spaces over dg manifolds form a category of fibrant objects. result Transitive L∞ algebroids over dg manifolds also form a category of fibrant objects. Homotopy theory of differentiable sheaves connects manifold properties to underlying homotopy types.
problem Understanding the homotopy type of manifolds using differentiable sheaves.
method Developed model structures and homotopical calculi on the ∞-category Diff∞ to compute and compare shapes. result The shape of any manifold coincides with various other notions of underlying homotopy types.
New smooth models for string groups defined in ∞-categories.
problem Defining string group models in smooth spaces.
method Homotopy-theoretic definition using singular complex functor.
result New smooth models for the string group.
The paper describes the topology of spaces of functions with specific singularities on surfaces.
problem Understanding the homotopy type and structure of spaces of functions with prescribed singularities.
method Analyzes the homotopy type of spaces of functions with specific singularities on surfaces, considering the action of coordinate transformations.
result Describes the homotopy type and decomposition of spaces of functions with prescribed singularities on surfaces.
Inverse function theorem and homotopy description for L-infinity bundles.
problem Inverse function theorem and homotopy description for L-infinity bundles.
method Local sections composed of elementary morphisms.
result Simple description of homotopy category of L-infinity bundles.
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.
We construct a functor from the derived category of homotopy Gerstenhaber algebras with finite-dimensional cohomology to the purely geometric category of so-called F∞-manifolds. The latter contains Frobenius manifolds as a subcategory (so that a pointed Frobenius manifold is itself a homotopy Gerstenhaber alg…
Develops derived differential geometry theory.
problem Homotopy and intersection in smooth manifolds.
method Using L∞[1]-algebras and homotopy transfer. result Derived manifolds form a category of fibrant objects.
The paper proves a category of dg manifolds with finite positive amplitude.
problem Understanding the structure of dg manifolds with finite positive amplitude.
method Using path spaces and homotopy transfer theorem for curved L∞[1]-algebras. result Proves that dg manifolds of finite positive amplitude form a category of fibrant objects.
The paper explores connections between dg manifolds and homotopy Lie algebras.
problem Understanding the relationship between dg manifolds and homotopy Lie algebras.
method Study of formal exponential maps, Atiyah classes, and Kapranov L-infinity algebras.
result Existence of formal exponential maps linked to vanishing of Atiyah classes.
Homotopy theory applied to singular foliations leads to new results.
problem Existence and uniqueness of universal L∞-algebroids for singular foliations. method Applied homotopy theory to left semi-model categories and L∞-algebroids. result Recovery of results similar to Laurent-Gengoux and al. about universal L∞-algebroids. New algebra structure derived from Lie pairs.
problem Constructing A∞-algebras from Lie pairs. method Using homotopy equivalence and Lie algebroids.
result Chevalley-Eilenberg cohomology gains an associative algebra structure.
We prove a categorified Poincaré lemma using A∞-structures.
problem Categorifying the Poincaré lemma for smooth homotopies.
method Using ∞-local systems and Chen's iterated integrals. result Homotopy equivalences induce quasi-equivalences on DG categories of ∞-local systems. 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.
The paper explores conditions for compactness and finiteness in stratified homotopy theory.
problem Conditions for compactness and finiteness in stratified homotopy theory.
method Analyzes conditions for compactness and finiteness in stratified homotopy theory, providing sufficient conditions and deducing results.
result Conditions for compactness and finiteness in stratified homotopy theory are established.
This research extends Lie algebra actions to singular foliations.
problem Understanding symmetries in singular foliations without additional assumptions.
method Equivalence of categories between Lie-Rinehart algebras and Lie ∞-algebroids. result Universal Lie ∞-algebroids for singular foliations. We introduce mappings between spaces of functions on (super)manifolds that generalize pullbacks with respect to smooth maps but are, in general, nonlinear (actually, formal). The construction is based on canonical relations and generating functions. (The underlying structure is a formal category, which is a "thickening…
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.
Develops homotopies for Lagrangian field theory using advanced algebraic structures.
problem Formulating a consistent framework for Lagrangian field theory.
method Introduces L∞ algebras and homotopies to enrich the Batalin-Vilkovisky framework. result Provides an explicit lift of the Batalin-Vilkovisky framework to local forms.
We study Maurer-Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term $L_\infty…
This Master Thesis is devoted to the study of n-plectic manifolds and the Strongly Homotopy Lie algebras, also called L∞-algebras, that can be associated to them. Since multisymplectic geometry and L∞-algebras are relevant in Theoretical Physics, and in particular in String Theory, we introduce th…
Homotopy equivalence of cotangent bundles' function algebras is shown.
problem Understanding homotopy equivalence in cotangent bundles and their function algebras.
method Using shifted Poisson algebras and homotopy equivalence of bundles.
result Homotopy equivalent bundles have equivalent Poisson algebras.
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.
Homotopy equivalence between formalities with different covariant derivatives.
problem Formality of Dolgushev depends on covariant derivative choice.
method Proved homotopy equivalence of L∞-morphisms twisted by gauge equivalent elements. result Globalized formalities with different covariant derivatives are homotopic.
