New proofs and refined theorems on bounded cohomology.
problem Properties of bounded cohomology and comparison map.
method Homotopy-theoretic properties and generalizations.
result New proofs and refined versions of vanishing and covering theorems.
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48 results for “homotopy colimits”
Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy ty…
We study Morse theory on noncompact manifolds equipped with exhaustions by compact pieces, defining the Morse homology of a pair which consists of the manifold and related geometric/homotopy data. We construct a collection of Morse data parametrized by cubes of arbitrary dimensions. From this collection, we obtain a fa…
Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Simil…
We summarize our axioms for higher categories, and describe the blob complex. Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$ to any n-manifold W. The 0-th homology of this chain complex recovers the usual topological quantum field theory invariants of W. The higher homology groups should …
We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted K-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural -Dirac operators. As an application we give new examples of non-trivial Ham…
We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any pretopology on the original category induces a canonical pretopology of generalised coverings on the free cocompletion. We show that with respect to these pretopologies the homot…
On the Homotopy Type and the Fundamental Crossed Complex of the Skeletal Filtration of a CW-Complexmath.GT
We prove that if is a CW-complex, then the homotopy type of the skeletal filtration of does not depend on the cell decomposition of up to wedge products with -disks , when the later are given their natural CW-decomposition with unique cells of order 0, and ; a result resembling J.H.C. Whi…
Khovanov homology for pro-tangles and spectral sequences
problem Developing a framework for Khovanov homology for pro-tangles and spectral sequences
method Using pro-tangles, simplicial presheaves, and spectral sequences
result Establishing a fully faithful embedding and an algebraic spectral sequence for pro-tangles
Localizes smooth spaces to study their homotopy properties.
problem Understanding the homotopy theory of smooth spaces.
method Model category localization, Quillen equivalences, fibrant replacement.
result Localisation of smooth spaces agrees with motivic-style -localisation.
Paper develops a finite dimensional approximation scheme for Riemannian manifolds.
problem Integration on Riemannian manifolds.
method New finite dimensional approximation scheme motivated by categorical colimit.
result Establishes a generalization for -functionals on Riemannian manifolds.
Study the moduli space of reducible 3-manifolds using prime decomposition.
problem Understanding the homotopy type of moduli spaces of reducible 3-manifolds.
method Construct a splitting map from to , yielding a prime decomposition fibre sequence.
result The fibre is a finite, connected cell complex, and the prime decomposition fibre sequence is effective for computations.
New colored knot Floer homology defined using infinite full twists.
problem Defining a new homology theory for knots.
method Defining colored knot Floer homology through colimit of link Floer homology with infinite full twists.
result Colored knot Floer homology is a module over the colored knot Floer homology of the unknot.
The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spacesmath.MG
Develops a new framework for large-scale geometry.
problem Characterizing large-scale models of metric spaces.
method Categorical framework for metric Rips filtration and universal quasigeodesic cones.
result Establishes universal properties and adjointness of the Rips colimit.
We compute the Khovanov lasagna module of S²×S², confirming a conjecture.
problem Computing the Khovanov lasagna module of S²×S².
method Interpreting Manolescu-Neithalath's formula as a homotopy colimit, using categorified projectors.
result The Khovanov lasagna module of S²×S² is trivial.
Model for assembly map of bordism-invariant functors.
problem Understanding assembly maps of bordism-invariant functors.
method Categorical model using oplax colimits of stable, hermitian, and Poincaré categories.
result Explicit description of the kernel of the assembly map.
Develops Morse homology with DG coefficients for manifolds and spaces.
problem Homology with DG coefficients for manifolds and spaces.
method Derived local systems, DG modules, twisting cocycles, Morse trajectories.
result Isomorphic to DG Tor and Ext functors, recovers homology of total space of fibrations.
The paper proves a Whitehead theorem for fine shape spaces.
problem Proving a Whitehead theorem for fine shape spaces.
method Using Steenrod-Sitnikov homotopy groups and ind-groups.
result Fine shape morphisms are equivalences if they induce isomorphisms on π_i.
We study diffeologies on locally convex spaces and their application to smooth multiplication of distributions.
problem Constructing smooth multiplication of distributions on locally convex spaces.
method Using diffeological colimits and wavefront-set criterion.
result Proving smooth multiplication of microlocally multipliable distributions.
Unified solution to Goodman-Pollack transversal problem using matroids and topology.
problem Existence of an affine k-dimensional transversal to convex sets.
method Matroidal joins and topological methods.
result Unified solution including colorful Helly theorem and Holmsen's theorem.
Study knot spaces and Atiyah duality in spectral categories.
problem Understanding the space of embeddings of a circle into higher-dimensional manifolds.
method Develop a cosimplicial model using Atiyah duality and prove a comodule version.
result Compute knot spaces in low degrees and establish isomorphisms on fundamental groups.
