arXiv research

Introduces fundamental heaps for surfaces, linking them to cocycle invariants.

Defined by Joyce and Matveev, the fundamental quandle is a complete invariant of oriented classical knots. We consider invariants of knots defined from quotients of the fundamental quandle. In particular, we introduce the fundamental Latin Alexander quandle of a knot and consider its Gröbner basis-valued invariants, wh…

We find the complete set of fundamental invariants for systems of ordinary differential equations of order $\ge 4$ under the group of point transformations generalizing similar results for contact invariants of a single ODE and point invariants of systems of the second and the third order. It turns out that starting fr…

The rho-invariant is an invariant of odd-dimensional manifolds with finite fundamental group, and lies in the representations modulo the regular representations (after tensoring with Q). It is a fundamental invariant that occurs in classifying lens spaces, their homotopy analogues, and is intimately related to the eta-…

Equivalent categories of groups and risandles for 3-manifolds.

We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow f…

Study virtual fundamental classes of derived manifolds, proving invariant vanishes.

The study identifies two sources of invariants in 2--nondegenerate CR geometries.

Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization …

Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.

We prove that the group-homological version of the generalized Goncharov invariant of finite-volume locally rank one symmetric spaces determines their generalized Neumann-Yang invariant, which is defined using ideal fundamental cycles.

Study proves higher-order conformal forms don't exist in odd dimensions.

Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang-Mills gauge theory from twisting by the Picard group of a four manifold in the case when the fundamental group is free abelian. We then generalize it to the…

Paper finds Hempel pairs distinguishable by Turaev-Viro invariants.

Heap theory applied to framed links yields new invariants.

Introducing the notion of stabilized fundamental group for the complement of a branch curve in $CP^2$, we define effectively computable invariants of symplectic 4-manifolds that generalize those previously introduced by Moishezon and Teicher for complex projective surfaces. Moreover, we study the structure of these inv…

New method classifies spin 4-manifolds using Kervaire-Milnor invariant.

We show that on a closed Riemannian manifold with fundamental group isomorphic to $\mathbb{Z}$, other than the circle, every isometry that is homotopic to the identity possesses infinitely many invariant geodesics. This completes a recent result of the second author.

We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. The main motivation comes from $3$-braid invariants and their application.

New bounds on Euler characteristics for certain manifolds with finite groups.

The paper defines and analyzes quandles of knotoids and linkoids.

A singular point of a smooth map F: M -> N of manifolds is a point in M at which the rank of the differential dF is less than the minimum of dimensions of M and N. The classical invariant of the set S of singular points of F of a given type is defined by taking the fundamental class [\bar{S}]\in H_*(M) of the closure o…

New invariant for virtual n-links defined and studied.

We consider the classical problem of a position of n-dimensional manifold M in R^{n+2}. We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting M to R^{n+2}. In particular, we show that for any fixed quandle, quandle coloring, and shadow quan…

Complete surgery obstructions for manifolds with finite fundamental group, disproving a conjecture.

Proves surface embedding theorem for 4-manifolds with good fundamental group.

We describe a presentation for the augmented fundamental rack of a link in the lens space $L(p,1)$. Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in $L(p,1)$. In this case, the counting rack invariants also include the informatio…

New cohomology theory shows compact Lie group actions are Morita invariant.

Classifies stable diffeomorphism of spin 4-manifolds with specific fundamental groups.

The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is non-negative for all closed simply connected manifolds of dimension $\ge 5$. We extend this to show that Yamabe invariant is non-negative for a…

Paper provides a condition to distinguish links up to 4-moves.

We survey various Alexander-type invariants of plane curve complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to complex plane curves. Also included are some new computations of higher-order degrees of curves, which are invariants defined in a previou…

Paper proves Legendrian knots with same GL-rack have similar invariants.

This paper explores tradeoffs between invariance and sensitivity in adversarial examples.

Generalizes Toledo invariant to singular klt varieties, proving Milnor-Wood inequality.

New exotic 4-manifolds with even $b_2^+$ and $\mathbb{Z}/2\mathbb{Z}$ fundamental group.

Proves conditions for Willmore surfaces to have finite ends or finite total curvature.

New methods distinguish exotic 4-manifolds using Heegaard Floer homology.

Study topological 4-manifolds with specific fundamental groups.

Given an affine isometry of $\R^3$ with hyperbolic linear part, its Margulis invariant measures signed Lorentzian displacement along an invariant spacelike line. In order for a group generated by hyperbolic isometries to act properly on $\R^3$, the sign of the Margulis invariant must be constant over the group. We show…

Conformal invariants of manifolds of non-positive scalar curvature are studied in association with growth in volume and fundamental group.

Infinitesimal calculations link fundamental groups to Lie algebras.

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

The paper introduces a new method to characterize cosmological models using observer-based invariants.

We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.

Proves conjecture on deformation invariance of big fundamental groups.

We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit an universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the considered surface.