The paper defines a new equivalence relation for knot projections and finds an infinite number of distinct classes.
problem Classifying knot projections based on weak homotopy equivalence.
method Defining weak (1, 2, 3) homotopy and using it to find an invariant.
result There are an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
New presentations found for classifying spaces of orbifolds and Lie groupoids.
problem Finding new presentations for the weak homotopy types of classifying spaces.
method Using weak homotopy types and machinery of higher stacks and Lie groupoids.
result Presentations for weak homotopy types of classifying spaces of Lie groupoids and orbifolds.
Proves PL cobordism category's homotopy type, analogous to smooth case.
problem Describes the homotopy type of PL cobordism category.
method Introduces PL Madsen-Tillmann spectrum and proves weak equivalence.
result Weak homotopy equivalence between PL cobordism category and Ω^∞-1 MTPL(d).
New homotopy types and invariants defined for knots.
problem Defining and characterizing different homotopy types of knot projections.
method Introducing strong and weak (1, 2) homotopies and defining new invariants.
result New necessary and sufficient conditions for homotopy equivalence of knot projections.
The paper defines new homotopy relations on knot projections and classifies certain knot types.
problem Defining and classifying knot homotopy relations.
method Introducing cross chord numbers and using them to define strong and weak (1, 3) homotopies.
result Complete classification of knot projections with trivializing number two.
Study the spaces of flat connections for classical Lie groups using Chern-Weil theory.
problem Understanding the weak homotopy type of spaces of flat connections for classical Lie groups.
method Use Chern-Weil theory and relate to the functorial map involving continuous families of representations.
result Relate the spaces of flat connections to the weak homotopy type of the spaces of representations.
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.
We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.
Paper defines weak (1, 3) homotopy for knot projections and classifies trivial knots.
problem Classifying knot projections under weak (1, 3) homotopy.
method Defines weak (1, 3) homotopy, introduces a map to knot isotopy classes, and determines trivial knots.
result Determines which knot projections are trivial under weak (1, 3) homotopy.
Study constraints on diffeomorphisms and homeomorphisms of 4-manifolds with boundary.
problem Constraints on smooth families of 4-manifolds with boundary.
method Use Manolescu's Seiberg-Witten Floer stable homotopy type.
result Inclusion map between diffeomorphisms and homeomorphisms is not a weak homotopy equivalence.
This paper studies PL cobordism categories and their homotopy types.
problem Understanding the homotopy types of PL cobordism categories.
method Introducing a bordism category and showing weak homotopy equivalence to an infinite loop space.
result The classifying space BCdPL is weak homotopy equivalent to an infinite loop space. We prove that if M is a CW-complex, then the homotopy type of the skeletal filtration of M does not depend on the cell decomposition of M up to wedge products with n-disks Dn, when the later are given their natural CW-decomposition with unique cells of order 0, (n−1) and n; a result resembling J.H.C. Whi…
Model structures on multicomplexes help study complex geometry.
problem Understanding homotopy types of complex manifolds.
method Model category structures on N-multicomplexes with weak equivalences induced by quasi-isomorphisms. result Establishes a basis for studying almost and generalized complex manifolds.
Constructs a weak Poisson bracket and lifts it to differential forms.
problem Creating a weak Poisson bracket over submanifolds and foliations.
method Encoding weak Poisson structure into homotopy Poisson structure and lifting to differential forms.
result Lifts a weak Poisson bracket to the algebra of forms with a direct physical interpretation.
We calculate the weak homotopy type of the group of contactomorphisms of the three-sphere which coincide with the identity on (a neighborhood of) an overtwisted disk.
The paper proves a weak equivalence for topological cobordism categories.
problem Understanding the homotopy type of topological cobordism categories.
method Using smoothing theory and excision in tangential structure, reducing to smooth cobordism categories.
result The classifying space of the cobordism category is weakly equivalent to a specific Thom spectrum.
Paper defines weak representations and their equivalence to VB-groupoids.
problem Understanding representations of Lie groupoids and VB-groupoids.
method Introduces weak representations and shows equivalences between categories.
result Equivalence between 2-term representations up to homotopy and weak representations of G.
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.
We give an interpretation of Yetter's Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter's invariant depends only on the homotopy type of M, and the weak homot…
We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that the weak homotopy type of this space is preserved by certain surgeries on the bou…
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.
The study proves a strong parametric h-principle for minimal surfaces.
problem Proving a parametric h-principle for minimal surfaces.
method Using a parametric h-principle due to Forstneric and Larusson.
result The space of complete nonflat conformal minimal immersions has the same homotopy type as the space of continuous maps.
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.
