We define a diffeology on Milnor's classifying space and prove existence of connections.
problem Classifying spaces and connections in diffeology.
method Definition of diffeology on Milnor's classifying space and proof of connection existence.
result Existence of diffeological connections on principal bundles.
Geometric framework for Milnor classifying spaces in diffeological spaces.
problem Milnor classifying spaces in diffeological spaces.
method Developed spherical and projective models with natural diffeological structures, constructed Riemannian metrics, defined differential forms, and introduced Clifford structures.
result Established a coherent geometric setting combining classifying spaces, diffeology, and higher geometric structures.
New spherical Milnor spaces for diffeological groups with geometric and topological properties.
problem Understanding higher topological structures in diffeological spaces.
method Spherical Milnor construction based on quadratic normalization.
result Provides a natural setting for studying principal bundles with Z2-twists and higher cohomology. Smooth classifying spaces for groups defined using diffeological spaces.
problem Classifying smooth principal bundles for a smooth group G. method Developed the theory of smooth principal bundles using diffeological spaces, defining D-numerable bundles and proving classification results. result Smooth structures on Milnor's spaces EG and BG classify all D-numerable principal bundles over any diffeological space. Study of concordances between links in 4-space, defining new invariants.
problem Investigate the set of all embedded concordances between two fixed links in 4-space.
method Define Milnor-type invariants of the set of concordances, modulo indeterminacy.
result For slice links, these invariants classify the set of concordances up to link-homotopy.
New method classifies spin 4-manifolds using Kervaire-Milnor invariant.
problem Classifying spin 4-manifolds up to stabilisation.
method Using Kervaire-Milnor invariant to compute Arf invariants and classify.
result New stable classification of spin 4-manifolds with 2-dimensional fundamental groups.
Constructs a universal Chern-Weil map for infinite dimensional Lie groups.
problem Universal Chern-Weil map for infinite dimensional Lie groups.
method Introduces smooth simplicial sets and constructs a new classifying space as a smooth Kan complex.
result Verifies a conjecture of Reznikov for compactly generated Hamiltonian symplectomorphisms.
Classifies welded string links up to specific moves.
problem Classifying welded string links up to moves.
method Using Milnor invariants, classify welded string links up to 2n-move and Vn-move. result Classifies welded string links up to 2n-move and Vn-move. Simplified Milnor-Schwarz lemma for geometric group theory.
problem Conditions for orbit maps to be quasi-isometries.
method Succinct treatment and applications to non-Archimedean groups.
result Sharpened results on mapping class groups and quasi-isometry classification.
Paper classifies 3D locally symmetric Riemannian Lie groups using Milnor bases.
problem Classifying 3D locally symmetric Riemannian Lie groups.
method Used Milnor bases to solve polynomial equations of structure constants.
result Identified E0(2) as the only non-symmetric locally symmetric Lie group. Classifies knotted annuli in 4-space up to a specific equivalence.
problem Classifying knotted annuli in 4-space up to a specific equivalence.
method Uses Milnor invariants and a 4-dimensional version of them, along with a Roseman-type result for immersed surfaces.
result Classifies 2-string-links up to link-homotopy.
New invariants for handlebody-links using Milnor's invariants.
problem Classifying HL-homotopy classes of handlebody-links.
method Constructing invariants using Milnor's link-homotopy invariants.
result A bijection between HL-homotopy classes and tensor product space.
Link concordance and Whitney towers linked to Milnor invariants.
problem Link concordance and Whitney towers classification.
method Clasper surgeries, Whitney towers, and Milnor invariants.
result Link concordance and Whitney towers classified in terms of Milnor invariants.
Study of surface singularities and their deformations.
problem Characterize and classify quotient surface singularities and their deformations.
method Computation of resolutions, dual graphs, dimensions, Milnor numbers, and Milnor fibers.
result 6 pairs of minimal symplectic fillings are shown to be diffeomorphic.
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
Survey uses Milnor fibrations to classify first integrals of differential systems.
problem Classifying first integrals of differential systems using geometric-topological methods.
method Utilizing Milnor fibrations and connections with harmonic morphisms to provide topological and geometric descriptions.
result Geometric-topological classifications of first integrals for both isolated and non-isolated singularities.
Generalizes Milnor's axiomatic homology theorems to Polish spaces.
problem Characterize (co)homology of Polish spaces.
method Eilenberg-Steenrod axioms, Milnor's map excision axiom, and a generalized additivity axiom.
result Unique characterization of (co)homology of Polish spaces.
Study shows infinitely many nonnegatively curved metric components on Milnor spheres.
problem Understanding the moduli space of nonnegatively curved metrics on Milnor spheres.
method Analyzes the moduli space of metrics with nonnegative sectional curvature and positive Ricci curvature on Milnor spheres.
result The moduli space of nonnegatively curved metrics on Milnor spheres has infinitely many path components.
