We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,742 papers · 148 categories
Trend · papers per month
48 results for “framed cobordism”
The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries…
We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically sl…
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…
Frame-spun knots are constructed by spinning a knot of lower dimension about a framed submanifold of S^n. We show that all frame-spun knots are slice (null-cobordant).
We use topological surgery in dimension four to give sufficient conditions for the zero framed surgery manifold of a 3-component link to be homology cobordant to the 3-torus, which arises from zero framed surgery on the Borromean rings, via a topological homology cobordism preserving the homotopy classes of the meridia…
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 give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were in…
Study cobordisms of nested manifolds and their invariants.
problem Understanding cobordisms of nested manifolds and their invariants.
method Identify a nested analog of the Pontryagin-Thom construction and find spaces homotopy equivalent to nested Pontryagin-Thom spaces.
result Discover nested cobordism invariants and provide an alternative proof of Wall's splitting result.
Topology change and selection rules for high-dimensional $\Spin(1, n)_0$-Lorentzian cobordismsmath.GT
We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\Spin(1, n)_0$. This extends a result of Gibbons-Hawking on $\Sl(2, \C)$-Lorentzian co…
Study knot invariants and surgeries, proving properties of instanton homologies.
problem Characterize 3-manifolds as Dehn surgery on knots.
method Establish conjugation symmetry, use cobordism maps, analyze homology cobordism group.
result Prove is always zero or odd.
Revisits Pontryagin's proof of stable stems 0, 1, and 2.
problem Computing stable stems in dimensions 0, 1, and 2.
method Introduction of homotopy theory concepts, framed cobordism, and Pontryagin-Thom construction.
result Corrects the result of \(\pi_2^S\) from \cite{Pont2} and computes stable stems in dimensions 0, 1, and 2.
Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are…
Constructs TQFTs for cobordisms with cohomology class decorations.
problem Creating TQFTs for cobordisms with cohomology class decorations.
method Starting from an abelian group and a factorizable ribbon Hopf -bialgebra , constructs a TQFT for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in .
result Our functor recovers a special case of Kerler-Lyubashenko TQFTs when restricted to trivial decorations.
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.
Given a null-cobordant oriented framed link in a closed oriented --manifold , we determine those links in which can be realized as the singular point set of a generic map that has as an oriented framed regular fiber. Then, we study the linking behavior between the sing…
This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
problem Detecting exotic diffeomorphisms on reducible 4-manifolds with odd b+.
method Using a families Bauer-Furuta invariant and pin-cobordism.
result Constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
We present a short proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in details: {\bf Theorem.} Let be a connected oriented closed smooth 3-manifold. Let be the set of framed links in up to a framed cobordism. Let be the…
We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauff…
Generalizes Pontryagin's construction for proper maps in stable dimensions.
problem Mapping submanifolds to homotopy classes of proper maps.
method Introduces a new bijection between cobordism sets and homotopy classes for proper maps.
result Provides a bijection for cobordism sets of submanifolds embedded in .
Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the r…
New complexity measure for shake-slice knots established.
problem Defining and measuring complexity for shake-slice knots.
method Using dualizable patterns and studying knot signatures.
result Existence of -shake-slice knots with specified complexity.
Study instanton homology and its connection to Froyshov's invariant.
problem Determining framed instanton homology for Dehn surgeries on knots.
method Calculating the homology groups and relating them to Froyshov's invariant.
result Shows a relationship between framed instanton homology and Froyshov's invariant.
Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.
problem Matching instanton Floer homology with Heegaard Floer for specific 3-manifolds.
method Utilizes lattice homology and a recent cobordism map decomposition theorem.
result Isomorphism between framed instanton Floer homology and Heegaard Floer for almost-rational plumbings.
