arXiv research

Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.

For all left-invariant Riemannian metrics on three-dimensional unimodular Lie groups, there exist particular left-invariant orthonormal frames, so-called Milnor frames. In this paper, for any left-invariant Riemannian metrics on any Lie groups, we give a procedure to obtain an analogous of Milnor frames, in the sense t…

We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point o…

The paper proves a conjecture linking two metrics on manifold cohomology.

Higher-dimensional Milnor frames are characterized and contrasted with 3D Heisenberg and 4D nilpotent Lie algebras.

Let $M$ be a Milnor sphere or, more generally, the total space of a linear $S^3$-bundle over $S^4$ with $H^4(M;\mathbb{Q})=0$. We show that the moduli space of metrics of nonnegative sectional curvature on $M$ has infinitely many path components. The same holds true for the moduli space of metrics of positive Ricci cur…

New findings on isospectral tori and harmonic maps between flat tori.

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 $O_n(\mathbb{R})$ invariant…

Simplified Milnor-Schwarz lemma for geometric group theory.

Geometric framework for Milnor classifying spaces in diffeological spaces.

The paper finds solutions for specific curvature conditions on 5D Lie groups.

Study magnetic curvature on Lie groups, extending Milnor's work.

Paper disproves Milnor's conjecture about manifold fundamental groups.

Several proofs of Fáry--Milnor theorem are presented.

Study Stein and Milnor fillings of links from surface singularities.

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.

Combines combinatorial method to extend Milnor invariants to welded links.

Given a clover link, we construct a bottom tangle by using a disk/band surface of the clover link. Since the Milnor number is already defined for a bottom tangle, we define the Milnor number for the clover link to be the Milnor number for the bottom tangle and show that for a clover link, if Milnor numbers of length k …

We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that this invariant can distinguish some links for which the ordinary Milnor invaria…

Extends Milnor's invariants to knots and links in 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…

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…

Explains Milnor invariants for links and surfaces.

Study the topology of Milnor boundaries for real analytic map germs.

We reconfigure the Milnor invariant of links in terms of central group extensions and unipotent Magnus embeddings. We also develop a diagrammatic computation of the invariant and compute the first non-vanishing invariants of the Milnor link and of several other links. Moreover, we refine the original Milnor invariants …

Extends Milnor's criterion to biharmonic functions.

Paper solves Dirichlet problem at infinity for Riemannian cones.

Extends Milnor's invariants to 3-manifolds, solving an open problem.

We apply the lifting theorem of Searle and the second author to put metrics of almost nonnegative curvature on the fake RP^{6}s of Hirsch and Milnor and on the analogous fake RP^{14}s.

We calculate the eta-invariant for the odd signature operator relative to a specific submersion metric on the Milnor fibration of a quasihomogeneous hypersurface singularity using certain global boundary conditions in terms of the data of its fibre intersection form, monodromy and variation mapping resp. the monomial d…

New combinatorial model for Milnor fibration using oriented matroids.

Defines Milnor number for foliations and shows its topological invariance.

Paper introduces simplified formulas for Milnor's triple linking number.

New results on geometry of area-preserving diffeomorphisms using braids.

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,…

In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. A…

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.

Proves non-solvability of concordance groups using Milnor invariants.

Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.

Study volume growth in Milnor fibers using real Lagrangians.

For an $n$-component link $L$, the Milnor's isotopy invariant is defined for each multi-index $I=i_1i_2...i_m (i_j\in\n)$. Here $m$ is called the length. Let $r(I)$ denote the maximam number of times that any index appears. It is known that Milnor invariants with $r=1$ are link-homotopy invariant. N. Habegger and X. S.…

New method determines arrangement combinatorics from Milnor fiber boundary.

New method normalizes Milnor fibrations for real analytic maps.

Floer homology linked to Milnor fibers for certain singularities.

Formula calculates homology groups of Milnor fibres for real hypersurface singularities.

In this paper we extend the characterization of trivial map-germs for the real Milnor fibrations started by Church and Lamotke. Our main result cover all cases on the three dimensional real Milnor fibers.

In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles (``The continuity conjecture''), and furthermore, the limit at a bou…