We prove the Farrell-Jones Conjecture for (non-connective) -theory with coefficients and finite wreath products for hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties su…
This paper introduces - and -fold vector bundles as special functors from the - and -cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of -fold vector bundles and we prove that any -fold vector bundle admits a non-canonical isomorphism to a decomposed …
The paper extends vector bundle theory to non-Hausdorff manifolds.
problem Generalizing vector bundle theory to non-Hausdorff manifolds.
method Using Čech cohomology to classify real non-Hausdorff line bundles.
result Vector bundles over non-Hausdorff manifolds can be constructed as colimits of standard vector bundles.
New algebraic framework for studying surfaces in 3-manifolds.
problem Understanding incompressible surfaces in 3-manifolds.
method Defining Bar-Natan modules and functors from Frobenius algebras.
result Geometric content of Bar-Natan modules is presented via tunneling graphs.
Diffeology explores -forms and bundles with more information than traditional differential forms.
problem Understanding -forms and bundles in diffeological spaces.
method Developed theory of diffeological vector pseudo-bundles, including limits and colimits, and various operations.
result Sections of bundles of -forms contain more information than differential forms.
The paper calculates a new invariant for 4-manifolds using handle decompositions and skein relations.
problem Computing invariants for 4-manifolds built from handles.
method Handle attachment formulas, cabled colimits, lasso relation.
result Explicit calculations and partial vanishing results for specific 4-manifolds.
Study on diffeologies on locally convex spaces and smooth multiplication of distributions.
problem Geometric characterization and smoothness of distribution multiplication.
method Investigation of canonical and -diffeologies on locally convex spaces, proving geometric characterizations, and comparing diffeologies.
result Established a framework for nonlinear distribution theory beyond manifolds, realizing microlocally multipliable distributions as a diffeological colimit.
Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a…
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …
New examples of manifolds that are homotopy but not simple homotopy equivalent.
problem Characterizing simple homotopy types of even dimensional manifolds.
method Using algebraic K-theory, surgery obstruction map, and homotopy automorphisms.
result Construction of infinite families of manifolds that are homotopy equivalent but not simple homotopy equivalent.
By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisify a stratified version of Whitehead's theorem. As an example, we introd…
Classifies colored links and spatial graphs up to colored link-homotopy.
problem Classifying colored links and spatial graphs up to colored link-homotopy.
method Using Habegger-Lin theory for colored string links, and extending to colored links and spatial graphs.
result Classification of colored links and spatial graphs up to colored link-homotopy.
On homotopy braidsmath.GR
Paper proves homotopy braid group properties over integers and three strands.
problem Understanding homotopy braid group properties.
method Proved linearity over integers and torsion freeness for three strands.
result Homotopy braid group on three strands is torsion free.
Open 2D TFTs extend to closed theories with circle value as Hochschild homology.
problem Extending open 2D TFTs to closed theories.
method Using symmetric monoidal ∞-categories and Hochschild homology.
result Open 2D TFTs admit initial open-closed extensions.
New examples of manifolds with similar homotopy but different simple homotopy types.
problem Characterizing groups for which high-dimensional manifolds can be homotopy equivalent but not simple homotopy equivalent.
method Construction of doubles of thickenings and use of a formula for Whitehead torsion.
result Examples of high-dimensional manifolds exist for any finitely presented group with a nontrivial Whitehead group involution.
V. Turaev introduced the theory of topology of words and phrases in 2005. This is a combinatorialy extension of the theory of virtual knots and links. In this paper we generalize the notion of homotopy of words and phrases and we give geometric meanings of the generalized homotopy of words. Moreover using the generaliz…
The study shows how stabilizing manifolds with projective spaces affects their homotopy structure.
problem Understanding the homotopy of manifolds stabilized by projective spaces.
method Trace the effect of surgery on product manifolds, showing a loop homotopy decomposition after localization.
result A loop homotopy decomposition of a manifold after stabilization by a projective space is provided.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
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.
Extends six operations to sheaves in any symmetric monoidal category.
problem Extending six operations to a broader class of sheaves.
method Develops formalism for sheaves in any closed symmetric monoidal ∞-category, proving properties of locally contractible geometric morphisms and relating pullbacks and colimits.
result Establishes the six functor formalism for a wider range of sheaves, including those with values in any closed symmetric monoidal ∞-category.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples…
Characterizes compact complex surfaces with finite homotopy rank-sum.
problem Compact complex surfaces with finite homotopy rank-sum.
method Characterization and proof of Steinness of universal cover.
result Smooth compact complex Kaehler surfaces with finite homotopy rank-sum.
Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.
problem Investigate the homotopy of blow ups in algebraic and symplectic geometry.
method Develops fibrewise surgery theory and a purely homotopy theoretic approach.
result Obtained homotopy decompositions of the based loop space on blow ups.
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.
Characterizes Stein surfaces with finite homotopy rank-sum.
problem Finite homotopy rank-sum in Stein spaces.
method Rational homotopy theory, classification of Stein surfaces.
result Affine Stein surfaces with finite fundamental group are either simply connected or of order 2.
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.