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Norbert A'Campo

Publications and source records attributed to Norbert A'Campo.

At least 19 recordsLinked to original sources
arXiv2025

Plane curve singularities via divides

Generic relative immersions of compact one-manifolds in the closed unit disk, i.e. divides, provide a powerful combinatorial framework, and allow a topological construction of fibered classical links, for which the monodromy diffeomorphism is explicitly given as a product of Dehn twists. Complex isolated plane curve singularities provide a classical fibered link, the Milnor fibration, with its Milnor monodromy, monodromy group, and vanishing cycles. This surveys puts together much of the work done on divides and their role in the topology of isolated plane curve singularities.

math.GT↗
arXiv2024

Geometry on surfaces, a source for mathematical developments

We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results for such structures. Higher-dimensional analogues are also discussed. Some constructions with Riemann surfaces lead, by analogy, to notions that hold for arbitrary fields, and not only the field of complex numbers. The Riemann sphere is also defined using surjective homomorphisms of real algebras from the ring of real univariate polynomials to (arbitrary) fields, in which the field with one element is interpreted as the point at infinity of the Gaussian plane of complex numbers. Several models of the hyperbolic plane and hyperbolic 3-space appear, defined in terms of complex structures on surfaces, and in particular also a rather elementary construction of the hyperbolic plane usingreal monic univariate polynomials of degree two without real roots. Several notions and problems connected with conformal structures in dimension 2 are discussed, including dessins d'enfants, the combinatorial characterization of polynomials and rational maps of the sphere, the type problem, uniformization, quasiconformal mappings, Thurston's characterization of Speiser graphs, stratifications of spaces of monic polynomials, and others. Classical methods and new techniques complement each other. The final version of this paper will appear as a chapter in the Volume Surveys in Geometry. II (ed. A. Papadopoulos), Springer Nature Switzerland, 2024.

math.CV↗
arXiv2019

Tête-à-tête twists, monodromies and representation of elements of Mapping Class Group

We study monodromies of plane curve singularities and pseudo-periodic homeomorphisms of oriented surfaces with boundary, following an original idea of the first author: tête-à-tête graphs and twists. We completely characterize mapping classes that can be represented by tête-à-tête twists, and generalize the notion to be able to represent any class of the mapping class group relative to the boundary which is boundary-free periodic. This improves previous work on the subject by C. Graf. Furthermore, we introduce the class of mixed tête-à-tête graphs and twists, and prove that mixed tête-à-tête twists contain monodromies of irreducible plane curve singularities. In a sequel paper, the fourth author and B. Sigurdsson have extended this to the reducible case.

math.GT↗
arXiv2017

Signatures of monic polynomials

To a univariate monic polynomial is attached a special planar forest that is called the picture of the polynomial. Isotopy classes of pictures are called signatures. All combinatorially possible signatures are realized and spaces of polynomials realizing a given signature are contractible. A finite cell complex for the cohomology of the braid groups is obtained.

math.AG↗
arXiv2016

A Commentary on Teichmüller's paper "Über Extremalprobleme der konformen Geometrie" (On extremal problems in conformal geometry)

We comment on the paper Über Extremalprobleme der konformen Geometrie (On extremal problems in conformal geometry) by Teichmüller, published in 1941. This paper contains ideas on a wide generalization of his previous work on the solution of extremal problems in conformal geometry. The generalization concerns at the same time the fields of function theory, topology and algebra. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichmüller theory. This volume is dedicated to the memory of Alexander Grothendieck.

math.GT↗
arXiv2016

Actions of the Absolute Galois Group

We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation came in part from his desire to understand the absolute Galois group. But he was also interested in Thurston's work on surfaces, and he expressed this in his Esquisse d'un programme, his R{é}coltes et semailles and on other occasions. He introduced the notions of dessin d'enfant, Teichm{ü}ller tower, and other related objects, he considered the actions of Γ Q on them or on their etale fundamental groups, and he made conjectures on some natural homomorphisms between the absolute Galois group and the automor-phism groups (or outer automorphism groups) of these objects. We mention several ramifications of these ideas, due to various authors. We also report on the works of Sullivan and others on nonlinear actions of Γ Q , in particular in homotopy theory. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{ü}ller theory. This volume is dedicated to the memory of Alexander Grothendieck.