Study configuration space integrals for Milnor invariants of string links and trivalent trees.
problem Understanding Milnor invariants of string links using combinatorial methods.
method Combinatorial analysis of trivalent homotopy link diagrams and configuration space integrals.
result Established a correspondence between Milnor invariants and linear combinations of trivalent trees.
Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.
problem Applying Milnor-Schwarz lemma to inverse monoids.
method Two proofs provided: elementary and using Vietoris-Rips complex.
result Generalization of Milnor-Schwarz lemma to inverse monoids.
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…
To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its charact…
Extends Milnor invariants to surface-links using cut-diagrams.
problem Tackles invariants for surface-links in 4-space.
method Introduces cut-diagrams and groups associated to them.
result Yields concordance and link-homotopy invariants for surface-links.
New theory classifies knotted spheres in 4D space.
problem Classifying knotted punctured spheres in 4D space.
method Diagrammatic theory of welded graphs, Tube map extension, Milnor invariants.
result Complete link-homotopy classification of knotted punctured spheres.
Discrete groups act properly on 3-space, solving Milnor's question.
problem Proper actions of discrete groups on 3-space.
method Historical review and recent progress.
result Existence of proper affine actions of free groups on 3-space.
String links classified up to 2n-moves and link-homotopy.
problem Classifying string links up to specific moves and link-homotopy.
method Using Milnor link-homotopy invariants and 2n-moves. result Equivalence classes of string links form a finite group.
The study explores conditions for realizing lens spaces as boundaries of Milnor fibers of hypersurface singularities.
problem Realizing other lens spaces as boundaries of Milnor fibers of hypersurface singularities.
method Examining necessary conditions for the realization of (L(p,q),ξ) as boundaries of Milnor fibers. result Obtained a series of necessary conditions for realizing other lens spaces as boundaries of Milnor fibers.
Formula for Milnor triple linking number in link diagrams with multiple crossings.
problem Calculating Milnor triple linking number for complex link diagrams.
method Polyak-Viro type formula with explicit computation of configuration space integral.
result Formula applicable to diagrams with triple or more crossings.
We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.
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 Milnor fiber conjecture is proven for splice type singularities.
problem Proving the Milnor fiber conjecture for a specific class of singularities.
method Combining techniques from tropical geometry, log geometry, and rounding of logarithmic spaces.
result The Milnor fiber conjecture is proven for splice type singularities.
Paper solves Dirichlet problem at infinity for Riemannian cones.
problem Solvability of Dirichlet problem at infinity in Riemannian cones.
method Separation of variables and comparison arguments for ODE's.
result Sufficient condition for solvability related to Milnor's classification.
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 (h−1f)/∣∣h−1f∣∣ defines a smooth locally trivial fibration on the sphere. 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…
Classifies links in 3-sphere using Whitney towers in rational homology 4-ball.
problem Classifying links in 3-sphere using geometric methods.
method Complete classifications of links using Whitney towers in rational homology 4-ball.
result Geometric characterization of Milnor invariants and higher order Arf invariants.
In this paper we propose a new treatment about infinite dimensional manifolds, using the language of category and functor. Our definition of infinite dimensional manifolds is a natural generalization of finite dimensional manifolds in the sense that de Rham cohomology and singular cohomology can be naturally defined an…
Study on mod 2 Betti numbers of complex hyperplane arrangements and Milnor fiber homology.
problem Determining mod 2 Betti numbers of complex hyperplane arrangement complements.
method Combining the mod 2 Aomoto complex and the transfer long exact sequence.
result First homology of Milnor fiber has non-trivial 2-torsion for the icosidodecahedral arrangement.
Generalizes classifying spaces for topological groups with torsion.
problem Classifying spaces for topological group actions with non-Hausdorff spaces.
method Generalizes Milnor's, Gelfand-Fuks', and Segal's theorems to non-Hausdorff spaces.
result Existence and uniqueness theorems for G-spaces over metric spaces. The space of positive Lagrangians in an almost Calabi-Yau manifold is an open set in the space of all Lagrangian submanifolds. A Hamiltonian isotopy class of positive Lagrangians admits a natural Riemannian metric Υ, which gives rise to a notion of geodesics. We study geodesics of positive On(R) invariant…
Real algebraic structures help classify overtwisted contact 3-spheres.
problem Classifying overtwisted contact structures on 3-spheres.
method Using real algebraic functions and open book decompositions.
result Most overtwisted contact structures are real algebraic.
We show that Orr's space's third homotopy group is infinitely generated.
problem Non-triviality of Orr's space's third homotopy group.
method Analyzing the structure of the third homotopy group of Orr's space.
result The third homotopy group is infinitely generated.
Study plane curve singularities to determine vanishing cycles and monodromy groups.
problem Understanding vanishing cycles and monodromy groups for plane curve singularities.
method Intrinsic description of geometric monodromy group, easy criterion for vanishing cycles, canonical framing.
result Monodromy groups are injective for singularities with Milnor fiber of genus at least 7.
This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new qua…
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…
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.
There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 loca…
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