We show that the theory of stable complex -cobordisms, for a torus , is embedded into the theory of stable complex -cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex -cobordism theory does not…
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48 results for “cobordism theory”
This paper classifies virtual string links up to cobordism using group theory.
problem Classifying virtual string links up to cobordism.
method Using group theory, specifically the group .
result Virtual string links up to cobordism are classified by elements of .
Paper shows string cobordism at 24 dims can be determined by elliptic genus.
problem Determining string cobordism at dimension 24.
method Using elliptic genus as an invariant.
result Shows string cobordism at 24 dims can be determined by elliptic genus.
Using methods inspired from algebraic -theory, we give a new proof of the Genauer fibration sequence, relating the cobordism categories of closed manifolds with cobordism categories of manifolds with boundaries, and of the Bökstedt-Madsen delooping of the cobordism category. Unlike the existing proofs, this approach…
Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.
problem Understanding the relationship between Legendrian knots through Lagrangian cobordisms.
method Combinatorial and geometric methods, including Heegaard Floer Homology and contact surgery.
result Construction of nondecomposable Lagrangian cobordisms between Legendrian knots.
Proves super-version of index theorem from algebraic cobordism invariants.
problem Cobordism invariants in supersymmetric quantum mechanics.
method Trace methods for deformation quantization.
result Recovery of cobordism invariant using trace methods.
Cobordisms of singular mapsmath.GT
Szűcs introduced cobordism of singular maps to compute groups of immersions and embeddings.
problem Computing cobordism groups of immersions and embeddings in dimensions where classical theory fails.
method Investigation of classifying spaces constructed by Szűcs and Rimányi.
result Collection and organization of results towards computation of cobordism groups of singular maps.
It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured F…
We interpret heterotic M-theory in terms of h-cobordism, that is the eleven-manifold is a product of the ten-manifold times an interval is translated into a statement that the former is a cobordism of the latter which is a homtopy equivalence. In the non-simply connected case, which is important for model building, the…
Functor connects sheaf categories of Legendrian submanifolds.
problem Connecting sheaf categories of Legendrian submanifolds.
method Constructs a functor between sheaf categories using Nadler-Shende's work.
result Provides a sheaf theory description of Lagrangian cobordism.
The study extends cobordism theory to complex sections, defining and calculating cobordism groups.
problem Understanding when almost complex manifolds can have complex sections.
method Defined complex section cobordism, determined groups, and introduced an obstruction.
result The obstruction vanishes for certain multiplicative generators in the complex cobordism ring.
In this paper, by use of techniques associated to cobordism theory and Morse theory,we give a simple proof of Poincare conjecture, i.e. Every compact smooth simply connected 3-manifold is homeomorphic to 3-sphere.
Combinatorial skeletons of 2-cobordism and annular categories with applications to equational logicmath.CT
Research categorizes 2-cobordisms and finds non-finitely axiomatizable equational theories.
problem Equational theories of local monoids in cobordism categories are not finitely axiomatizable.
method Introduced combinatorial data encoding -cobordisms, constructed von-Neumann-regular extensions, and studied various quotients.
result Local monoids of do not have finitely axiomatizable equational theories.
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.
The paper explores connections between braids, links, and cobordisms using algebraic methods.
problem Investigating functions on manifolds and their connections to braids, links, and cobordisms.
method Algebraic methods including group theory, sheaves, and formal groups.
result Constructs Lazard's one-dimensional universal commutative formal group and applies it to cobordism theory.
A new spectrum recovers cobordism cut and paste groups of manifolds with boundary.
problem Defining and studying cobordism cut and paste groups of manifolds with boundary.
method Constructing a spectrum that recovers the cobordism cut and paste groups of manifolds with boundary.
result Construction of a spectrum that recovers the cobordism cut and paste groups of manifolds with boundary.
New discrete cobordism category for nested manifolds and relations to algebraic structures.
problem Discrete cobordism category for nested manifolds.
method Stratified Morse theory, Cyl-objects, doubling construction, cylindrical bar construction.
result Relations between Cyl-objects and algebraic structures like Temperley-Lieb algebras.
Study defines PFH cobordism maps from Lefschetz fibrations.
problem Indirectly defined PFH cobordism maps using Seiberg-Witten theory.
method Defined PFH cobordism maps using holomorphic curves.
result Computed PFH cobordism maps for specific cases.
Functor connects sheaves on Lagrangian cobordisms, proving equivalence and action decreasing properties.
problem Understanding sheaf equivalences and actions on Lagrangian cobordisms.
method Analyzing sheaf quantizations and Legendrian lifts, proving functorial properties.
result Lagrangian cobordism functor is action decreasing and Morita equivalent to sheaf categories of Legendrians.