math.GT↗
arXiv2016

On Grothendieck's tame topology

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surface topology and moduli spaces of Riemann surfaces, Grothendieck calls there for a recasting of topology, in order to make it fit to the objects of semialgebraic and semianalytic geometry, and in particular to the study of the Mumford-Deligne compactifications of moduli spaces. A new conception of manifold, of submanifold and of maps between them is outlined. We review these ideas in the present chapter, because of their relation to the theory of moduli and Te-ichm{ü}ller spaces. We also mention briefly the relations between Grothendieck's ideas and earlier theories developed by Whitney, Lojasiewicz and Hironaka and especially Thom, and with the more recent theory of o-minimal structures. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{ü}ller theory. This volume is dedicated to the memory of Alexander Grothendieck.

math.GT↗
arXiv2016

On Grothendieck's construction of Teichmüller space

In his 1944 paper Veränderliche Riemannsche Flächen , Teichmüller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichmüller space. In the same paper, Teichmüller introduced the so-called universal Teichmüller curve -- a space over Teichmüller space where the fiber above each point is a Riemann surface representing that point. In fact, Teichmüller proved the existence of the Teichmüller curve as a space of Riemann surfaces parametrized by an analytic space, with an existence and uniqueness theorem establishing this analytic structure. This result was later reformulated and proved by Grothendieck in a series of ten lectures he gave at Cartan's seminar in 1960-1961. In his approach , Grothendieck replaced Teichmüller's explicit parameters by a general construction of fiber bundles whose base is an arbitrary analytic space. This work on Teichmüller space led him to recast the bases of analytic geometry using the language of categories and functors. In Grothendieck's words, the Teichmüller curve becomes a space representing a functor from the category of analytic spaces into the category of sets. In this survey, we comment on Grothendieck's series of lectures. The survey is primarily addressed to low-dimensional topologists and geometers. In presenting Grothendieck's results, we tried to explain or rephrase in more simple terms some notions that are usually expressed in the language of algebraic geometry. However, it is not possible to short-circuit the language of categories and functors. The survey is also addressed to those algebraic geometers who wish to know how the notion of moduli space evolved in connection with Teichmüller theory. Explaining the origins of mathematical ideas contributes in dispensing justice to their authors and it usually renders the theory that is surveyed more attractive. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichmüller theory. This volume is dedicated to the memory of Alexander Grothendieck.

math.GT↗
arXiv2016

On the early history of moduli and Teichm{ü}ller spaces

We survey some major contributions to Riemann's moduli space and Teichm{ü}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichm{ü}ller space. We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{ü}ller's results were correct.This paper was written for the book "Lipman Bers, a life in Mathematics," edited by Linda Keen , Irwin Kra and Rubi Rodriguez (Amercian Mathematical Society, 2015). It is dedicated to the memory of Lipman Bers who was above all a complex analyst and spent a large part of his life and energy working on the analytic structure of Teichm{ü}ller space. His work on analysis is nevertheless inseparable from geometry and topology. In this survey, we highlight the relations and the logical dependence between this work and the works of Riemann, Poincar{é}, Klein, Brouwer, Siegel, Teichm{ü}ller, Weil, Grothendieck and others. We explain the motivation behind the ideas. In doing so, we point out several facts which seem to be unknown to many Teichm{ü}ller theorists.

math.HO↗
arXiv2015

A Commentary on Teichm{ü}ller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl{ä}chen''

