Homological stability aids in computing group homology.
problem Computing homology of families of groups.
method Proving homological stability theorems and computing stable homology.
result Computation of Higman-Thompson groups' homology.
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48 results for “homology groups”
Study homology groups of mapping and Torelli groups for surfaces with abelian covers.
problem Understanding the homology of mapping and Torelli groups for surfaces with specific topologies.
method Examined the first homology group of mapping and Torelli groups with coefficients in the first rational homology group of the universal abelian cover of the surface.
result For surfaces with one boundary component, the twisted homology groups are finite-dimensional, but for surfaces with one puncture, they are infinite-dimensional.
Maps quandle homology to relative group homology.
problem Understanding the relationship between quandle and group homology.
method Introducing a chain map and constructing quandle cocycles.
result Relates quandle homology to relative group homology through triangulations.
Study on homological Dehn functions of groups of type .
problem Understanding the homological Dehn functions of groups of type .
method Proved foundational results, studied homological Dehn functions of Leary's groups, and provided methods to obtain groups with specific homological Dehn functions.
result Found groups of type with quartic homological Dehn function and unsolvable word problem.
Researchers found the second homology of Torelli groups for large g.
problem Calculating the second rational homology of Torelli groups.
method Explicit calculations for .
result The second rational homology group is determined for Torelli groups with g >= 6.
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 …
The paper connects Brauer algebra homology to symmetric group homology.
problem Understanding the homology of Brauer algebras.
method Interpreting Brauer algebras as Tor-groups and comparing to symmetric group homology.
result Isomorphism of homology groups under specific conditions.
We give a Dehn-Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by Levine, of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some…
Homological stability fails for Cremona groups, rational varieties, and function fields.
problem Homological stability in Cremona groups fails in both possible ways.
method Explained the failure of homological stability for Cremona groups.
result Homological stability fails for Cremona groups in both possible ways.
Computes quandle associated groups using group homology.
problem Computing associated groups of quandles.
method Using group homology theory to describe and compute associated groups.
result Computed second quandle homology groups of specific quandle families.
The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor's link invariants an…
Compute Bredon homology for a specific type of Artin groups.
problem Calculate Bredon homology for Artin groups of dihedral type.
method Compute Bredon homology groups of the classifying space for virtually cyclic subgroups with K-theory coefficients.
result Computed Bredon homology groups for Artin groups of dihedral type.
Paper proves vanishing homology groups for certain hyperbolic groups.
problem Understanding homology groups of specific hyperbolic groups.
method Using twisted Wirtinger presentations to prove homology group vanishing.
result Second homology groups vanish for certain Gromov hyperbolic groups.
Study homology groups for non-orientable surfaces with boundary.
problem Determine the first homology group for mapping class groups of non-orientable surfaces.
method Calculate the first homology group with twisted coefficients.
result Explicitly determined the first homology group for various mapping class groups.
Study shows infinite-rank summand in homology concordance group of knots.
problem Homology concordance of knots in integer homology three-spheres.
method Knot Floer homology to construct homology concordance homomorphisms.
result Homology concordance group modulo knots from S^3 contains an infinite-rank summand.
Homology cobordism group of homology cylinders and invariants related to lower central seriesmath.GT
The study examines invariants of homology cylinders and their relations to free nilpotent groups.
problem Understanding invariants of homology cylinders and their connections to free nilpotent groups.
method Extensions of Johnson homomorphisms, Milnor invariants, and Orr invariants of links to homology cylinders; establishment of a combined filtration.
result Determination of the image of the filtration under the invariants and investigation of relations among the invariants.
We determine the second homology group of the homological Goldman Lie algebra for an oriented surface.
Study the first homology group for a specific mapping class group.
problem Calculating the first homology group for a specific mapping class group.
method Determined using coefficients in for a non-orientable surface.
result Determined the first homology group for the mapping class group of a specific surface.
