A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. …
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48 results for “functorial maps”
Study normal bundle and deformation to get new pushforward maps.
problem Construct pushforward maps in various homology theories.
method Use deformation Lie groupoids to construct pushforward maps.
result Functoriality of pushforward maps recovers and generalizes previous cases.
The paper describes new types of picture-valued invariants and their applications.
problem Understanding and categorizing picture-valued invariants on diagrams.
method General description and classification of derivations and functorial maps.
result Two new examples of functorial maps are introduced, including the order and lifting maps.
We prove that Morrison and Nieh's categorification of the su(3) quantum knot invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified su(2) theory, which was not functorial as originally defined. We use methods of Bar-Natan to construct explicit chain maps for each variation of…
Functoriality proved for higher rho invariants of elliptic operators.
problem Computing higher rho invariants of elliptic operators.
method Functoriality proved through finite-propagation argument.
result Maximal higher rho invariants behave functorially under quotient maps.
In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functor…
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
We construct various functorial maps (projections) from virtual knots to classical knots. These maps are defined on diagrams of virtual knots; in terms of Gauss diagram each of them can be represented as a deletion of some chords. The construction relies upon the notion of parity. As corollaries, we prove that the mini…
Investigates properties of moment maps and stratifications on Lie groups.
problem Understanding moment maps and stratifications on real reductive Lie groups.
method Functorial, algebraic approach to moment map and Kirwan-Ness stratification.
result Properties and properties of moment maps and stratifications established.
Functorial maps and weak parities are equivalent descriptions of rules of substitution virtual crossings for classical in diagrams of a knot in a way compatible with Reidemeister moves. We introduce the notion of maximal weak parity and describe it for knots in a given closed oriented surface. This weak parity defines …
Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators.
We define compactifications of vector spaces which are functorial with respect to certain linear maps. These "many-body" compactifications are manifolds with corners, and the linear maps lift to b-maps in the sense of Melrose. We derive a simple criterion under which the lifted maps are in fact b-fibrations, and identi…
We discuss two kinds of functorial prolongations of the functional bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base. We study the prolongation of vector fields in both cases and we prove that the bracket is preserved. Our proof is based on several new res…
Functor connects symplectic and contact structures via cutting and blowups.
problem Establishing a functorial relationship between symplectic and contact structures.
method Developed a cutting procedure and its inverse for manifolds with boundary and equivariant transverse maps, then applied it to non-symplectic and non-contact structures.
result Obtained an inverse functor for equivariant radial-squared blowups.
Extends Khovanov homology to surfaces with singularities.
problem Computing invariants for surfaces with singularities.
method Functorial extension of Khovanov homology to surfaces with double points.
result Induces a map between Khovanov homology groups of boundary links.
Mapping spaces of supermanifolds are usually thought as exclusively in functorial terms (i.e. trough the Grothendieck functor of points). In this work we provide a geometric description of such mapping spaces in terms of infinite-dimensional super-vector bundles.
Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. Fo…
It is a deep fact that the homotopy classification of topological manifolds is convariantly functorial. In other words, a map from a topological manifold M to another N naturally induces a map from the structure set S(M) to S(N). We extend the fact to the isovariant structure set S_G(M, rel M_s) of G-equivariant topolo…
We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction, and prove invariance. We also make a comparison with the graph TQFT for Heegaard Floer homology.
We adapt the notion of Jacobi diagrams on surfaces (considered by Andersen-Mattes-Reshetikhin), and construct a LMO-like map that we use to compare some functoriality properties of WRT and LMO invariants.
Khovanov spectra are shown to be functorial under certain conditions.
problem Understanding functoriality of Khovanov spectra.
method Proving functoriality up to homotopy and sign for Khovanov spectra.
result Khovanov spectra are functorial under specific conditions.
We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We gi…
Clarifies sign ambiguity in Khovanov homology functoriality.
problem Sign ambiguity in Khovanov homology functoriality.
method Blanchet's oriented model and cobordisms with singularities.
result Strict functoriality established for the oriented model.