We provide an alternative proof that Koschorke's -invariant is injective on the set of link homotopy classes of -component homotopy Brunnian links . The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techni…
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further,…
The notion of the geometrical --control of self-intersection of a skew-framed immersion and the notion of the -structure (the cyclic structure) on the self-intersection manifold of a $\D_4$-framed immersion are introduced. It is shown that a skew-framed immersion $f:M^{\frac{3n+q}{4}…
We give a presentation of the -dimensional oriented cobordism category with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor from the category of smooth oriented manifolds and diffeomorphisms to an arbitrary cat…
This paper studies the interplay between the N=2 gauge theories in three and four dimensions that have a geometric description in terms of twisted compactification of the six-dimensional (2,0) SCFT. Our main goal is to construct the three-dimensional domain walls associated to any three-dimensional cobordism. We find t…
New proof for surface groups using Reeb graphs and Morse functions.
problem Classifying epimorphisms onto free groups for surface groups.
method Constructing a correspondence between epimorphisms and systems of hypersurfaces, using Morse functions and Reeb graphs.
result Any epimorphism onto a free group for a surface can be represented by a Reeb epimorphism.
Instantons prove knots are fibered and strongly quasipositive.
problem Proving properties of knots using instantons and L-space surgeries.
method New cobordism maps in framed instanton Floer homology.
result Proves fibered and strongly quasipositive knots have specific homomorphisms.
New 2-representations link spectral enhancements in link homology.
problem Spectral enhancements in link homology.
method Skew Howe duality, frames, and multifunctors.
result Spectral 2-representations of categorified quantum groups.
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.
The paper introduces a new invariant for cobordism classes of manifolds and extends cobordism groups.
problem Developing a new invariant for cobordism classes of manifolds.
method Introducing an invariant for null-cobordant -manifolds and constructing cobordism groups .
result The invariant is a complete invariant of -cobordism classes of null-cobordant -manifolds.
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 study explores conditions for -cobordisms between smooth 4-manifolds.
problem Determining when smoothly -cobordant 4-manifolds are also -cobordant.
method Providing new conditions for positive answers to the -cobordism theorem.
result Conditions under which standard methods to construct -cobordisms fail.
Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.
problem Understanding the relationship between Khovanov homology and decomposable Lagrangian cobordisms.
method Utilized previously defined filtered invariants to give obstructions.
result Partial answer to Ekholm, Honda, and Kálmán's question about Khovanov homology and decomposable Lagrangian cobordisms.
We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any compact Lie group G. In the special case when G is finite or a torus, we show that ou…
The article examines how many stabilizations are needed to transform 5D s-cobordisms into product cobordisms.
problem Understanding the number of stabilizations required to convert 5D s-cobordisms into product cobordisms.
method Utilizes Gabai's 4D light bulb theorem and Freedman Quinn invariant.
result Determines the number of stabilizations needed for 5D s-cobordisms.
Extends cobordism groups of immersions to projections with new results.
problem Understanding cobordism groups of projected immersions.
method Classifying space construction by Szűcs and Terpai, Salomonsen's exact sequence.
result Obtains results on cobordism groups in small and large codimensional cases.
We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a c…
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 uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over , and , the Kervaire invariant, the -invariant, and the -invariant. This requires either a framing or a corner structure. The canonical framing pro…
New results on algebraic knots with Brieskorn polynomials.
problem Understanding cobordisms of algebraic knots defined by Brieskorn polynomials.
method Analyzing Fox--Milnor type relations, decomposing algebraic cobordism classes, and studying cyclic suspensions.
result Spherical algebraic knots associated with Brieskorn polynomials have infinite order in the knot cobordism group.
On -cobordisms of complexity math.GT
Study -cobordisms of complexity 2 in 5D, finding obstructions and examples.
problem Understanding -cobordisms of complexity 2 in 5D.
method Compute monopole Floer homology and action of twisting involution.
result Obtained obstructions and constructed examples of high complexity.
Determine pairs of torus knots with genus one cobordisms, with exceptions.
problem Identify pairs of torus knots with genus one cobordisms.
method Combine obstructions from Heegaard Floer knot complex with explicit constructions.
result Determine pairs of torus knots with genus one cobordisms, with exceptions.
Study finds conditions for existence of specific pseudo-Riemannian cobordisms.
problem Existence of Spin-dimensional cobordisms with specific signature.
method Analyzes Spin-dimensional cobordisms with signature .
result Computes cobordism groups and necessary/sufficient conditions for existence.
We study the problem of defining maps on link Floer homology induced by unoriented link cobordisms. We provide a natural notion of link cobordism, disoriented link cobordism, which tracks the motion of index zero and index three critical points. Then we construct a map on unoriented link Floer homology associated to a …