We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of …
In this article, we reformulate the cobordism map of embedded contact homology, which is induced by exact symplectic cobordism and defined as direct limit of homomorphisms called filtered ECH cobordism map. The filtered ECH cobordism map is defined by counting embedded holomorphic curves with zero ECH index and we prov…
We reconsider the su(3) link homology theory defined by Khovanov in math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of (complexes of) `cobordisms with seams' (actually, a `canopo…
New cobordism maps for instanton knot homology help compute surgeries on knots.
problem Computing surgeries on knots using instanton knot homology.
method Construct cobordism maps for instanton knot homology.
result Compute the framed instanton Floer homology of certain knots.
We classify higher-SPTs and their anomalies via cobordism theory.
problem Understanding higher symmetries and anomalies in quantum field theories.
method Developed a generalized cobordism theory using advanced mathematical tools.
result Classified higher-SPTs and their boundary anomalies.
Study exact Lagrangian cobordisms in cotangent bundles, proving bounds on sheaf interleaving distance and shadow distance.
problem Understanding Lagrangian cobordisms and their properties in cotangent bundles.
method Use microlocal theory of sheaves, sheaf quantization, and cone decompositions.
result Interleaving distance of sheaves is bounded by the shadow distance of the cobordism.
We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect…
We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we pr…
Author defines products of elements in cobordism-like modules induced from generic maps.
problem Generalizing cobordism modules for negative codimension generic maps.
method Defining products for pairs of elements in cobordism-like modules.
result Suitable elements for products defined in cobordism-like modules.
Foam cobordism groups linked to interval exchange automorphisms.
problem Understanding cobordism groups of foams.
method Using the Sah-Arnoux-Fathi invariant and K-theory.
result Identified cobordism groups with homology and K-groups.
Study one-dimensional topological theories with linear generating functions.
problem Understanding one-dimensional topological theories with defects.
method Construct bases of hom spaces for decorated unoriented one-dimensional cobordisms.
result Gram determinant and linear generating functions constructed.
This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into the three space. Secondly, we extend the und…
We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits…
We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stac…
Paper connects algebraic K-theory to foam geometry.
problem Identifying K-theory groups with foam cobordisms.
method Using stratified manifolds with flat bundles over facets, and gluing conditions.
result First K-theory group identified with 1-foam cobordism group.
TQFTs and quantum computingquant-ph
Program connects quantum computing and topological field theories.
problem Connecting quantum computing and topological field theories.
method Formalizes the connection using cobordisms and parallel transport.
result Realizes quantum circuits as cobordisms in a double category.
New proof of index theorem for topological manifold bundles.
problem Index theorem for fiber bundles of compact topological manifolds.
method Use of a convenient framework for bivariant theories and recent results on the homotopy type of the topological cobordism category.
result Refinement of the assembly map for an extended A-theory characteristic.
Frobenius extensions play a central role in the link homology theories based upon the sl(n) link variants, and each of these Frobenius extensions may be recast geometrically via a category of marked cobordisms in the manner of Bar-Natan. Here we explore a large family of such marked cobordism categories that are releva…
Two constructions of classifying spaces for singular maps are connected.
problem Understanding the classifying spaces of singular maps.
method Homotopy theoretical connection between two constructions.
result Some classifying spaces have a simple product structure.
For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q…
Paper extends cobordism maps in Khovanov and instanton homologies.
problem Define and prove compatibility of cobordism maps in Khovanov and instanton homologies.
method Extend embedded cobordism map to immersed cobordisms and prove compatibility.
result Induced map on Khovanov homology is injective for certain concordances.
We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding is at least two. This is a topological counterpart of the algebraic splitting theorem for embedded cobordisms of the first author, A. Nemethi…
Cobordism of disk knotsmath.GT
We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant r…
The paper explores how the unit inclusion affects topological quantum field theories in non-semisimple categories.
problem Understanding the effects of unit inclusion in non-semisimple braided tensor categories on topological quantum field theories.
method Analyzes the dualizability of the unit inclusion morphism in Morita 4-category of braided tensor categories and applies the Cobordism Hypothesis.
result Shows that the unit inclusion in non-semisimple modular categories leads to non-compact relative 3D topological quantum field theories.
We study here compact manifolds with positive scalar curvature metrics. We use the relative Yamabe invariant from math.DG/0008138 to define the conformal cobordism relation on the category of such manifolds. We prove that corresponding conformal cobordism groups $\Pos_n^{\conf}(γ)$ are isomorphic to the cobordism group…
In [Duke Math. J. 101 (1999) 359-426], Mikhail Khovanov constructed a homology theory for oriented links, whose graded Euler characteristic is the Jones polynomial. He also explained how every link cobordism between two links induces a homomorphism between their homology groups, and he conjectured the invariance (up to…
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.
For a fixed closed manifold , we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type . Our main theorem identifies the homotopy type of the classifying space of this cobordism category with that of the infinite loop-space of a certain spectrum related to the spectr…
Algebraic Poincare cobordismmath.AT
This paper is an introduction to the use of the cobordism of chain complexes with Poincaré duality in surgery theory. It is a companion to the author's paper "An introduction to algebraic surgery" math.AT/0008071 (to appear in Volume 2 of Surveys in Surgery Theory, Ann. of Maths. Studies, Princeton, 2001) which is an i…