arXiv research

Unified framework recovers and improves classical Brouwer homeomorphism results.

We prove three theorems giving fixed points for orientation preserving homeomorphisms of the plane following forgotten results of Brouwer.

New tiles in higher dimensions are shown to be homeomorphic to balls.

Recent proofs of classical theorems in polynomial algebra and functional analysis are discussed, which use tools from the topology of real manifolds. Simpler proofs were discovered in the new century, of the Hilbert Nullstellensatz, and the Gelfand-Mazur Theorem. We give a related proof that an irreducible real polynom…

Study Kazdan-Warner equations on graphs using Brouwer degree theory.

M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and som…

We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.

We give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set (the set of "walls") which splits the plane into maximal translation areas and irreducible areas. We then focus on Brouwer mapping classes relatively to four orbits and describe them explicitly by adding to Handel's diagr…

The proof of Brouwer's fixed-point theorem based on Sperner's lemma is often presented as an elementary combinatorial alternative to advanced proofs based on algebraic topology. The goal of this note is to show that: (i) the combinatorial proof of Sperner's Lemma can be considered as a cochain-level version, written in…

Extends Brouwer fixed point theorem with new conditions for continuous maps.

Paper extends Brouwer Fixed Point Theorem with amiable and almost amiable fixed sets.

We investigate the fixed point property of the group actions on a coarse space and its Higson corona. We deduce the coarse version of Brouwer's fixed point theorem.

New proof shows no flat embedding for Petersen family graphs.

We give an answer to the question given by T.Y.Kong in his article "Can 3-D Digital Topology be Based on Axiomatically Defined Digital Spaces?" In this article he asks the question, if so called "good pairs" of neighborhood relations can be found on the set Z^n such that the existence of digital manifolds of dimension …

A new technique for the study of geodesic connectedness in a class of Lorentzian manifolds is introduced. It is based on arguments of Brouwer's topological degree for the solution of functional equations. It is shown to be very useful for multiwarped spacetimes, which include different types of relativistic spacetimes.

This is a mathematical commentary on Teichm{ü}ller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl{ä}chen'' (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces). This paper is among the last (and may be the last one) that Teich…

A new method calculates the minimum volume swept by a sphere's homotopy in 3D space.

The Brouwer fixed point theorem says that any continuous function from disc to itself has a fixed point. By using simple geometrical technique we have generalized the result in manifold and proved that any continuous function on the boundary of a bounded convex domain of a $2$-dimensional Riemannian manifold with a pol…

Walraswap solves batch auction pricing by finding optimal AMM swaps.

Paper proves a new criterion for time-like geodesics in flat spacetimes.

Surgery triangles are an important computational tool in Floer homology. Given a connected oriented surface $Σ$, we consider the abelian group $K(Σ)$ generated by bordered 3-manifolds with boundary $Σ$, modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show that $K(Σ)$ is a f…

We answer the question of when a new point can be added in a continuous way to configurations of $n$ distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered configuration of $n$ points if and only if $n \neq 1$. On the other hand, when the points are not ordered and the d…

We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that the intersection of A and B is H and the union is G and such that the complementary graphs A,B are both d-balls. The graph theoretic de…

Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.

Study on homeomorphism groups of manifolds using set theory.

New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.

Uniform interpretation of group theory in manifold homeomorphisms.

Proves existence of sentences to identify homeomorphic manifolds.

We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian manifold whose Pfaffian form is a positive multiple of the volume form, and $y\in C^{1,α}(M;\mathbb{R}^{n+1})$ is an isometric immersion with $n/(n+1)< α\leq 1$, then $y(M)$ is a surface of bounded extrinsic curvature. This is proved by showi…

New homeomorphism found in Klein bottle group.

Study weak conjugacy in surface homeomorphisms.

Paper explains dynamics of homeomorphisms to mapping tori geometry.

Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.

The study of periodic subgroups in homeomorphism groups of manifolds.

Adapts pivoting technique to circle homeomorphisms for proofs.

No algorithm exists to decide 4-manifold homeomorphism.

Proof shows homeomorphism problem is unsolvable.

Study covers of sphere with homeomorphisms lifting property.

Goldman bracket distinguishes surface homeomorphisms.

Study shows similar result to Margulis for Cantor set homeomorphisms.

In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal s…

The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.

Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invari…

Study shows certain surface homeomorphisms groups can't be precompact.

We prove that the set of non-properness of a polynomial mapping of the three dimensional space which is a local homeomorphism cannot be homeomorphic to the real line $\R.$

For each $n > 0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations `conjugated by $z \to z^n$'. We show that these families are free of relations, which determines the structure of `the group of homeomorphisms of finite type'. We also discuss a numbe…

Study classifies 2-manifolds with special homeomorphism groups.

Study chaotic behavior in homeomorphism groups of countable products of spaces.