Khovanov-Rozansky homology shows periodic links have group actions.
problem Understanding periodic links through homology.
method Showed Khovanov-Rozansky -homology has group actions on -periodic links.
result Proved an analog of periodicity criterion using -homology.
We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-ty…
Milnor-Thurston homology theory is a construction of homology theory that is based on measures. It is known that it is equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor-Thurst…
Finite group action on a surface yields a special homology subspace.
problem Finite group action on a surface.
method Invariant Lagrangian subspace in homology.
result Existence of a G-invariant Lagrangian subspace in H1(S).
Homology of partition algebras matches symmetric group homology under certain conditions.
problem Understanding homology of partition algebras and comparing it to symmetric groups.
method Inductive resolution and high acyclicity arguments, parallel to earlier work on Brauer algebras.
result Homology of partition algebras is isomorphic to symmetric group homology under specific conditions.
Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "offici…
We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an -module.
We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute t…
Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
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.
Study Heegaard Floer homology and word metric on Torelli group.
problem Relationship between Heegaard Floer homology and Torelli group word metric.
method Analyzes Heegaard Floer homology correction terms and word metric properties of Torelli group.
result Cayley graph of Torelli group has infinite diameter in word metric.
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
Algorithm calculates knot Floer homology for a specific knot type.
problem Computing knot Floer homology for knots.
method Algorithm based on fundamental group of knots.
result Algorithm successfully computes knot Floer homology.
The paper extends a knot invariant to graphs and connects it to homology cylinders.
problem Understanding the structure of homology cobordism groups.
method Using tangle Floer homology, the authors define a new invariant for embedded graphs and prove a concatenation formula.
result The new invariant induces a homomorphism on the homology cobordism group of homology cylinders.
In this survey paper, we give a complete list of known results on the first and the second homology groups of surface mapping class groups. Some known results on higher (co)homology are also mentioned.
By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the …
Proves homology of mapping class groups for infinite-type surfaces.
problem Homology of mapping class groups for infinite-type surfaces.
method Modification of Mather's argument and homological stability result.
result Homology of mapping class groups determined for binary tree surfaces.
The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel…
Spaces are classified as almost homology n-manifolds if their homology groups are trivial for all but the last dimension.
problem Classifying spaces as almost homology n-manifolds.
method Providing a necessary and sufficient condition for locally compact homogeneous ANR-spaces or strongly locally homogeneous ANR-spaces to be almost homology n-manifolds.
result Spaces are classified as almost homology n-manifolds based on their homology groups.
A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and knotted surfaces. In this paper, a homology theory is defined that unifies group an…
We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.
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.
We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of finite-type invariants of 3-manifolds. We study the structure of this group from several d…
Investigates local indicability of groups with circle homology presentations.
problem Conditions for local indicability in groups with circle homology presentations.
method Generalizes results for two-relator presentations to circle homology presentations.
result Extends results on local indicability to LOT groups and non-cycle-free Adian presentations.
Torelli group's second homology is finite for large genus surfaces.
problem Determining the finiteness of Torelli group's homology.
method Analyzing rational homology groups of Torelli groups for surfaces of genus g ≥ 51.
result Second rational homology of Torelli group is finite for large genus surfaces.
Let be a 3-dimensional handlebody of genus . We determine the twisted first homology group of the mapping class group of with coefficients in the first integral homology group of the boundary surface for .
For an orbifold M we define a homology group, called t-singular homology group t-H_q(M), which depends not only on the topological structure of the underlying space of M, but also on the orbifold structure of M. We prove that it is a b-homotopy invariant of orbifolds. If M is a manifold, t-H_q(M) coincides with the usu…
According to seminal work of Kontsevich, the unstable homology of the mapping class group of a surface can be computed via the homology of a certain lie algebra. In a recent paper, S. Morita analyzed the abelianization of this lie algebra, thereby constructing a series of candidates for unstable classes in the homology…
The only finite nonabelian simple group acting on a homology 3-sphere - necessarily non-freely - is the dodecahedral group (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group ). In the present pa…
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.