In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday-Pirashvili construction of higher Hochschild homology; in f…
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48 results for “representation homology”
Direct formula found for ADO invariants from homological representations.
problem Computing ADO invariants from quantum group representations.
method Direct homological formula for ADO invariants using partial traces of homological representations.
result Direct formula for ADO invariants without further truncations.
Polynomial representations found in surface braid and mapping class groups.
problem Homological representations of surface braid and mapping class groups.
method Study of homological representation functors and short exact sequences.
result Many homological representation functors are polynomial.
Researchers link knot Floer homology, Burau representation, and quantum gl(1|1).
problem Understanding the Burau representation and its relation to knot Floer homology.
method Developed a Heegaard Floer homology theory and associated a bordered sutured Heegaard Floer homology group to any tangle.
result Established a connection between the Burau representation and quantum gl(1|1), leading to a geometric proof of the braid representation.
Unified study of homological representations of mapping class groups.
problem Fundamental relationships between different homological representations.
method Analysis of twisted homology of configuration spaces.
result Unified account of non-degenerate pairings, embeddings, and isomorphisms.
Study mapping class groups' action on surface homology.
problem Characterize mapping class groups' homology representation.
method Precise characterization of induced homology representation.
result Characterized the image of the homology representation.
Unified approach to homological representations of topological groups.
problem Constructing homological representations of topological groups.
method Functorial approach using topological enrichment of Quillen bracket construction.
result Unified construction of homological representations for mapping class groups and motion groups.
Proves SU(2) representations for certain 3-spheres with embedded tori.
problem Characterizing SU(2) representations for specific 3-manifolds.
method Instanton Floer homology, surgery exact triangle, holonomy perturbations, non-vanishing results, and cable surgery results.
result Fundamental groups of certain 3-manifolds admit irreducible SU(2) representations.
New results show how complex representations are needed to realize taut sutured handlebodies as homology products.
problem Understanding the complexity of representations needed for taut sutured handlebodies as homology products.
method Illustrates the relationship between taut sutured handlebody topology and the complexity of realizing them as homology products.
result Illustrates the complexity of representations needed for taut sutured handlebodies as homology products.
In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the -dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least , and we proved that the homology of such a subcomplex is concentrated in degree . This homology group supports a natural act…
Heegaard Floer homology connects to polynomial representations of Hecke algebras.
problem Understanding polynomial representations of double affine Hecke algebras.
method Using higher-dimensional Heegaard Floer homology and topological interpretations.
result Recovery of polynomial representations from Heegaard Floer homology.
The paper calculates asymptotic Betti numbers and homology multiplicities for graph configuration spaces.
problem Understanding the homology of ordered configuration spaces of graphs.
method Explicit formulas for asymptotic Betti numbers and homology multiplicities in characteristic zero.
result Explicit formulas for asymptotic multiplicities in homology of irreducible representations of the symmetric group.
We classify SL(2;C)-representations of a Brieskorn homology 3-sphere. We show any irreducible representation into SL(2;C) is conjugate to that into either SU(2) or SL(2;R). We also give a construction of SL(2;R)-representations for a Brieskorn homology 3-sphere from PSL(2;R)-representations of the base orbifold fundame…
A new model calculates colored Jones polynomials using homology.
problem Calculating colored Jones polynomials for different colors.
method Using a homological model based on the Lawrence representation and Kohno's result.
result Colored Jones polynomials are described as a graded intersection pairing of homology classes.
In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equ…
The study connects knot representations to Seifert hypersurfaces and instanton Floer homology.
problem Existence of Seifert hypersurfaces and irreducible representations for 2-knots.
method Quantitative instanton Floer homology and related techniques.
result Existence of at least four irreducible SU(2) representations for certain knots.
Defines odd Khovanov homology via categorification of q-Schur algebra.
problem Constructing odd Khovanov homology via representation theory.
method Supercategorification of q-Schur algebra, odd foams, tensor product on chain complexes.
result Odd Khovanov homology defined via categorification.
New findings on mapping class groups and their subgroups in homological representations.
problem Understanding subgroups within mapping class groups through homological representations.
method Analyzing the Johnson filtration and finite covers of mapping class groups.
result No faithful homological representations of mapping class groups exist.
New Khovanov homology for marked links collapses spectral sequence.
problem Understanding instanton theory and its relation to Khovanov homology.
method Introducing marked Khovanov homology and studying its properties.
result Khovanov homology for alternating links does not depend on marking data for non-split links.
Study of modular representations in homology of congruence subgroups.
problem Understanding modular representations in homology of congruence subgroups.
method Analysis of sequences of modular representations of symplectic and special linear groups over finite fields.
result Established periodic representation stability in the sense of Church--Farb.
We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence's representation. This formula generalizes the famous Burau representation formula of the Alexander polynomial.
Enhances knot Floer homology with algebraic representation theory.
problem Compatibility between summands in bordered knot Floer homology.
method Categorifies intertwining property of higher representations.
result New algebraic reformulation of compatibility property.
A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their …
Floer homology connects to quiver Hecke algebras in Coulomb branches.
problem Understanding the representation theory of quiver Hecke algebras.
method Using Lagrangian Floer homology in multiplicative Coulomb branches.
result Cylindrical KLRW algebras are realized by Floer homology.
Quantum groups created from disk configuration space homologies.
problem Creating quantum groups from algebraic structures.
method Reconstructing quantum groups from homologies of configuration spaces of disks.
result New combinatorics and actual submanifolds of configuration spaces.
Graph manifolds with small homology have non-trivial SU(2) representations.
problem Characterizing graph manifolds with small homology.
method Topological methods avoiding gauge theory.
result Existence of irreducible SU(2) representations for certain graph manifolds.
New computations show sl(N) homology is related to SU(N) representations of knots.
problem Computing colored sl(N) homology for nontrivial knots and links.
method Using SU(N) representations of knot complements, we compute homology and show isomorphisms.
result Colored sl(N) homology is isomorphic to the cohomology of SU(N) representations of knot complements.
Witt algebra acts on Khovanov-Rozansky homology of links.
problem Understanding algebraic structures in knot theory.
method Construction of Witt algebra action on Khovanov-Rozansky homology.
result Induced maps between twists of the homology via link cobordisms.
Curved Rickard complexes extend link homologies to arbitrary representations.
problem Link homologies for arbitrary representations.
method Definition and study of curved Rickard complexes.
result Deformations of link homologies generalize previous work.
Extends Lawrence's representations to integral Verma-modules and braid groups.
problem Integrating Lawrence's representations into Verma-modules and braid groups.
method Defining homological operators and showing they provide a representation for , establishing isomorphisms and preserving key properties.
result Recovering an integral version of Kohno's theorem for Verma-modules and braid group representations.
Study Heisenberg homology on surface configurations, revealing new representations of mapping class groups.
problem Homology of surface configurations with Heisenberg group representations.
method Analysis of unordered configurations in a surface, using Heisenberg group actions and representations.
result Obtained genuine and projective representations of mapping class groups from Heisenberg group actions.
The paper explores subrepresentations in graph homology.
problem Characterizing homology classes in graph covers.
method Study of subrepresentations in .
result Induced subrepresentations are not sufficient to characterize homology classes.
The homology groups of many natural sequences of groups (e.g. general linear groups, mapping class groups, etc.) stabilize as . Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequ…
Enhances stability ranges for Torelli and congruence subgroup homologies.
problem Improving stability ranges for specific subgroup homologies.
method Analyzes H2(Torelli subgroup of Aut(Fn)'s), H2(Torelli subgroup of mapping class groups), and Hk(congruence subgroups of GL_n(R)'s).
result Improved central stability ranges for various subgroup homologies.
We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical t…
Study irreducible SU(2) representations for knots in 3D.
problem Existence of irreducible SU(2) representations for cyclic branched covers of knots.
method Combining covering space techniques, SICUP matrices, and instanton Floer homology.
result Irreducible SU(2) representations exist for certain knots and branched covers.
New method proves representation stability for linear groups, resolving homological questions.
problem Proving representation stability for linear groups over fields of characteristic zero.
method Introducing a technique for quantitative representation stability theorems.
result Vanishing result for higher syzygies of VIC- and SI-modules.
Study on mapping class groups for low genus surfaces.
problem Arithmetic image of homological representations for low genus mapping class groups.
method Examined centralizers of finite groups acting on surfaces with genus at most 3.
result Complete answer to the arithmetic image question for low genus surfaces.
Researchers compute knot invariants in Seifert manifolds using Wilson loops.
problem Calculating BPS invariants for knots in Seifert manifolds.
method Analytically continued Chern-Simons level and representation from Wilson loop operator.
result Homological blocks for knots in Seifert manifolds and Seifert integer homology spheres.
We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations …
Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb]. Applying this to the trivial representation, we obtain as a corollary that the un…
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.
New method for group representation presentations.
problem Constructing presentations for group representations.
method Developed a new technique for generating relations.
result Important for calculating Torelli group's homology.
Proves complex homology three-spheres can be bounded by many handles.
problem Complex homology three-spheres and their bounding four-manifolds.
method Uses homology cobordism invariant Γ from instanton Floer homology.
result Any bounding four-manifold must be built out of many handles.
Central stability for the homology of congruence subgroups and the second homology of Torelli groupsmath.AT
We prove a representation stability result for the second homology groups of Torelli subgroups of mapping class groups and automorphism groups of free groups. This strengthens the results of Boldsen-Hauge Dollerup and Day-Putman. We also prove a new representation stability result for the homology of certain congruence…
We study the structural properties of colored Kauffman homologies of knots. Quadruple-gradings play an essential role in revealing the differential structure of colored Kauffman homology. Using the differential structure, the Kauffman homologies carrying the symmetric tensor products of the vector representation for th…
We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting structures and symmetries in the colored homology of general knots. We also give a si…
Representations of the Iwahori-Hecke algebra of type A_{n-1} are equivalent to representations of the braid group B_n for which the generators satisfy a certain quadratic relation. We show how to construct such representations from the natural action of B_n on the homology of configuration spaces of the punctured disk.…