We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the repre…
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48 results for “Milnor lattices”
Paper calculates upper bounds for Betti numbers of Milnor fibers of hyperplane arrangements.
problem Determining Betti numbers of Milnor fibers for hyperplane arrangements.
method Combinatorial upper bounds using intersection lattice.
result Improved bounds for characteristic polynomial of Milnor fibers.
Study of Fukaya-Seidel categories for surface Lefschetz fibrations.
problem Understanding Fukaya-Seidel categories for surface Lefschetz fibrations.
method Proving exact Lefschetz fibrations and deriving Fukaya-Seidel categories.
result Derived Fukaya-Seidel categories are independent of symplectic structure and contain more information than Milnor lattices.
We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodes…
New volume invariant for cocycles of hyperbolic lattices, proving rigidity results.
problem Volume calculation for cocycles of hyperbolic lattices.
method Introducing a new volume invariant and proving Milnor-Wood type inequalities.
result Characterization of maximal cocycles and proof of rigidity results.
New findings on isospectral tori and harmonic maps between flat tori.
problem Determining if isospectral tori are isometric using harmonic maps.
method Examined harmonic maps between flat tori, focusing on Milnor's isospectral tori.
result Milnor's isospectral tori cannot be distinguished by harmonic maps from lower-dimensional tori, but can be distinguished by higher-dimensional ones.
We encode the variation structure of a quasihomogeneous polynomial with an isolated singularity as introduced by Nemethi in a set of spectral flows of the signature operator on the Milnor bundle by varying global elliptic boundary conditions in a specific way using the quasihomogeneous circle action on the Brieskorn la…
We consider the problem of existence of representations of topological groupoids on a principal bundle and the classification of such representations up to gauge transformation. Such representations naturally occur in various contexts such as gauge theory, lattice gauge fields, equivariant bundles, etc. In the course o…
We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as wel…
Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groupsmath.DG
Maximal representations in exceptional Hermitian Lie groups classified for complex hyperbolic lattices.
problem Classifying maximal representations of complex hyperbolic lattices in exceptional Hermitian Lie groups.
method Unified approach using cominuscule representation of complexified groups, focusing on exceptional cases.
result Complete description of maximal representations for and .
We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: …
Planar multilinks prove rational singularities in surface geometry.
problem Characterizing surface singularities using planar multilinks.
method Combining topological and combinatorial approaches, including Min--Roy--Wang's work.
result Planar multilinks imply rational singularities and sandwiched singularities.
Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian typemath.DG
Let G be either SU(p,2) with p>=2, Sp(2,R) or SO(p,2) with p>=3. The symmetric spaces associated to these G's are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs b…
Study symplectic fillings of lens spaces, focusing on virtually overtwisted contact structures.
problem Classify symplectic fillings of virtually overtwisted contact structures on lens spaces.
method Use curve configurations on surfaces, algebraic properties of integer lattices, geometric slicing of solid tori, and connections to algebraic geometry.
result Find necessary conditions for Stein fillings to be Milnor fibers of hypersurface singularities.
The paper classifies clover links using Milnor invariants and edge-homotopy.
problem Classifying clover links using Milnor invariants and edge-homotopy.
method Constructing a bottom tangle from a clover link, defining Milnor numbers for bottom tangles, and proving equivalence based on vanishing Milnor numbers.
result Two clover links are equivalent up to edge-homotopy and -equivalence if their Milnor numbers of length 2k+1 or less are equal.
Study Milnor invariants for covering links to distinguish certain links.
problem Distinguish links using Milnor invariants for covering links.
method Generalize Hartley and Murasugi's covering linkage invariants and use cobordism invariants.
result First non-vanishing Milnor invariants of a Brunnian link modulo 2 equals a sum of linking numbers of covering links.
Study Stein and Milnor fillings of links from surface singularities.
problem Comparing Stein and Milnor fillings of links from surface singularities.
method Analyzing the topology and obstructions of Stein fillings and Milnor fillings.
result Milnor fillings have bounded topology, while Stein fillings can be more varied.
