# 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)) ||R_{W}|| where R_{W} 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 L^{2}-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 L^{2}-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 L^{2}-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 L^{2}-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 M_{Cd(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 L^{2}-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).

## Subprojects

- B1: Singular spaces and foliations
- B2: Geometric evolution equations
- B3: Geometry of scalar curvature
- B4: Reductive groups and combinatorial structures
- B5: Rigidity and K-theory
- B6: Equivariant homotopy and homology
- B7: Measurable group theory and L
^{2}-invariants - B8: Symplectic Geometry - Theory and Applications to Dynamics
- B9: Cobordism categories and applications to geometric topology

## Principal investigators

- Prof. Dr. Peter Albers
- Prof. Dr. Arthur Bartels
- Prof. Dr. Christoph Böhm
- Prof. Dr. Johannes Ebert
- Prof. Dr. Linus Kramer
- Prof. Dr. Clara Loeh
- Prof. Dr. Joachim Lohkamp
- Prof. Dr. Wolfgang Lueck
- Prof. Dr. Roman Sauer
- Prof. Dr. Michael Weiss
- Prof. Dr. Burkhard Wilking
- Prof. Dr. Frederik Witt

## Participating scientists

- Youngjin Bae
- Dr. Noe Barcenas Torres
- Franziska Beitz
- Dr. Gabriele Benedetti
- Paul Bubenzer
- Lukas Buggisch
- Dr. Esther Cabezas-Rivas
- Dr. Brian Clarke
- Georg Frenck
- Dr. Walter Freyn
- Urs Fuchs
- Dr. Fernando Galaz Garcia
- Dr. Gregor Giesen
- William Gollinger
- Dr. Pilar Herreros
- Tuan Khan Hoang Nguyen
- Sebastian Hoelzel
- Florian Jaeger
- Dorothea Jansen
- apl. Prof. Dr. Michael Joachim
- Dr. Jungsoo Kang
- Daniel Kasprowski
- Dr. Martin Kerin
- Svenja Knopf
- Nils Leder
- Jun Young Lee
- Dr. habil. Alexander Lytchak
- PhD Arjun Malhotra
- Rupert McCallum
- Matthias Meiwes
- Adam Mole
- Artem Nepechiy
- Manuel Patzelt
- Dr. Gereon Quick
- Dr. Marco Radeschi
- Christian Rausse
- Raphael Reinauer
- Dr. Nena Roettgen
- Prof. Dr. Peter Schneider
- apl. Prof. Dr. Joerg Schuermann
- Dr. Petra Schwer
- Dr. Daniel Skodlerack
- PhD Iva Spakulova
- Wolfgang Spindeler
- Prof. Dr. Dr. Katrin Tent
- Olga Varghese
- Prof. Dr. Boris Vertman
- Cora Welsch
- Dr. Christoph Winges
- Dr. Stefan Witzel
- Dr. Andreas Woerner

## Publications and preprints of the SFB

- Galaz Garcia, Spindeler, : Nonnegatively curved fixed point homogeneous 5-manifolds
- Wilking, : A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities
- Wilking, Vitali Kapovitch: Structure of fundamental groups of manifolds with Ricci curvature bounded below
- Kerin, Galaz Garcia, : Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension
- Lytchak, Andreas Kollross: Polar actions on symmetric spaces of higher rank
- Lytchak, Marcos Alexandrino: On smoothness of isometries between orbit spaces
- Lytchak, Claudio Gorodski: On orbit spaces of representations of compact Lie groups
- Lytchak, : On contractible orbifolds
- Kerin, : A note on totally geodesic embeddings of Eschenburg spaces into Bazaikin spaces
- Galaz Garcia, Luis Guijarro: Isometry groups of Alexandrov spaces
- Wilking, Cabezas-Rivas, : How to produce a Ricci flow via Cheeger-Gromoll exhaustion
- Witt, Hartmut Weiss: Energy functionals and solution equations for G_2-forms
- Witt, Hartmut Weiss: A heat flow for special metrics
- Böhm, Ramiro Lafuente, Miles Simon: Optimal curvature estimates for homogeneous Ricci flows
- Böhm, Ramiro Lafuente: Immortal homogeneous Ricci flows
- Böhm, Ramiro Lafuente: Real geometric invariant theory
- Böhm, Ramiro Lafuente: The Ricci flow on solvmanifolds of real type
- Wilking, Richard Bamler, Esther Cabezas-Rivas: The Ricci flow under almost non-negative curvature conditions
- Vertman, : Ricci flow on singular manifolds
- Vertman, Eric Bahuaud: Long-time existence of the edge Yamabe flow
- Markus Stroppel: Kernels of Linear Representations of Lie Groups, locally Compact Groups, and Pro-Lie Groups
- Kramer, Theo Grundhoefer, Hendrik Van Maldeghem, Richard M. Weiss: Compact Totally Disconnected Moufang Buildings
- Kramer, : On the local structure and the homology of CAT$(\kappa)$ spaces and euclidean buildings
- Kramer, : The topology of a simple Lie group is essentially unique
- Kramer, : Metric properties of euclidean buildings
- Skodlerack, : The centralizer of a classical group and Bruhat-Tits buildings
- Skodlerack, : Embeddings of local fields in simple algebras and simplicial structures on the Bruhat-Tits building
- Witzel, Kai-Uwe Bux, Ralf Gramlich: Higher finiteness properties of reductive arithmetic groups in positive characteristic: the Rank Theorem
- Kramer, Lytchak, : Homogeneous compact geometries
- Kramer, Karl H. Hofmann: Transitive actions of locally compact groups on locally contractible spaces
- Skodlerack, : Field embeddings which are conjugate under a p-adic classical group
- Skodlerack, : On intertwining implies conjugacy for classical groups
- