Related papers: An introduction to Handel's homotopy Brouwer theor…
Brouwer homeomorphisms are fixed-point-free, orientation-preserving homeomorphisms of the plane. In recent years, their dynamics have been mostly studied through two complementary approaches, one introduced by Handel and the other by Le…
Michael Handel has proved in [Ha] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that turned out to be an efficient tool in the study of the dynamics of surface homeomorphisms. The present article…
We use the homotopy Brouwer theory of Handel to define a Poincar{\'e} index between two orbits for an orientation preserving fixed point free homeomorphism of the plane. Furthermore, we prove that this index is almost additive.
A Brouwer homeomorphism is a fixed-point free, orientation-preserving homeomorphism of the plane. A foundational result of Le Calvez establishes that every such homeomorphism $f$ admits an oriented planar foliation $\mathcal{F}$ such that…
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…
We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of "adjoint logic" in which the discretization and codiscretization modalities are characterized using a judgmental formalism of "crisp…
In 1912 Brouwer published his translation theorem, which implies, for example, that an orientation preserving homeomorphism of the plane having a periodic point also has a fixed point. This theorem has given rise to a number of…
We introduce a notion of graph homeomorphisms which uses the concept of dimension and homotopy for graphs. It preserves the dimension of a subbasis, cohomology and Euler characteristic. Connectivity and homotopy look as in classical…
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…
This paper extends the decorated Teichm\"uller theory developed before for punctured surfaces to the setting of ``bordered'' surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. The main new result…
We prove three theorems giving fixed points for orientation preserving homeomorphisms of the plane following forgotten results of Brouwer.
Brouwer's fixed point theorem from 1911 is a basic result in topology - with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, related to the game of HEX and to the piercing of multiple…
The Epstein-Baer theory of curve isotopies is basic to the remarkable theorem that homotopic homeomorphisms of surfaces are isotopic. The groundbreaking work of R. Baer was carried out on closed, orientable surfaces and extended by D. B. A.…
We show that if an orientation-preserving homeomorphism of the plane has a topologically chain recurrent point, then it has a fixed point, generalizing the Brouwer plane translation theorem.
In the present paper we study some homotopy invariants which can be defined by means of bundles with fiber a matrix algebra. We also introduce some generalization of the Brauer group in the topological context and show that any its element…
This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations.…
Michael Handel proved in [7] the existence of a fixed point for an orientation preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links…
A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping $f : (X \times Y) \to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff…
More than a century ago, L. E. J. Brouwer proved a famous theorem, which says that any orientation preserving homeomorphism of the plane having a periodic point must have a fixed point. In recent years, there are still some authors giving…
In condensed matter physics and related areas, topological defects play important roles in phase transitions and critical phenomena. Homotopy theory facilitates the classification of such topological defects. After a pedagogic introduction…