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Jean-Claude Hausmann

Publications and source records attributed to Jean-Claude Hausmann.

At least 19 recordsLinked to original sources
arXiv2022

The cell-dispensability obstruction for spaces and manifolds

We compare two properties for a CW-space $X$ of finite type: (1) being homotopy equivalent to a CW-complex without $j$-cells for $k\leq j\leq \ell$ (($k,\ell$)-cellfree) and (2) $H^j(X;R)=0$ for any $\mathbb Zπ_1(X)$-module $R$ when $k\leq j\leq \ell$ (cohomogy ($k,\ell$)-silent). Using the technique of Wall's finiteness obstruction, we show that a connected CW-space $X$ of finite type which is cohomogy ($k,\ell$)-silent determines a "cell-dispensability obstruction'' $w_k(X)\in\tilde K_0(\mathbb Zπ_1(X))$ which vanishes if and only if $X$ is ($k,\ell$)-cellfree ($k\geq 4$). Any class in $\tilde K_0(\mathbb Zπ)$ may occur as the cell-dispensability obstruction $w_k(X)$ for a CW-space $X$ with $π_1(X)$ identified with $π$. Using projective surgery, a similar theory is obtained for manifolds, replacing "cells" by "handles" (antisimple manifolds).

math.AT↗
arXiv2018

A simplification problem in manifold theory

Two smooth manifolds M and N are called R-diffeomorphic if their product with the real line are diffeomorphic. We consider the following simplification problem: does R-diffeomorphism imply diffeomorphism or homeomorphism? For compact manifolds, analysis of this problem relies on some of the main achievements of the theory of manifolds, in particular the h- and s-cobordism theorems in high dimensions and the spectacular more recent classification results in dimensions 3 and 4. This paper presents what is currently known about the subject as well as some new results about classifications of R-diffeomorphisms.

math.GT↗
arXiv2013

Simple Hamiltonian manifolds

A simple Hamiltonian manifold is a closed connected symplectic manifold equipped with a Hamiltonian action of a torus T with moment map Phi: M-->t^*, such that the fixed set M^T has exactly two connected components, denoted M_0 and M_1. We study the differential and symplectic geometry of simple Hamiltonian manifolds, including a large number of examples.

math.SG↗
arXiv2013

Triangles on planar Jordan $C^1$-curves

We prove that a Jordan $\calc^1$-curve in the plane contains any non-flat triangle up to translation and homothety with positive ratio. This is false if the curve is not $C^1$. The proof uses a bit configuration spaces, differential and algebraic topology as well as the Schoenflies theorem. A partial generalization holds true in higher dimensions.

math.MG↗
arXiv2010

Conjugation spaces and 4-manifolds

We show that 4-dimensional conjugation manifolds are all obtained from branched 2-fold coverings of knotted surfaces in Z/2-homology 4-spheres.

math.GT↗
arXiv2009

Equivariant Bundles and Isotropy Representations

We introduce a new construction, the isotropy groupoid, to organize the orbit data for split $Γ$-spaces. We show that equivariant principal $G$-bundles over split $Γ$-CW complexes $X$ can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex $A=Γ\backslash X$ is a graph, with all edge stabilizers toral subgroups of $Γ$, we obtain a purely combinatorial classification of bundles with structural group $G$ a compact connected Lie group. If $G$ is abelian, our approach gives combinatorial and geometric descriptions of some results of Lashof-May-Segal and Goresky-Kottwitz-MacPherson.

math.GT↗
arXiv2008

Conjugation spaces and edges of compatible torus actions

Duistermaat introduced the concept of ``real locus'' of a Hamiltonian manifold. In that and in others' subsequent works, it has been shown that many of the techniques developed in the symplectic category can be used to study real loci, so long as the coefficient ring is restricted to the integers modulo 2. It turns out that these results seem not necessarily to depend on the ambient symplectic structure, but rather to be topological in nature. This observation prompts the definition of ``conjugation space'' in a paper of the two authors with V. Puppe. Our main theorem in this paper gives a simple criterion for recognizing when a topological space is a conjugation space.