Extends Khovanov bracket to link cobordisms, proving functoriality up to scalars.
problem Proving functoriality of Khovanov homology under link cobordisms.
method Extending generalized Khovanov bracket to smooth link cobordisms in R^3×I and proving functoriality up to global invertible scalars.
result Generalized Khovanov bracket is functorial up to global invertible scalars.
Simplifies fixing Khovanov homology functoriality.
problem Fixing the sign indeterminacy in Khovanov homology.
method Adjusting signs of Reidemeister and Morse moves.
result Functoriality of Khovanov homology is fixed.
For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as homological representations. Representations of this kind have proved th…
We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in ) is functoria…
Defines cobordism maps connecting Khovanov and instanton homologies.
problem No direct correspondence between Khovanov and instanton homology cobordism maps.
method Defines a cobordism map on the instanton cube complex as a filtered chain map.
result Proves the cobordism map recovers both Khovanov and instanton homology cobordism maps.
Proves Khovanov homology functoriality and positivity for gl2 webs.
problem Proving functoriality and positivity of Khovanov homology.
method Categorical proof using linear complexes stability.
result Strong positivity result for surface skein algebras.
Clarifies properties of Ozsvath-Szabo contact invariant.
problem Properties and functoriality of Ozsvath-Szabo contact invariant.
method Explains and proves functoriality properties of the invariant.
result Strong functoriality under Stein cobordisms.
Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere's functorial notion…
Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of…
Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with sm…
Functorial semi-norms on singular homology give refined "size" information on singular homology classes. A fundamental example is the l^1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the l^1-semi-norm.
We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov homology of the link; (2) this filtered complex behaves nicely under planar isotopy…
Extends odd Khovanov bracket to link cobordisms and proves functoriality up to sign.
problem Proving functoriality of odd Khovanov homology up to sign.
method Extending odd Khovanov bracket to link cobordisms and proving functoriality up to sign.
result Functoriality of odd Khovanov homology up to sign.
We prove that the bigraded colored Khovanov-Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.
In this paper, we prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds.
We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…
Quantum field theory uses Lorentzian bordisms to describe time evolution.
problem Describing the time evolution of quantum field theories.
method Defines a functorial field theory on Lorentzian bordism pseudo-category.
result Lorentzian bordisms naturally arise in algebraic quantum field theory.
Alternative proof of Khovanov's up-to-sign functoriality for odd Khovanov homology.
problem Functoriality of Khovanov's odd Khovanov homology.
method Extending Hochschild (co)homology to quasi-associative algebras and applying it to Khovanov's homology.
result First proof of functoriality of Naisse and Putyra's tangle theory up to unit.
We verify that the formula of X. Ma for the analytic torsion form of an iterated fibration implies that Lott's secondary analytic index is functorial.
We describe a modification of Khovanov homology (math.QA/9908171), in the spirit of Bar-Natan (math.GT/0410495), which makes the theory properly functorial with respect to link cobordisms. This requires introducing `disorientations' in the category of smoothings and abstract cobordisms between them used in Bar-Natan's …
Establishes functoriality of Baum-Bott residues under specific conditions.
problem Functoriality of Baum-Bott residues under certain conditions.
method Establishes functoriality of Baum-Bott residues under specific conditions.
result The singular set of a foliation has dimension at least \( k-1 \).
In this paper, we define the equivariant eta form of Bismut-Cheeger for a compact Lie group and establish a formula about the functoriality of equivariant eta forms with respect to the composition of two submersions.
The paper defines a category of Lagrangian correspondences in super Hilbert spaces and constructs a functorial field theory.
problem Understanding composition of Lagrangian correspondences in Hilbert spaces.
method Study of Lagrangian correspondences, construction of a category, and functorial field theory.
result Well-defined composition law in a category of Lagrangian correspondences.
This work draws inspiration from three important sources of research on dissimilarity-based clustering and intertwines those three threads into a consistent principled functorial theory of clustering. Those three are the overlapping clustering of Jardine and Sibson, the functorial approach of Carlsson and Mémoli to par…