Study knotted surfaces in simply-connected 4-manifolds with trivial boundary.
problem Locally flat, compact, genus g surfaces with boundary a fixed knot in simply-connected 4-manifolds with boundary the 3-sphere.
method Analyzes homology and isotopy of surfaces with trivial fundamental group complements.
result Homologous surfaces are topologically ambiently isotopic rel. boundary.
Study 2-complexes' homology properties and torsion growth.
problem Quantitative connections between 1-cycle filling inequalities and homology complexities.
method Geometric lower bounds on first homology of finite covers.
result Geometric lower bound on first homology size of finite covers.
New framework for quantum invariants of 3-manifolds using homology.
problem Quantum invariants of 3-manifolds.
method Homological construction of mapping class group representations extended to cobordisms.
result Construction of a (2+1)-dimensional TQFT assigning twisted homology to surfaces. The paper extends Johnson's result on Torelli group homology.
problem Understanding the homology of the Torelli group and its subgroups.
method Analyzing the pushforward homomorphism on higher homology groups induced by Dehn twists.
result The pushforward homomorphism is injective for certain subgroups of higher homology groups.
Extend Kauffman bracket skein module to homology theory using Heegaard splittings
problem Extend Kauffman bracket skein module to homology theory
method Combinatorial approach using Heegaard splittings
result Homology theory depends on Heegaard splittings
Rational homology ribbon cobordism defines a partial order on 3-manifolds.
problem Classifying 3-manifolds based on their homology
method Proving a partial order on homeomorphism classes of 3-manifolds
result Closed, connected, oriented 3-manifolds are partially ordered by rational homology ribbon cobordism.
Torelli groups' homology is finitely generated in stable range.
problem Whether the homology groups of Torelli subgroups are finitely generated in stable range.
method Using unipotency condition and Tavgen's theorem.
result Homology groups are finitely generated in stable range.
Computes second homology groups of orbifold groups, proving profinite rigidity and Grothendieck pairs.
problem Computing the second homology group of orbifold groups.
method Effective method for computing H2 in specific cases of orbifold groups. result Explicit computations of second homology groups, including H2(ΓO,Z)≅Z/2Z. Two codimension-one submanifolds are cobordant if they have the same homology class.
problem Understanding cobordism equivalence of codimension-one submanifolds.
method Using handle decompositions and surgeries, we relate cobordisms to homology classes and prove equivalence for Seifert surfaces.
result Seifert surfaces are related by tube attachments and removals, providing a conceptual proof.
Homology of abelian differentials stabilizes with more zeros.
problem Understanding the homology of abelian differentials with many simple zeros.
method Developed an h-principle for these strata, valid in a range of homological degrees.
result Homology stabilizes in a range where the number of simple zeros is large.
The study examines conditions for Haken 3-manifolds and their fundamental groups.
problem Conditions for Haken 3-manifolds and their fundamental groups.
method Conditions and finite-sheeted covers of integral homology three-spheres.
result Lower bound of 4 for first Betti number of certain covers.
Study of Dehn twists on a disc with 3 points, solving conjugacy problem.
problem Solving conjugacy problem for Dehn twists on a disc with 3 marked points.
method Explicit description of orbits of Dehn twists on the Dynnikov plane, relating to homology dynamics.
result Explicit solution to conjugacy problem for Dehn twists, presenting an untwisting algorithm.
Researchers debunk a generalized Property R conjecture for 2-component links.
problem The generalized Property R conjecture for 2-component links in S3. method Disproved a further generalization of the GPRC using specific counterexamples.
result 2-component framed links in S3 surging to a connected sum of homology S1imesS2's are not handleslide equivalent to a split link. The paper improves bounds on how many squares can fit in a rectangle and still have stable homology.
problem Homological stability in the space direction of square configurations.
method Analyzing the ordered configuration space of squares in a rectangle.
result Most rectangles can be almost entirely filled with squares and still have stable homology.
The paper studies Goeritz equivalence in lens spaces, describing actions and obstructions.
problem Understanding Goeritz equivalence in lens spaces L(p,1) for genus two Heegaard splittings. method Describes the Goeritz group action on the homology of the Heegaard surface and provides obstructions.
result Homology and homotopy obstructions for Goeritz equivalence of curves in the Heegaard surface.
Linear bound on Betti numbers of negatively curved orbifolds.
problem Bounding Betti numbers of negatively curved orbifolds.
method Quantitative bound on the homology of spherical quotients.
result Linear growth of Betti numbers with volume, arbitrary field coefficients.
New knot homology invariant grows exponentially with color.
problem Constructing and understanding colored torus knot homology.
method Invariant construction and recursive formula for reduced HOMFLY homology.
result Doubly-graded invariant of positive torus knots grows exponentially in color.
Study rational homology of moduli space via Morse functions, proving stability phenomena.
problem Homology of Deligne--Mumford compactification of moduli space of stable curves.
method Using a family of Morse functions, specifically the sys_T functions, and exploiting geometric and Morse properties.
result Homology of Deligne--Mumford compactification is supported entirely on the boundary in low degrees, and rational homology is finite generated and stable across all genera and marked points.
Study on torsion in homology of Torelli group for surfaces.
problem Torsion in homology of Torelli group for surfaces.
method Analysis of abelian cycles and Dehn twists.
result Subgroup of homology generated by abelian cycles is a finite-dimensional Z/2Z-vector space for k=2 and g≥4. New framework captures non-autonomous IFS limit set topology.
problem Understanding topological properties of non-autonomous IFS limit sets.
method Homological framework applied to fractal square.
result Provides insights into fractal topology, answering Mandelbrot's percolation problem.
Minimal complexes for two-strand braids defined directly.
problem Determining minimal complexes for braids.
method Explicit formulas derived from educated guesswork and reverse engineering.
result Construction of minimal complexes homotopy equivalent to Rickard complexes.
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.
New proof for 3-manifolds with specific homology groups.
problem Characterizing 3-manifolds based on their homology groups.
method Holonomy perturbation techniques in instanton Floer homology.
result Irreducible representations exist unless the manifold is a specific type.
Two obstructions found to keep a foliation transverse.
problem Ensuring a foliation remains transverse to a submanifold.
method Two obstructions: determinant-line and twisted homology/degree.
result Obstructions can be used to determine when a foliation is not transverse.
Study initiates homology theory for Bol-Moufang quasigroups.
problem Developing a homology theory for a specific type of quasigroups.
method Using extensions by affine quasigroups, define boundary operations and compute homology groups.
result Computed second homology groups for examples, speculating on relation to functorial homology.
New algorithm calculates S-invariants for links efficiently.
problem Cannot be done by standard spectral sequence methods.
method Developed an algorithm for S-invariants of links. result Demonstrated efficiency and applicability to sl(3)-link homology.
Study equivariant isotopy in higher dimensions, finding exceptions.
problem Under what conditions are equivariant isotopic diffeomorphisms also equivariantly isotopic?
method Construct an invariant valued in the homology of an infinite cover to distinguish isotopic from equivariantly isotopic diffeomorphisms.
result Many equivariant diffeomorphisms are isotopic but not equivariantly isotopic in higher dimensions.
New Morse theory for path homology with coefficients.
problem Defining operations on path homology with differential graded coefficients.
method Using tools from Morse theory and string topology.
result Morse-theoretic description of a product on path homology.
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.