Given a compact manifold Nn, an integer k∈N∗ and an exponent 1≤p<∞, we prove that the class C∞(Qm;Nn) of smooth maps on the cube with values into Nn is dense with respect to the strong topology in the Sobolev space Wk,p(Qm;Nn) when the homotopy group $π_…
Extends Lie bialgebroids to homotopy theory.
problem Characterize Lie bialgebroids and their morphisms.
method Interprets Lie bialgebroids in terms of odd symplectic dg-manifolds.
result Introduces L∞-bialgebroids and their morphisms. Lie algebroids linked to L∞ spaces in derived geometry.
problem Relating Lie algebroids to L∞ spaces in derived geometry. method Constructing a faithful functor from Lie algebroids to L∞ spaces and showing the relationship between representations and vector bundles. result Lie algebroids provide an essentially unique L∞ space, and a shifted-symplectic structure on a dg Lie algebroid produces a shifted-symplectic structure on the associated L∞ space. Study shows solutions of differential inclusions are homotopy equivalent in W1,p-topology.
problem Homotopy properties of solutions in differential inclusions.
method Analyzes differential inclusion with specific assumptions on corank one distribution.
result Solutions are homotopy equivalent to loop spaces in W1,p-topology. The paper proves asphericity of configuration spaces and covers them with entire functions.
problem Proving asphericity of configuration spaces and classifying covering spaces.
method Using algebraic topology and covering spaces theory.
result Locally finite infinite configuration spaces are aspherical and coverings can be realized by entire functions.
The paper constructs infinite rank summands in diffeomorphism groups via Seiberg-Witten theory.
problem Constructing infinite rank summands in diffeomorphism groups.
method Using Seiberg-Witten theory, the paper constructs spherical families and computes invariants.
result Infinite rank summands in homotopy and homology groups of diffeomorphism groups.
Paper constructs L∞-algebroids from homotopy Poisson structures.
problem Homotopy Poisson structures and their algebraic properties.
method Introduces thick morphisms and L∞-morphisms. result Establishes an L∞-algebra structure on forms. Geometric Hopf invariant unifies double point treatments in equivariant homotopy theory.
problem Capturing double points in equivariant homotopy theory.
method Explicit construction of geometric Hopf invariant and its equivariant version.
result Unified homotopy theoretic treatments of double points.
Smooth functions on Klein bottle split it into two Möbius bands.
problem Understanding the homotopy types of orbits of smooth functions on Klein bottle.
method Analyzing the right action of diffeomorphisms on smooth functions and computing orbit path components.
result Orbit of a special class of functions on Klein bottle is homotopy equivalent to the product of orbits on two Möbius bands.
Homotopy theory for Lie ∞-groupoids aids integrating L-infinity algebras.
problem Integrating L-infinity algebras and compatibility with homotopy theory.
method Developed a homotopy theory for Lie ∞-groupoids, showing they form an incomplete category of fibrant objects.
result Henriques' integration functor is exact with respect to quasi-split fibrations.
Generalizes Lie bialgebroids to supermanifolds with homotopy Poisson structures.
problem Relating Lie bialgebroids to homotopy Poisson structures on supermanifolds.
method Introduces L-infinity bialgebroids and higher Koszul brackets to connect these structures.
result Shows that (TM,T∗M) has an L-infinity bialgebroid structure for homotopy Poisson structures. The paper studies cobordism categories with Morse theory applied.
problem Understanding the homotopy type of cobordism categories with specific conditions.
method Using parametrized Morse theory to determine weak homotopy equivalences.
result Proves weak homotopy equivalences between cobordism categories and Thom spectra.
Stability of Khovanov homotopy types of torus links as m increases.
problem Stability of Khovanov homotopy types of torus links as m increases.
method Construction of Khovanov homotopy types and analysis of stabilization.
result Khovanov homotopy types of (n,m) torus links become stably equivalent as mightarrow∞. We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A_infty functor from the representations up to homotopy of a Lie algebroid to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several exampl…
Given a Lie group acting on a manifold M preserving a closed n+1-form ω, the notion of homotopy moment map for this action was introduced in Callies-Fregier-Rogers-Zambon [6], in terms of L∞-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This descr…
Smooth approximations lead to homotopy equivalences in manifold spaces.
problem Understanding homotopy equivalences in spaces of smooth maps.
method Proving weak homotopy equivalences for spaces of C^r maps and isotopies.
result Inclusions of specific spaces are weak homotopy equivalences.
This article reviews ∞-bundles and their applications in geometry and physics.
problem Understanding higher bundles in geometry and physics.
method An ∞-categorical formulation of higher bundles. result Identification of higher bundles in various contexts.
Paper studies symmetries in singular foliations using Lie ∞-morphisms.
problem Understanding symmetries in singular foliations.
method Analyzes Lie ∞-morphisms induced by Lie algebra actions on singular foliations. result Deduces geometrical consequences, including examples of non-extendable symmetries.