Louis Funar

On the geometric simple connectivity of open manifolds

A manifold is said to be geometrically simply connected if it has a proper handle decomposition without 1-handles. By the work of Smale, for compact manifolds of dimension at least 5, this is equivalent to simple connectivity. We prove that there exists an obstruction to an open simply connected n-manifold of dimension n ≥ 5 being geometrically simply connected. In particular, for each n ≥ 4, there exist uncountably many simply connected n-manifolds which are not geometrically simply connected. We also prove that for n ≠ 4, an n-manifold proper homotopy equivalent to a weakly geometrically simply connected polyhedron is geometrically simply connected (for n = 4, it is only end-compressible). We analyze further the case n = 4 and Poenarus conjecture.

Torus bundles not distinguished by TQFT invariants

We show that there exist arbitrarily large sets of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in SL.2;Z/ and its congruence quotients, the classification of SOL (polycyclic) 3‐manifold groups and an elementary study of a family of Pell equations. A key ingredient is the congruence subgroup property of modular representations, as it was established by Coste and Gannon, Bantay, Xu for various versions of TQFT, and by Ng and Schauenburg for the Drinfeld doubles of spherical fusion categories. In particular, we obtain non-isomorphic 3‐manifold groups with the same pro-finite completions, answering a question of Long and Reid. On the other side we prove that two torus bundles over the circle with the same U.1/ and SU.2/ quantum invariants are (strongly) commensurable. In the appendix (joint with Andrei Rapinchuk) we show that these examples have positive density in a suitable set of discriminants. 20F36, 57M07; 20F38, 57N05

ON THE QUOTIENTS OF CUBIC HECKE ALGEBRAS

Between the rank 3 quotients of cubic Hecke algebras there is essentially one of maximal dimension. We prove it has a unique Markov trace having values in a torsion module. Therefore the description of a Markov trace on the cubic Hecke algebra corresponding tox3+1 and having the parameters (1, 1) is derived. Thus we obtain a numerical link invariant of finite degree, and define a whole sequence of 3rd order Vassiliev invariants.

Cubulations, immersions, mappability and a problem of habegger

Abstract The aim of this paper (inspired from a problem of Habegger) is to describe the set of cubical decompositions of compact manifolds mod out by a set of combinatorial moves analogous to the bistellar moves considered by Pachner, which we call bubble moves. One constructs a surjection from this set onto the the bordism group of codimension-one immersions in the manifold. The connected sums of manifolds and immersions induce multiplicative structures which are respected by this surjection. We prove that those cubulations which map combinatorially into the standard decomposition of R n for large enough n (called mappable), are equivalent. Finally we classify the cubulations of the 2-sphere.

On the wgsc and qsf tameness conditions for finitely presented groups

A finitely presented group is weakly geometrically simply connected (wgsc) if it is the fundamental group of some compact polyhedron whose universal covering is wgsc, i.e., it has an exhaustion by compact connected and simply connected sub-polyhedra. We show that this condition is almost-equivalent to Bricks qsf property, which amounts to finding an exhaustion approximable by finite simply connected complexes, and also to the tame comba- bility introduced and studied by Mihalik and Tschantz. We further observe that a number of standard constructions in group theory yield qsf groups and analyze specific examples. We show that requiring the exhaustion be made of metric balls in some Cayley complex is a strong constraint, not satisfied by general qsf groups. In the second part of this paper we give sufficient conditions under which groups which are extensions of finitely presented groups by finitely generated (but infinitely presented) groups are qsf. We prove, in particular, that the finitely presented HNN extension of the Grigorchuk group is qsf.

The braided Ptolemy-Thompson group is finitely presented

Pursueing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group

On smooth maps with finitely many critical points

T^*

Two questions on mapping class groups

(and its further generalizations) which is an extension of the Ptolemy-Thompson group

Asymptotically rigid mapping class groups and Thompson groups

T

Central extensions of the Ptolemy-Thompson group and quantized Teichmuller theory

by means of the full braid group