arXiv research

Hanner's theorem is a classical theorem in the theory of retracts and extensors in topological spaces, which states that a local ANE is an ANE. While Hanner's original proof of the theorem is quite simple for separable spaces, it is rather involved for the general case. We provide a proof which is not only short, but a…

Classifies 4-manifolds with elementary amenable groups and their boundaries.

The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matousek and Wagner, and of Dotterrer, Kaufman and Wagner). We also discuss a simple generalization in which the vert…

The paper shows how to rearrange arcs to form closed curves.

We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also…

Study of elementary planes in Apollonian orbifold with unusual equidistribution.

We present simple examples of finite-dimensional connected homogeneous spaces (they are actually topological manifolds) with nonhomogeneous and nonrigid factors. In particular, we give an elementary solution of an old problem in general topology concerning homogeneous spaces.

Mednykh proved that for any finite group G and any orientable surface S, there is a formula for #Hom(pi_1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius and Schur. Both of these proofs use c…

This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to students familiar with just the fundamentals of algebraic topology.

We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential Topology/Geometry, including orbibundles, differential forms, integration and (equivariant) …

In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. …

The paper explores metrics on tree moduli spaces and a new topological group.

We construct an elementary, combinatorial kind of topological quantum field theory, based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda--Kazez--Matic. This topological field theory stores inf…

Using a characterization of parabolics in reductive Lie groups due to Furstenberg, elementary properties of buildings, and some algebraic topology, we give a new proof of Tits' classification of 2-transitive Lie groups.

We present a new, more elementary proof of the Freedman-Teichner result that the geometric classification techniques (surgery, s-cobordism, and pseudoisotopy) hold for topological 4-manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reform…

New proof shows a link problem is hard without complex links.

Nielsen reduction is an algorithm which decomposes any automorphism of a free group into a product of elementary Nielsen transformations. While this may be applied to a mapping class of a surface $S_{g,1}$ with one boundary component, the resulting decomposition in general will not have a topological interpretation. In…

Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic $3$-manifold such as a hyperbolic knot complement. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic De…

This is a recreational paper showing that certain linked graphs cannot be separated. The proofs employ elementary covering space theory, an appeal to a theorem of Scharlemann (concerning the band sums of two unknots), and a Jones polynomial calculation.

Constructing VAE Latent Spaces with Prescribed Topology

Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central role played by hyperbolic geometry within three-dimensional topology. We will also point out the striking difference wi…

Using basic topology and linear algebra, we define a plethora of invariants of boundary links whose values are power series with noncommuting variables. These turn out to be useful and elementary reformulations of an invariant originally defined by M. Farber.

Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f: O -> P between smooth orbifolds O and P. We show that Sard's theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence o…

We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.

In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that thi…

In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of 3-manifolds. The main part is devoted to the construc…

Sequences of Nilsson type appear in abundance in Algebraic Geometry, Enumerative Combinatorics, Mathematical Physics and Quantum Topology. We give an elementary introduction on this subject, including the definition of sequences of Nilsson type and the uniqueness, existence, and effective computation of their asymptoti…

We give a "soft" proof of Alberti's Luzin-type theorem in [1] (G. Alberti, A Lusintype theorem for gradients, J. Funct. Anal. 100 (1991)), using elementary geometric measure theory and topology. Applications to the $C^2$-rectifiability problem are also discussed.

Let $(M,g)$ be a compact conformally flat manifold of dimension $n\geq4$ with positive scalar curvature. According to a positive mass theorem by Schoen and Yau, the constant term in the development of the Green function of the conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the sphere. On sp…

Applying the classical Serre-Swan theorem, as this is extended to topological (non-normed) algebras, one attains a classification of elementary particles via their spin-structure. In this context, our argument is virtually based on a ``correspondence principle'' of S. A. Selesnick, formulated herewith in a sheaf-theore…

Surveying topological complexity of graph configurations, unifying traditional and modern approaches.

Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these invariants. We discuss the the relationship of our work to stabilization problems in cl…

We prove that there exists no branched cover from the torus to the sphere with degree 3h and 3 branching points in the target with local degrees (3,...,3), (3,...,3), (4,2,3,...,3) at their preimages. The result was already established by Izmestiev, Kusner, Rote, Springborn, and Sullivan, using geometric techniques, an…

We prove that there are no pseudoholomorphic theories of anything other than curves, even if one allows more general spaces than almost complex manifolds. The proof is elementary, except for theories of pseudoholomorphic hypersurfaces, where topological techniques are needed. Surprisingly, hypersurface theories exist `…

We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and combinatorics. We define a mod2-valued self-intersection invariant (i.e. the van Kampe…

New class of maps restricts manifolds strongly in algebraic topology.

The study compares lamplighter graphs up to quasi-isometry using coarse topology.

The goal of this paper is to describe and clarify as much as possible the 3-dimensional topology underlying the Helmholtz cuts method, which occurs in a wide theoretic and applied literature about Electromagnetism, Fluid dynamics and Elasticity on domains of the ordinary space. We consider two classes of bounded domain…

This article presents a new and more elementary proof of the main Seiberg-Witten-based obstruction to the existence of Einstein metrics on smooth compact 4-manifolds. It also introduces a new smooth manifold invariant which conveniently encapsulates those aspects of Seiberg-Witten theory most relevant to the study of R…

This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the calculus of continued fractions. We characterize the class of strongly invertib…

Proves constraints on groups extending Möbius transformations on spheres.

New method realizes planar graphs as Reeb graphs of algebraic functions.

If X is a proper CAT(-1)-space and $Γ$ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y=X/$Γ$ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary $\partial X$ and the non-wanderin…

We prove that a dense subgroup of $\mathrm{Homeo}_{+}(I)$ is not elementary amenable. We also show that the topological group $\mathrm{Homeo}_{+}(I)$ does not satisfy the Stability of the Generators Property, moreover, any finitely generated subgroup of $\mathrm{Homeo}_{+}(I)$ admits a faithful discrete representation …

This is an informal set of lecture notes on moduli spaces of curves based on a set of lectures given at the ICTP last summer. It begins at an elementary level and discusses the genus 1 case in detail. The notes then give an informal treatement of Teichmuller space and higher genus moduli spaces. The point of view is ge…

Paper proves curves can be smoothed to reduce self-intersection by exactly 1.

We give a general treatment of the somewhat unfamiliar operation on manifolds called Connected Sum at Infinity, or CSI for short. A driving ambition has been to make the geometry behind the well definition and basic properties of CSI as clear and elementary as possible. CSI then yields a very natural and elementary pro…

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Fi…