Quotients of Bing spaces

A Bing space is a compact Hausdorff space whose every component is a hereditarily indecomposable continuum. We investigate spaces which are quotients of a Bing space by means of a map which is injective on components. We show that the class of such spaces does not include every compact space, but does properly include the class of compact metric spaces.     

Quotients of uniform spaces

This paper develops the basic theory of quotients of uniform spaces via sufficiently nice group actions. We generalize and unify two fundamental constructions: quotients of topological groups via closed normal subgroups and quotients of metric spaces via actions by isometries. Basic results about inverse limits of topological groups are extended to inverse limits of group actions on uniform spaces, and notions of prodiscrete action and generalized covering map are introduced.     

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a closed semi-algebraic set ${X \subset \mathbb{R}^n}$ and a C ² semi-algebraic function ${f : \mathbb{R}^n \rightarrow\mathbb{R}}$ such that ${f_{\vert X}}$ has a finite number of critical points. We relate the topology of X to the topology of the sets ${X \cap \{ f * \alpha \}}$ , where ${* \in \{\le,=,\ge \}}$ and ${\alpha \in \mathbb{R}}$ , and the indices of the critical points of ${f_{\vert X}}$ and ${-f_{\vert X}}$ . We also relate the topology of X to the topology of the links at infinity of the sets ${X \cap \{ f * \alpha\}}$ and the indices of these critical points. We give applications when ${X=\mathbb{R}^n}$ and when f is a generic linear function.     

Separating families for semi-algebraic sets

A new invariantp(V) is defined for real algebraic varietiesV which measures the complexity of semi-algebraic sets inV.p(V) is the least integer such that every semi-algebraic setS ?-V can be separated from its compliment byp(V) polynomials. This is a very natural invariant to consider. Using results of Bröcker [4–8] and generalizations of Bröcker’s results found in [16,17], upper bounds forp(V) are computed. The proof is simpler than the proof of similar results in [5–9],[15–18] since the complicated local-global formula for the stability index and the various pasting techniques are not needed. Lower bounds forp(V) are also computed in some special cases, the technique here being to first study the corresponding invariantp(X, G) for a finite space of orderings (X, G) [13,14].     

Cohomology of locally closed semi-algebraic subsets

Let k be a non-Archimedean field, let ? be a prime number distinct from the characteristic of the residue field of k. If χ is a separated k-scheme of finite type, Berkovich’s theory of germs allows to define étale ?-adic cohomology groups with compact support of locally closed semi-algebraic subsets of χ ^an . We prove that these vector spaces are finite dimensional continuous representations of the Galois group of k ^sep /k, and satisfy the usual long exact sequence and Künneth formula. This has been recently used by E. Hrushovski and F. Loeser in a paper about the monodromy of the Milnor fibration. In this statement, the main difficulty is the finiteness result, whose proof relies on a cohomological finiteness result for affinoid spaces, recently proved by V. Berkovich.     

Quotients of gravitational instantons

A classification result for Ricci-flat anti-self-dual asymptotically locally Euclidean 4-manifolds is obtained: they are either hyperk?hler (one of the gravitational instantons classified by Kronheimer), or they are a cyclic quotient of a Gibbons–Hawking space. The possible quotients are described in terms of the monopole set in \mathbbR³{\mathbb{R}^3} , and it is proved that every such quotient is actually K?hler. The fact that the Gibbons–Hawking spaces are the only gravitational instantons to admit isometric quotients is proved by examining the possible fundamental groups at infinity: most can be ruled out by the classification of three-dimensional spherical space form groups, and the rest are excluded by a computation of the Rohklin invariant (in one case) or the eta invariant (in the remaining family of cases) of the corresponding space forms.     

On the cardinality of a semi-algebraic set

It is shown that the cardinality of a finite semi-algebraic subset over a real closed field can be computed in terms of signatures of effectively constructed quadratic forms.     

Open book structures on semi-algebraic manifolds

Given a C ² semi-algebraic mapping \({F} : {\mathbb{R}^N \rightarrow \mathbb{R}^p}\), we consider its restriction to \({W \hookrightarrow \mathbb{R^{N}}}\) an embedded closed semi-algebraic manifold of dimension \({n-1 \geq p \geq 2}\) and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection \({\frac{F}{\Vert F \Vert}:W{\setminus} F^{-1}(0) \to S^{p-1}}\). Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering W as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of F with the canonical projection \({\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}}\) and prove that the fibers of \({\frac{F}{\Vert F \Vert}}\) and \({\frac{\pi \circ F}{\Vert \pi \circ F \Vert}}\) are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection \({\frac{F}{\Vert F \Vert}}\) and \({W \cap F^{-1}(0)}\). Similar formulae are proved for mappings obtained after composition of F with canonical projections.     

Quotients of peck posets

An elementary, self-contained proof of a result of Pouzet and Rosenberg and of Harper is given. This result states that the quotient of certain posets (called unitary Peck) by a finite group of automorphisms retains some nice properties, including the Sperner property. Examples of unitary Peck posets are given, and the techniques developed here are used to prove a result of Lovász on the edge-reconstruction conjecture.Supported in part by a National Science Foundation research grant.     

Computing the dimension of a semi-algebraic set

In this paper, we consider the problem of computing the real dimension of a given semi-algebraic subset of R ^k, where R is a real closed field. We prove that the dimension k′ of a semi-algebraic set described by s polynomials of degree d in k variables can be computed in time

Quantitative results on quadratic semi-algebraic sets

Quantitative semi-algebraic geometry studies accurate bounds on topological invariants (such as the Betti numbers) of semi algebraic sets in terms of the number of equations, their degree and their number of variables. For general semialgebric sets, these bounds have an exponential dependance in the number of variables. In contrast, for semi-algebraic sets defined by quadratic equation, the dependance is polynomial in the number of variables. The talk will include a survey of the main results known for general semi-algebraic sets before concentrating on the quadratic case. The lecture will use material from joint work with Saugata Basu and Dimitri Pasechnik.