# Topologie

### Robert Oliver

Université de Paris XIII, Villetaneuse, France### Cameron Gordon

University of Texas at Austin, USA### Wolfgang Lück

Universität Bonn, Germany

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## Abstract

This was the most recent in a long series of annual conferences in Oberwolfach covering all areas of algebraic and geometric topology, and the last such conference before going over to the new two-year cycle of meetings. According to the records kept in the library of the institute, the first topology meeting was held in 1963, and meetings have been held every year since then except for 1968. None of the participants in the first meeting is still active in research. Of the people present at this meeting, it was Rainer Vogt who has been attending this series for the longest period: since 1969. Every year for the last twelve years, a ``keynote speaker'' has been chosen to give some focus to the topology meeting. Thus while we do have talks which cover all areas of algebraic and geometric topology, we try to focus on one particular area of current interest. This year, the keynote speaker was Yair Minsky, who talked about the classification of non-compact hyperbolic 3-manifolds $N$ with finitely generated fundamental group. The two main conjectures (now proved) in this area are Marden's Tameness Conjecture and Thurston's Ending Lamination Conjecture. The Tameness Conjecture is about the topology of $N$, and asserts that any end of $N$ is {\em topologically tame}, i.e., is homeomorphic to $S \times \mathbb{R}$ for some closed surface $S$. The Ending Lamination Conjecture is about the geometry of $N$, and concerns the data needed to determine $N$ up to isometry. The most interesting case is that of a geometrically infinite (and topologically tame) end $\varepsilon$ of $N$. Thurston showed how to associate to such an $\varepsilon$ a geodesic lamination $\lambda$ on the surface $S$, the {\em ending lamination} of $\varepsilon$, and conjectured that $\varepsilon$ is determined up to isometry by $\lambda$. The Ending Lamination Conjecture was recently proved by Minsky, partly in joint work with Brock and Canary, and making use of the earlier result of Masur and Minsky that the curve complex of a surface is hyperbolic in the sense of Gromov. At the meeting, Minsky gave the keynote series of three lectures on the background to and proof of the Ending Lamination Conjecture. Within the last year, the Tameness Conjecture has also been proved, by Agol (an independent proof has also been announced by Calegari and Gabai). In his talk at the meeting Agol sketched some of the ideas of the proof and outlined several applications to other problems in 3-dimensional topology. The Tameness and Ending Lamination Conjectures together give a complete para\-metrization of the set $AH(M)$ of (non-compact) hyperbolic 3-manifolds homotopy equivalent to a given compact 3-manifold $M$ with non-empty boundary. However, if $AH(M)$ is given the natural {\em algebraic topology}, i.e., that coming from its containment in the $\operatorname{PSL}_2(\mathbb{C})$ character variety of $\pi_1(M)$, then the classifying data is not continuous. As a consequence, the topological structure of $AH(M)$, in other words the deformation theory of these hyperbolic structures, is not completely clear. This was the topic of Canary's talk. He described how, although (in the case that $\partial M$ is incompressible) the components of the interior of $AH(M)$ are open topological cells, their closures can intersect in unexpected and wild ways. Probably the main problem left in this whole area is to better understand the topology of this deformation space $AH(M)$. For example, what is the Hausdorff dimension of its boundary? In contrast, if $M$ is a closed hyperbolizable 3-manifold, then $AH(M)$ is simply a pair of points, by Mostow rigidity. However, even in this case the relation between the geometry of $M$ (i.e., its hyperbolic metric) and its topology is not well understood. Souto's talk addressed an interesting question in this context, namely the relation between the Heegaard genus of $M$ and the lengths of geodesics in $M$. Specifically, if $S$ is a genus $g$ Heegaard surface in $M$, then, although it is easy to see that there is no lower bound on the lengths of closed geodesics in $M$ that depends only on $g$, Souto showed that there exists $\varepsilon_g > 0$ such that the set of primitive closed geodesics in $M$ of length $\le \varepsilon_g$ is unknotted in the sense that it can be isotoped to lie on parallel copies of $S$. The other talks were chosen to cover as many different areas of topology as possible, and hence it is difficult to find an overall theme to describe them. On the more geometric side, Nathalie Wahl described her work on diffeomorphism groups of 3-manifolds obtained by attaching certain types of handles to $S^3$, and their connection to groups of self equivalences of certain graphs. Among the applications of this work are new proofs of homological stability of $\textup{Aut}(F_n)$ and $\textup{Out}(F_n)$ ($F_n$ a free group), the vanishing of $H_*(\textup{Aut}(F_n);\mathbb{Z}^n)$ in a range, and the construction of an infinite loop map from $\mathbb{Z}\times{}B\Gamma_\infty^+$ (the limit of mapping class groups of surfaces) to $\mathbb{Z}\times{}B\textup{Aut}_\infty^+$ (the limit of the $\textup{Aut}(F_n)$). Ian Hambleton described his recent proof that for any pair of finite periodic groups $G$ and $G'$, the product $G\times{}G'$ acts freely and smoothly on $S^n\times{}S^n$ for some $n$ --- even in the cases (already well known) when $G$ and $G'$ themselves do not act freely on any spheres. Stefan Bauer described his recent work on invariants of 4-manifolds, including a refinement of the Seiberg-Witten invariants due to him and Furuta. Hyam Rubinstein talked about an interesting generalization of the class of small Seifert fibered 3-manifolds, in which the three solid tori whose union is the manifold are replaced by handlebodies of genus~~2. In a more algebraic direction, Jesper Grodal described some of the latest developments in the field of 2-compact groups --- spaces which are complete at the prime 2, and whose loop space has finite mod 2 cohomology (i.e., looks like the 2-completion of a finite complex). The goal is to classify all simply connected 2-compact groups (this has already been done at odd primes), and understand how close they are to being 2-completions of classifying spaces of compact connected Lie groups. In his talk, Grodal focused on the problem of defining root systems for 2-compact groups, and some of the problems which arise at the prime 2 and did not arise for odd primes. Among the other algebraic talks, Kathryn Hess described new algorithms for describing the Hopf algebra structure on the homology of the loop space of a space~~$X$, in terms of an appropriate model for chains on $X$. In the field of geometric group theory, Martin Bridson talked about subgroups of direct products of hyperbolic groups, and described his counterexample with Grunewald to a conjecture of Grothendieck, where they construct a homomorphism of finitely presented, residually finite groups which induces an equivalence of representation categories but is not an isomorphism.