Suppose M is a noncompact connected smooth 2-manifold without boundary and let D(M)_0 denote the identity component of the diffeomorphism group of M with the compact-open C^infty-topology. In this paper we investigate the topological type of D(M)_0 and show that D(M)_0 is a topological ell_2-manifold and it has the hom…
Atiyah classes of DG manifolds of positive amplitude are invariant under weak equivalences.
problem Defining and studying Hochschild cohomology of DG manifolds of positive amplitude.
method Using poly-differential operators and derived intersection, proving invariance under weak equivalences.
result Hochschild cohomology of DG manifolds of positive amplitude is invariant under weak equivalences.
New methods prove weak homotopy inequivalence for manifold symmetries.
problem Understanding symmetries of smooth 4-manifolds.
method Using Pin(-2)-monopole equations.
result New examples of 4-manifolds where symmetries are not fully represented.
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.
Study the topology of stable vector fields and Lyapunov functions on R^n.
problem Topology of stable vector fields and Lyapunov functions on R^n.
method Differential topology, Lyapunov theory, and results on diffeomorphism groups of discs.
result Path-connected and simply connected spaces of stable vector fields for n≠4,5 and weakly contractible for n≤3.
Let M be the cotangent bundle of S^2, with the standard symplectic structure. By adapting an argument of Gromov we determine the weak homotopy type of the group S of those symplectic automorphisms of M which are trivial at infinity. It turns out that S is weakly homotopy equivalent to \Z. π_0(S) is generated by the cla…
We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2-category of pointed ho…
Classifies crossings in tangles on surfaces, finding no nontrivial indices.
problem Classifying crossings in tangles on surfaces.
method Divides crossings into tribes compatible with Reidemeister moves and analyzes components, orders, and homotopy types.
result No nontrivial indices on classical knot diagrams.
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.
This paper refines homotopy theory for cubical sets and uniform spaces.
problem Classical homotopy theory limitations in cubical sets and uniform spaces.
method Develops a uniform-theoretic refinement for cubical sets and uniform spaces, lifting to a full and faithful embedding.
result Lifts classical homotopy categories to new uniform homotopy categories, generalizing cohomology theories.
Engel knots map to formal knots without restrictions.
problem Classifying Engel knots and their properties.
method Weak homotopy equivalence of Engel knots to formal knots.
result Engel knots map to formal knots without restrictions.
Study of diffeomorphisms groups on lens spaces with Morse-Bott foliations.
problem Computing homotopy types of diffeomorphism groups of specific foliations.
method Analysis of leaf-preserving and foliated diffeomorphisms on lens spaces.
result Inclusion of leaf-preserving groups into foliated groups is a homotopy equivalence.
Paper proves strong h-principle for minimal surfaces and null curves in higher dimensions.
problem Proving the parametric h-principle for minimal surfaces and null curves in higher dimensions.
method Using complex analysis and homotopy theory, the paper proves the parametric h-principle for minimal surfaces and null curves in higher dimensions.
result The inclusion of real parts of nonflat null holomorphic immersions into nonflat conformal minimal immersions is a weak homotopy equivalence.
Study shows weak homotopy equivalences for complete minimal surfaces.
problem Understanding complete minimal surfaces and their properties.
method Analyzes algebraic null immersions and conformal minimal immersions.
result Inclusion and differential mappings are weak homotopy equivalences.
This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants -- the homotopy groups, and (2) an analog o…
Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable …
Let f:M→N be a smooth area decreasing map between two Riemannian manifolds $(M,\gm)$ and $(N,\gn)$. Under weak and natural assumptions on the curvatures of $(M,\gm)$ and $(N,\gn)$, we prove that the mean curvature flow provides a smooth homotopy of f to a constant map.
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.
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.
Study homology manifolds using spectral sheaves and spectral six functor formalism.
problem Characterize and understand homology manifolds through spectral sheaves.
method Adapt six functor formalism to spectral sheaves on locally compact Hausdorff spaces.
result Prove that compact ANR homology manifolds are Poincaré duality complexes.
An eε-Lipschitz and co-Lipschitz map, as a metric analogue of an ε-Riemannian submersion, naturally arises from a sequence of Alexandrov spaces with curvature uniformly bounded below that converges to a space of only weak singularities. In this paper we prove its homotopy lifting property and its homotopy stabilit…
Study shows infinite real homotopy types for complex nilmanifolds.
problem Understanding real homotopy types of complex nilmanifolds.
method Analyzing 2n-dimensional nilmanifolds with generalized complex structures. result Infinitely many real homotopy types exist for 2n-dimensional nilmanifolds. 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.
The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.