Researchers use Gysin sequence to show sl(N) homology of T(2,m) is cohomology of SU(N) representations.
problem Calculating sl(N) homology of T(2,m) explicitly.
method Used Gysin exact sequence to show isomorphism without explicit calculation.
result sl(N) homology of T(2,m) is isomorphic to cohomology of SU(N) representations.
Part 2 discusses compatibility between analytic and topological Gysin maps.
problem Compatibility between analytic and topological Gysin maps.
method Detailed discussion using Thom's transversality theorem and Jakob's K-homology theory.
result Equivalence of analytic K-homology theory with the former Gysin maps.
Generalizes double transgression formulas on complex manifolds.
problem Currential double transgression formulas on complex manifolds.
method General framework using Bott-Chern Duality.
result Complements existing transgression formulas and provides new applications.
We use the theory of pseudo-holomorphic quilts to establish a counterpart, in symplectic Floer homology, to the Gysin sequence for the homology of a sphere-bundle. In a motivating class of examples, this "symplectic Gysin sequence" is precisely analogous to an exact sequence describing the behaviour of Seiberg-Witten m…
New Smith-Gysin sequence for non-semi-free actions without semi-free condition.
problem Constructing a Smith-Gysin sequence for non-semi-free actions.
method Developed a new Smith-Gysin sequence that includes an exotic term.
result Inclusion of an exotic term that depends on the subset MS1. In this paper we construct Cech cohomology groups that form a Gysin-type long exact sequence for principal torus bundles. This sequence is modeled on a de Rham cohomology sequence published in earlier work by Bouwknegt, Hannabuss and Mathai, which was developed to compute the global properties of T-duality in the prese…
New method defines Gysin maps for stratified spaces, preserving signatures.
problem Defining and preserving signatures in stratified spaces.
method Smooth atlas stratified spaces, fiber bundles, Gysin maps, KK-theory.
result Gysin maps preserve analytic signature classes of Witt spaces.
The paper proves a pointwise Gysin formula for vector bundles and applies it to show positivity of polynomials.
problem Positivity of polynomials in Chern forms of Griffiths semi-positive vector bundles.
method Develops a pointwise Gysin formula for hermitian vector bundles and applies it to show positivity.
result Positivity of several polynomials in Chern forms of Griffiths semi-positive vector bundles.
We give two generalizations of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action: 1) replacing the torus action by a compact connected Lie group action, 2) replacing the manifold having a torus action by an equivariant map. This provides a systematic method for calcula…
Study spherical T-duality and Massey products in iterated sphere bundles.
problem Understanding spherical T-duality and Massey products in iterated sphere bundles.
method Analyzing Gysin sequences and Massey products to find T-dual iterated sphere bundles.
result For certain iterated sphere bundles, spherical T-duality can be represented by Massey products.
We develop a new approach to the pulling back fixed point theorem of W. Browder and use it in order to prove various generalizations of this result.
In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen's geometric description of cobordism groups for finite dimensional smooth manifolds \cite{Q}. Quillen stresses the fact that this construction allows the definition of Gysin ma…
The blow-down map is studied in Lie algebroid cohomology.
problem Computing Lie algebroid cohomology of blowups.
method Developed a Gysin sequence for Lie algebroids and used it to compute cohomology.
result Generalized Mazzeo-Melrose theorem to Lie algebroids.
This paper introduces persistent equivariant cohomology and applies it to circle actions.
problem Understanding the cohomology of filtered spaces with group actions.
method Persistent Borel equivariant cohomology, Serre spectral sequence, Gysin homomorphism.
result Explicit description and cohomology computation for circle actions.