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 Teichm{ü}ller wrote on the theory of moduli. It contains the proof of the so-called Teichm{ü}ller existence theorem for a closed surface of genus at least 2. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichm{ü}ller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). After that, he defines a map between the latter and the Euclidean space of dimension 6g-6 Using Brouwer's theorem of invariance of domain, he shows that this map is a homeomorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincar{é} and Klein's use of the so-called ``continuity principle'' in their attempts to prove the uniformization theorem.

math.GT↗
arXiv2014

Transitional geometry

We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric entities like points, lines, dis-tances, triangles, angles, area, curvature, etc. as well as trigonometric formulae and other properties transit in a continuous manner from one geometry to another. AMS classification: 01-99 ; 53-02 ; 53-03 ; 53A35.

math.GT↗
arXiv2014

On Klein's So-called Non-Euclidean geometry

In two papers titled "On the so-called non-Euclidean geometry", I and II, Felix Klein proposed a construction of the spaces of constant curvature -1, 0 and and 1 (that is, hyperbolic, Euclidean and spherical geometry) within the realm of projective geometry. Klein's work was inspired by ideas of Cayley who derived the distance between two points and the angle between two planes in terms of an arbitrary fixed conic in projective space. We comment on these two papers of Klein and we make relations with other works.

math.MG↗
arXiv2012

A commentary on Teichmüller's paper "Veränderliche Riemannsche Flächen" (Variable Riemann Surfaces)

This is a commentary on Teichmüllers' paper "Veränderliche Riemannsche Flächen" (Variable Riemann Surfaces), published in 1944. This paper is the last one that Teichmüller wrote on the problem of moduli. At most places the paper contains ideas and no technical details. The author presents a completely new approach to Teichmüller space, compared to the approach he took in his first seminal paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" and its sequel "Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen" in which he completed some of the the results stated in the former. In the paper "Extremale quasikonforme ...", Teichmüller led the foundations of what we call today Teichmüller theory (but without the complex structure), defining its metric and introducing in that theory the techniques of quasiconformal mappings and of quadratic differentials as essential tools. In the present paper, the approach is more abstract, through complex analytic geometry. Teichmüller space, equipped with its complex-analytic structure, is characterized here by a certain universal property. Among the other ideas and results contained in the paper, we mention the following: (1) The existence and uniqueness of the universal Teichmüller curve, rediscovered later on by Ahlfors and by Bers. At the same time, this introduced the first fibre bundle over Teichmüller space. (2) The proof of the fact that the automorphisms group of the univeral Teichmüller curve is the extended mapping class group. (3) The idea of a fine moduli space. (4) The idea of using the period map to define a complex structure on Teichmüller space.

math.GT↗
arXiv2003

TQFT computations and experiments

With the help of a new program, we do computations concerning the Witten-Reshetikhin-Turaev representations of mapping class groups. In particular we distinguish some mutant fibered knots. The program can be downloaded from http://www.geometrie.ch/TQFT

math.GT↗
arXiv2003

A natural construction for the real numbers

A new construction of the real number system, that is built directly upon the additive group of integers and has its roots in the definition due to Henri Poincaré of the rotation number of an orientation preserving homeomorphism of the circle.

math.GN↗
arXiv2003

Monodromy of real isolated singularities

For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced by complex conjugation and another involution. This topological property holds for all isolated complex plane curve singularities. Using real morsifications, we compute the action of complex conjugation and of the other involution on the Milnor fiber of real plane curve singularities. These involutions have nice descriptions in terms of divides for the singularity.

math.AG↗
arXiv2000

A combinatorial property of generic immersions of curves

A divide is a relative generic immersion of a finite union of copies of the unit interval in the unit disk. A divide defines a classical link in the 3- sphere, which is a fibered link if the image of the immersion is connected. We prove in this paper, that the Lefschetz number of the monodromy is 0. This result was known for divides, which correspond to real morsifications of complexe plane curve singularities. The proof uses the space of gradient lines of a morse function.

math.AG↗