Project B: Geometry and topology


The fundamental topics in research area B are curvature, Ricci flow, rigidity, actions of groups on manifolds and buildings, and homotopy theory.
Curvature is one of the most important notions in Riemannian geometry. One may investigate the curvature properties for a specific Riemannian metric, the stability of certain curvature conditions under perturbations of the Riemannian metric or ask what topological consequences the existence of a Riemannian metric with specific curvature conditions (e.g., positive, non-positive, negative sectional curvature, positive scalar curvature) has. This is the central topic in B1, B2 and B3.
In B1 one studies classes of manifolds in the Gromov Hausdorff topology as a whole. The understanding of phenomena that occur when passing to limit spaces give first hints on structure results and finiteness results in the class under consideration. If the limit of a sequence of n-manifolds has dimension < n one says the sequence collapses. It is conjectured for some classes and known for others that collapse occurs along singular (metric) foliations. This is one of the reasons why singular Riemannian foliations and isometric group actions are also studied in their own right in B1.
In B2 Ricci flow invariant curvature conditions are studied and attempts are made to classify the manifolds in the various classes using the Ricci flow (with surgery).
In B3 manifolds with positive scalar curvature are investigated. There is a link to B1, where we plan to prove the Lott conjecture which predicts that almost non-negatively curved spin manifolds should have a vanishing Â-genus. There is also a link of B3 to B2 since several methods from scalar curvature geometry (conformal change of metric, stable minimal hypersurfaces) can be applied to investigate one of the most important curvature conditions in B2, namely scal ≥ √(2(n-1)(n-2)) ||RW|| where RW denotes the Weyl curvature. However, the main objective in B3 is to study manifolds with positive scalar curvature in their own right. Moreover the hypersurface singularities treated in B3 appear in different guises in higher dimensional geometric flows. Thus the recently developed coarse smoothing of such singularities may be fruitful for B2 as well. In turn some flow arguments in B2 may be treated also by a dimensional reduction argument utilizing minimal hypersurfaces.
Their is still a deep lack of understanding of what positive scalar curvature means: different from sectional or Ricci curvature a lower bound on scalar curvature has relations to global metric properties which become visible only at a large scale. The positive mass theorem is a prototypical result. In particular, when scalar curvature is analyzed via (usually singular) minimal hypersurfaces, one tool shows up many times: one considers positive solutions of elliptic equations on domains of some model space. This is nothing but the Martin boundary and usually investigated by stochastic processes as in C5. The topic of finding obstructions for the existence of positive scalar curvature also appears in B6. There, the unstable Gromov-Lawson Rosenberg Conjecture is investigated which is linked to calculations of ko-homology of classifying spaces of groups.
The notion of curvature has been generalized from Riemannian manifolds to metric spaces and thus, via the Cayley graph, to groups. Examples are the notions of hyperbolic or CAT(0) metric spaces and groups. For groups such curvature conditions can be formulated by asking for specific actions on spaces satisfying the right curvature conditions. Answers to the question what kind of actions a group allows for spaces with a certain geometry reveal a lot about the structure of the group itself. In B4 buildings and actions of groups on them are investigated. The fruitful relationship between Lie groups and symmetric spaces is transferred to a relationship between p-adic groups and buildings. Geometric properties of groups such as being hyperbolic or a CAT(0)- group are used in B5 to prove prominent conjectures due to Bass, Borel, Farrell-Jones, Novikov and Serre for those groups.
A natural object relating geometry and group theory is the fundamental group. One may ask, for instance, what geometric properties of a given space are reflected in its fundamental group. It is well known that the existence of a Riemannian metric of positive, non-negative, non-positive, or negative sectional curvature imposes conditions on the fundamental group. The Singer Conjecture (see B7) makes predictions about the possible L2-Betti numbers of the fundamental group under the assumption that the manifold carries a Riemannian metric of negative or non-positive sectional curvature. The question which elements in the fundamental group come from loops within a ball of a given diameter around the base point is analyzed in B1 for closed Riemannian manifolds whose Ricci curvature is bounded by -1. The question whether a given group is the fundamental group of a closed topological manifold will be addressed in B5.

Via the fundamental group and its action on the universal covering one can attach a lot of interesting theories to a space, i.e., the algebraic K-and L-theory of the group ring, the topological K-theory of the group C*-algebra, the L2-homology, equivariant homology and homotopy, and bounded homology. Interesting and prominent invariants take values in these theories or can be derived from these theories as with finiteness obstructions, torsion invariants, surgery obstructions, characteristic classes, indices of operators, and L2-invariants. They do carry interesting information about the fundamental group and the space itself. The computation of K-and L-groups and equivariant homology and homotopy groups associated to the fundamental group or some group are in the focus of B6, whereas B7 deals with L2-invariants and other homology theories with coefficients in functional analytic spaces. This yields a connection to the project C1, C2, C3 and C5, in particular via the Baum-Connes Conjecture, the theory of von Neumann algebras and random walks.

Rigidity questions are also in the focus of this research area. One of the questions is which properties of a group or space are quasiisometric invariants, a basic topic of B4. Notions like boundaries of hyperbolic groups and asymptotic cones come into play. The measure theoretic analog of quasi-isometry is investigated in B7. A basic problem in B5 is topological rigidity, i.e., the question of when the fundamental group of a topological manifold determines the manifold up to homeomorphism and when a homotopy equivalence between two given topological manifolds is always homotopic to a homeomorphism. These questions are linked to the Farrell-Jones Conjecture.

Classification problems of manifolds lead in the ideal case to a rigidity result, but in general the situation is more complicated. One needs much more refined invariants than the fundamental group. In B6 invariants of equivariant or singular spaces such as indices of operators, characteristic classes, surgery obstructions will be investigated. In B6 the problem of classifying torus bundles over lens spaces is considered. The answer is expected to be rather complicated and involved and not at all a rigidity result. Often structural or finiteness results about members of a class of manifolds comes from the investigation of its closure in the Gromov-Hausdorff topology and the collapsing, a basic topic of B1. In B2 a certain class of Riemannian manifolds MCd(n) is introduced which is described in terms of the Riemannian metric with certain properties and is conjectured to be invariant under the Ricci flow. An open problem is whether one can find topological invariants which give necessary and/or sufficient conditions for a smooth manifold to admit a Riemannian metric such that it lies in this class. Interesting invariants of groups and group actions come from L2-methods, the main topic of B7.
The study of actions and the investigation of the Baum-Connes and the Farrell-Jones Conjecture also requires methods from homotopy theory and its equivariant versions. The equivariant versions are well understood for finite groups and shall be extended to proper actions of discrete groups in B6. A basic test case whether these extensions have the desired properties is to give a proof of the Segal Conjecture in this setting. Classifying spaces attached to groups give interesting links between groups and geometry. They are studied in B6. Buildings (see A4 and B4) often provide nice models for classifying spaces for proper actions. The methods developed in B6 will also be useful for the projects C1, C2 and C3.
It is a natural question to investigate the isometry or diffeomorphism group of a (Riemannian) manifold. The homotopy groups of such spaces are linked via pseudo-isotopy to the Farrell-Jones Conjecture (see B6). The dynamics of automorphism of affine buildings is a topic of B4. On the other hand certain investigations, which are in general very hard, are within reach under the presence of symmetries. The study of the Ricci flow for homogeneous spaces or cohomogeneity one manifolds are typical examples (see B1).

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