Several proofs of Fáry--Milnor theorem are presented.
problem Fáry--Milnor theorem
method Sketches several proofs
result Proofs of Fáry--Milnor theorem
Combines combinatorial method to extend Milnor invariants to welded links.
problem Extending Milnor invariants to welded links.
method Combinatorial approach.
result Invariance of extended Milnor invariants for welded links.
In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere.
Reconfigures Milnor invariants using unipotent Magnus embeddings.
problem Computing and understanding Milnor invariants of links.
method Central group extensions and unipotent Magnus embeddings.
result First non-vanishing invariants and refinements of higher degree invariants computed.
Study new invariants for string links using Milnor and Morita-Milnor homomorphisms.
problem Developing invariants for string links.
method Define and study total Milnor and infinitesimal Morita-Milnor homomorphisms.
result Introduced new invariants for string links.
Higher-dimensional Milnor frames are characterized and contrasted with 3D Heisenberg and 4D nilpotent Lie algebras.
problem Characterizing higher-dimensional Milnor frames and their properties.
method Definition and classification of higher-dimensional Milnor frames and their relationship to known Lie algebras.
result Higher-dimensional Milnor frames are isomorphic to direct sums of 3D Heisenberg and 4D nilpotent Lie algebras and an abelian Lie algebra.
Defines linking number and Milnor invariants, their properties.
problem None explicitly stated; definitions and properties are discussed.
method Definitions and properties of linking number and Milnor invariants are discussed.
result Definitions and properties of linking number and Milnor invariants are established.
Extends Milnor's invariants to knots and links in 3-manifolds.
problem Detecting higher-order linking phenomena in 3-manifolds.
method Study lower central quotients of link groups in 3-manifolds.
result Unified and generalized Milnor's invariants for 3-manifolds.
We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. suppor…
The study finds conditions for a special vector field in singularities.
problem Existence of Milnor vector field for new singularities.
method Introducing sufficient conditions for the existence of the Milnor vector field.
result Conditions guarantee the existence of the Milnor vector field for new classes of singularities.
New stability result for Milnor fibers of pure braid groups.
problem Understanding the stable homology of Milnor fibers.
method Representation stability in the context of Milnor fibers of pure braid groups.
result Computed stable integral homology of Milnor fibers.
Explains Milnor invariants for links and surfaces.
problem Milnor invariants of links and surfaces.
method Expository viewpoint of research.
result Translation and explanation of Milnor invariants.
Study the topology of Milnor boundaries for real analytic map germs.
problem Topology of Milnor boundaries for real analytic map germs.
method Prove Milnor boundaries are double of Milnor tubes and use generalized open-book decompositions.
result Prove Euler characteristic formulae connecting Milnor boundaries and links.
Extends Milnor's criterion to biharmonic functions.
problem Deciding surface type for biharmonic functions.
method Generalizes Milnor's criterion to biharmonic functions.
result Characterizes whether a surface is hyperbolic or parabolic for biharmonic functions.
Extends Milnor's invariants to 3-manifolds, solving an open problem.
problem Extract transfinite Milnor invariants for 3-manifolds.
method Develops a theory of transfinite invariants for 3-manifold groups.
result Realizes nontrivial values of transfinite Milnor invariants.
Fox-Milnor theorem extended to knots in thickened surfaces.
problem Extending classical knot theory to knots in thickened surfaces.
method Using Milnor torsion to prove a Fox-Milnor theorem for knots in a thickened surface.
result A Fox-Milnor theorem for concordant knots in a thickened surface.
New combinatorial model for Milnor fibration using oriented matroids.
problem Understanding the homotopy type of Milnor fibers of complexified real arrangements.
method Introducing a poset quasi-fibration based on a subdivision of the Salvetti complex and an oriented matroid.
result Homotopy type of Milnor fiber depends only on the combinatorial structure of the oriented matroid.