math.AT↗
arXiv2007

On the conjecture of Kevin Walker

In 1985 Kevin Walker in his study of topology of polygon spaces raised an interesting conjecture in the spirit of the well-known question "Can you hear the shape of a drum?" of Marc Kac. Roughly, Walker's conjecture asks if one can recover relative lengths of the bars of a linkage from intrinsic algebraic properties of the cohomology algebra of its configuration space. In this paper we prove that the conjecture is true for polygon spaces in R^3. We also prove that for planar polygon spaces the conjecture holds is several modified forms: (a) if one takes into account the action of a natural involution on cohomology, (b) if the cohomology algebra of the involution's orbit space is known, or (c) if the length vector is normal. Some of our results allow the length vector to be non-generic, the corresponding polygon spaces have singularities. Our main tool is the study of the natural involution and its action on cohomology. A crucial role in our proof plays the solution of the isomorphism problem for monoidal rings due to J. Gubeladze.

math.AT↗
arXiv2006

Holonomy orbits of the snake charmer algorithm

The snake charmer algorithm permits us to deform a piecewise smooth curve starting from the origin in R^d, so that its end follows a given path. When this path is a loop, a holonomy phenomenon occurs. We prove that the holonomy orbits are closed manifolds diffeomorphic to real Stiefel manifolds. A survey of the snake charmer algorithm is given in the paper.

math.DG↗
arXiv2005

Conjugation spaces

There are classical examples of spaces X with an involution tau whose mod 2-comhomology ring resembles that of their fixed point set X^tau: there is a ring isomorphism kappa: H^2*(X) --> H^*(X^tau). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism kappa is part of an interesting structure in equivariant cohomology called an H^*-frame. An H^*-frame, if it exists, is natural and unique. A space with involution admitting an H^*-frame is called a conjugation space. Many examples of conjugation spaces are constructed, for instance by successive adjunctions of cells homeomorphic to a disk in C^k with the complex conjugation. A compact symplectic manifold, with an anti-symplectic involution compatible with a Hamiltonian action of a torus T, is a conjugation space, provided X^T is itself a conjugation space. This includes the co-adjoint orbits of any semi-simple compact Lie group, equipped with the Chevalley involution. We also study conjugate-equivariant complex vector bundles (`real bundles' in the sense of Atiyah) over a conjugation space and show that the isomorphism kappa maps the Chern classes onto the Stiefel-Whitney classes of the fixed bundle.

math.AT↗
arXiv2002

Contrôle des bras articulés et transformations de Moebius (Control of robot arms and Moebius transformations)

For a m-tuple a=(a_1,...,a_m) of positive real numbers, the robot arm of type a in R^d is the map f^a:(S^{d-1})^m -> R^d defined by f^a(z_1,...,z_m) to be the sum of the a_jz_j's. Our aim is to attack the inverse problem via the horizontal liftings for the distribution Delta^a orthogonal to the fibers of f^a. One shows that the connected components by horizontal curves are the orbits of an actiion on (S^{d-1})^m by a product of groups of Moebius transformations. In several cases, the holonomy orbits of the distribution Delta^a are also described.

math.DG↗
arXiv2002

The space of clouds in an Euclidean space

We study the space $\nua{m}{d}$ of clouds in $\bbr^d$ (ordered sets of $m$ points modulo the action of the group of affine isometries). We show that $\nua{m}{d}$ is a smooth space, stratified over a certain hyperplane arrangement in $\bbr^m$. We give an algorithm to list all the chambers and other strata (this is independent of $d$). With the help of a computer, we obtain the list of all the chambers for $m\leq 9$ and all the strata when $m\leq 8$. As the strata are the product of a polygon spaces with a disk, this gives a classification of $m$-gon spaces for $m\leq 9$. When $d=2,3$, $m=5,6,7$ and modulo reordering, we show that the chambers (and so the different generic polygon spaces) are distinguished by the ring structure of their ${\rm mod} 2$-cohomology.

math.DG↗
arXiv2002

Maximal Hamiltonian tori for polygon spaces

We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.

math.SG↗
arXiv1999

Théorie de jauge et groupo\"ıdes (Gauge theory and groupoids)

We consider the problem of existence of representations of topological groupoids on a principal bundle and the classification of such representations up to gauge transformation. Such representations naturally occur in various contexts such as gauge theory, lattice gauge fields, equivariant bundles, etc. In the course of the proofs, some new facts about Milnor's classifying spaces and gauge groups are established.

math.DG↗