In this paper we study the topological T-dual of spaces with a non-free circle action mainly using the stack theory method of Bunke and co-workers \cite{Bunke1}. We first compare three formalisms for obtaining the Topological T-dual of a semi-free S1-space in a simple example. Then, we calculate the T-dual of genera…
The theory of differential characters is developed completely from a de Rham - Federer viewpoint. Characters are defined as equivalence classes of special currents, called sparks, which appear naturally in the theory of singular connections. There are many different spaces of currents which yield the character groups. …
Reciprocity laws for line bundles on circle fibrations over complex manifolds.
problem Analytic reciprocity laws for complex line bundles on fibrations in oriented circles.
method Study of Gysin maps and first Chern classes in cohomology.
result Sum of Gysin maps of Chern classes equals zero in H3(B,Z) under specific conditions. Spherical T-duality for iterated sphere bundles
problem T-duality for iterated sphere bundles
method Repackaging cohomological data into Massey products
result Found T-dual iterated sphere bundles associated to Massey products
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.
In this paper, we investigate the behaviour of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specificially with the Chas-Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fibre bundl…
Develops a new theory of localization in algebraic geometry.
problem Understanding localizations in cohomological theories with open-closed structures.
method Categorical and algebro-geometric approach, focusing on torsors and refinements.
result Establishes compatibility with various algebraic operations and recovers classical results.
Paper develops equivariant basic cohomology for Lie groupoids.
problem Equivariant cohomology for Lie groupoids with weak actions.
method Using Kan fibrations and fiber structures, constructing models and comparing with existing theories.
result Equivariant basic cohomology theory for orbifolds and Lie groupoids.
Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
problem Determining when cosymplectic manifolds are diffeomorphic based on their cosymplectomorphism groups.
method Characterized Reeb flow, used to descend isomorphism to symplectic base manifolds, preserved monodromy class ensuring bundle equivalence.
result Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
Proves a Thom isomorphism for foliated differential forms.
problem Transverse Lie algebra actions on foliated manifolds and vector bundles.
method Thom isomorphism theorem for differential forms in foliated settings.
result Established a new Thom isomorphism theorem.
New knot quandles distinguish ribbon knots with isomorphic groups.
problem Distinguishing knots with isomorphic fundamental groups.
method Examined knot quandles of Suciu's ribbon knots and computed their types.
result Knot quandles of Suciu's ribbon knots are mutually non-isomorphic.
Geometric duality connects graph isomorphism and knot equivalence.
problem Understanding the equivalence of graph isomorphism and knot equivalence.
method Observation of geometric duality in planar graphs and links.
result The equivalence relation defined by isomorphisms of checkerboard graphs is the same as 2-isomorphisms of checkerboard graphs.
In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a ch…
The paper explores isomorphisms on isoparametric hypersurfaces in spheres, leading to new geometric structures.
problem Investigating isomorphisms between principal distributions on isoparametric hypersurfaces.
method Constructing vector bundle isomorphisms and nearly Kähler structures.
result Explicit construction of a global vector bundle isomorphism for all odd multiplicities.
We construct examples of knots that have isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms, showing that the linking forms provide more information than the modules alone. This generalizes work of Trotter, who found examples of knots that have isomorphic classical Alexander modules, but…
Proves regularity of isomorphisms between hyperbolic 3-manifolds.
problem Understanding the regularity of isomorphisms between hyperbolic 3-manifolds.
method Strengthened a previous result by proving regularity in the sense of Boileau and Friedl.
result Profinite isomorphisms of hyperbolic 3-manifolds are regular.
We classify connected Lie groups which are locally isomorphic to generalized Heisenberg groups. For a given generalized Heisenberg group N, there is a one-to-one correspondence between the set of isomorphism classes of connected Lie groups which are locally isomorphic to N and a union of certain quotients of noncom…
Non-isomorphic groups with similar profinite completions found.
problem Finding non-isomorphic groups with similar profinite completions.
method Exhibited infinitely many pairs of non-isomorphic groups with specific properties.
result Groups with Property FA and non-trivial actions on trees have isomorphic profinite completions.