Defines Milnor number for foliations and shows its topological invariance.
problem Defining and proving invariance of Milnor number for non-isolated singularities of holomorphic foliations.
method Defining Milnor number as intersection number of sections; proving invariance via topological equivalences.
result Milnor number is invariant under topological equivalences.
Proof of Milnor conjecture in 3 dimensions using limit spaces.
problem Milnor conjecture in dimension 3
method Cheeger-Colding theory on limit spaces of manifolds with bounded Ricci curvature
result Proof of Milnor conjecture in dimension 3
Paper introduces simplified formulas for Milnor's triple linking number.
problem Computational difficulty in calculating Jones polynomial for topological polymers.
method Developed Gauss diagram formulas for Milnor's Vassiliev invariants.
result Introduced non-torsion valued Milnor's triple linking number.
Milnor fibrations have been studied since 1960's. In this paper, we study singular points of differentiable maps, called Milnor fibration product maps, obtained by several Milnor fibrations. We give a characterization of singular points of such product maps, and for the case of certain weighted homogeneous polynomials,…
Investigates Milnor invariants of clover links, defining certain invariants under specific conditions.
problem Investigating indeterminacy in Milnor invariants of links.
method Introduced clover links to study Milnor invariants, showing well-definedness and non-well-definedness conditions.
result Well-defined Milnor numbers for certain lengths of sequences in clover links, with explicit ranges for specific sequences.
We give formulas expressing Milnor invariants of an n-component link L in the 3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariant \barμ_J(L) vanishes for any sequence J with length at most k, then any Milnor \barμ-invariant \barμ_I(L) with length between 3 and 2k+1 can be represented as a c…
This paper characterizes Milnor invariants using diagrammatic methods.
problem Classifying links and string links based on their Milnor invariants.
method Diagrammatical characterization using welded knot theory and arrow calculus.
result Conditions for equality of Milnor invariants and triviality of certain lengths.
Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.
problem Computing the first homology group of the Milnor fiber boundary for generic hyperplane arrangements.
method Analyzing the Milnor fiber boundary for hyperplane arrangements in C^3.
result Affirmative answer to the conjecture of Suciu and example of arrangements with non-trivial torsion.
For an -component link , the Milnor's isotopy invariant is defined for each multi-index $I=i_1i_2...i_m (i_j\in\n)$. Here is called the length. Let denote the maximam number of times that any index appears. It is known that Milnor invariants with are link-homotopy invariant. N. Habegger and X. S.…
Study volume growth in Milnor fibers using real Lagrangians.
problem Volume growth in Milnor fibers of Brieskorn polynomials.
method Investigate real Lagrangians, use Smith inequality for involutions in wrapped Floer homology.
result Uniform lower bound of volume growth for a class of Brieskorn polynomials.
Proves non-solvability of concordance groups using Milnor invariants.
problem Non-solvability of concordance groups of 2-string links and strongly invertible knots.
method Using Milnor invariants to prove non-solvability.
result Proves non-solvability of and equivariant concordance groups of strongly invertible knots.
New method normalizes Milnor fibrations for real analytic maps.
problem Existence of normalized Milnor fibrations for real analytic maps.
method Introducing a homeomorphism to transform a non-normalized Milnor fibration into a normalized one.
result Normalized map defines a smooth locally trivial fibration on the sphere.
New method determines arrangement combinatorics from Milnor fiber boundary.
problem Determining arrangement combinatorics from Milnor fiber boundary.
method Explicit method using plumbing graph in normal form.
result Milnor fiber boundary determines arrangement combinatorics.
Floer homology linked to Milnor fibers for certain singularities.
problem Understanding Floer homology of singularities using Milnor fibers.
method Combining Floer theory with Milnor fibers, using combinatorial formulas.
result Equality of Casson invariants in Donaldson and Seiberg-Witten theories.