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.
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 L1-functionals on Riemannian manifolds. 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.
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.
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.
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 …
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 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.
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 prove the Farrell-Jones Conjecture for (non-connective) A-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 n-fold vector bundles as special functors from the ∞- and n-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of n-fold vector bundles and we prove that any n-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.
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 Spinc-Dirac operators. As an application we give new examples of non-trivial Ham…
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
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.
Morse theory extended to noncompact manifolds with complex geometric data.
problem Extending Morse theory to noncompact manifolds with intricate geometric and homotopy data.
method Defining Morse homology for pairs of manifolds and related geometric/homotopy data, constructing a homotopy coherent diagram of linear maps, and showing it computes Morse homology.
result Morse homology can be computed using a chain complex derived from a homotopy coherent diagram.
Diffeology explores k-forms and bundles with more information than traditional differential forms.
problem Understanding k-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 k-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 c∞-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.
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 …
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.
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.
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.
Study homotopy sheaves on categories and their presheaves, proving descent properties.
problem Homotopy sheaves on categories and their presheaves.
method Homotopy right Kan extension, pretopologies, Yoneda embedding.
result Preserves homotopy sheaves and induces equivalence between sheaves and colimit-preserving sheaves.
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 BextrmDiff+(M) to BextrmDiff+(P1⊔⋯⊔Pn), yielding a prime decomposition fibre sequence. result The fibre Hg(P1,…,Pn) is a finite, connected cell complex, and the prime decomposition fibre sequence is effective for computations. 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 R-localisation. 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.
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…
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.
The paper classifies extensions of Yang-Mills-type theories and their spaces.
problem Classifying extensions of Yang-Mills-type theories with arbitrary pairings.
method Using a unified approach, the space of extensions is classified and compared with Yang-Mills theories.
result An upper bound to the rank of the space of extensions is given and compared with Yang-Mills theories.
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.
Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the_Roe (C*-)algebras_ C*(X) and their K…