Paper tackles NP-complete subgraph isomorphism counting problem.
problem Counting subgraph isomorphisms in large graphs.
method Learning framework that augments representation learning architectures and iteratively attends pattern and target graphs.
result Scalable learning approach counts subgraph isomorphisms in linear time.
Paper solves isomorphism problem for specific Baumslag-Solitar groups.
problem Isomorphism problem for small rose non-ascending generalized Baumslag-Solitar groups.
method Analyzed group actions on trees with specific stabilizers.
result Isomorphism problem solvable for the specified groups.
We consider the notion of stable isomorphism of bundle gerbes. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H^3(M, Z). Stable isomorphism sheds light on the local theory of bundle gerbes and enables us to develop a classifying theory …
New benchmarks improve model performance by accounting for isomorphism classes in multi-relational datasets.
problem Synthetic multi-relational datasets lack isomorphism class awareness, leading to overestimation of model performance.
method Proposed isomorphism-aware synthetic benchmarks and a prioritisation scheme to improve model performance and stability.
result Isomorphism classes can be utilised to improve model performance, stability during training, and reduce training time.
This is the fourth of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an a…
Undecidability proved for DG algebras problems.
problem Stable isomorphism, quasi-isomorphism, and Morita equivalence problems for semifree DG algebras.
method Essentially autonomous solutions by Gemini Deep Think and Aletheia.
result Proved undecidability of problems for semifree DG algebras.
A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that …
The paper disproves a conjecture about isomorphic subgroups in finite groups.
problem The existence of non-isomorphic subgroups that are isomorphic in extensions of finite groups.
method Constructing extensions of finite groups to show non-isomorphic pre-images of subgroups.
result Subgroups of finite groups that are isomorphic in extensions are not conjugate.
Jet isomorphism theorems for conformal geometry are discussed. A new proof of the jet isomorphism theorem for odd-dimensional conformal geometry is outlined, using an ambient realization of the conformal deformation complex. An infinite order ambient lift for conformal densities in the case in which harmonic extension …
Unique vertical isomorphisms between Fedosov dg manifolds are proven for Lie pairs.
problem Vertical isomorphisms of Fedosov dg manifolds associated with Lie pairs.
method Construction of Fedosov dg manifolds via splitting and connection, proving unique isomorphisms using iteration formula.
result Existence and uniqueness of vertical isomorphisms between Fedosov dg manifolds.
Proves regularity of isomorphisms between hyperbolic 3-manifolds.
problem Regularity of isomorphisms between cusped hyperbolic 3-manifolds.
method Proves regularity of profinite completions of fundamental groups.
result Proves A-polynomial is a profinite invariant. If M is a riemannian manifold, then the inclusion of the complex of coclosed harmonic forms into the de Rham complex induces a linear isomorphism in cohomology. If M has at most countably many connected components, this linear isomorphism is a Frechet isomorphism.
Projective manifolds with specific bundles are isomorphic to simpler spaces.
problem Characterizing projective manifolds with tangent bundles containing strictly nef subsheaves.
method Analyzing the structure of the tangent bundle and using properties of strictly nef subsheaves.
result Projective manifolds with the described bundles are isomorphic to projective bundles over hyperbolic manifolds or projective spaces.
The paper characterizes generalized Alexander quandles and counts them up to 127.
problem Classifying isomorphism classes of generalized Alexander quandles.
method Characterization through underlying groups and automorphisms, computation of counts.
result Computed the number of generalized Alexander quandles up to 127.
Survey on proof of homology isomorphism between two complex theories.
problem Proof of isomorphism between Heegaard Floer homology and embedded contact homology.
method Survey of proof methods from multiple papers.
result Established isomorphism between Heegaard Floer homology and embedded contact homology.
Homologies of Jones and partition algebras match cyclic and symmetric groups.
problem Matching homologies of specific algebras to cyclic and symmetric groups.
method Proving isomorphisms between algebras' homologies and group homologies.
result Homologies of Jones and partition algebras are isomorphic to cyclic and symmetric groups.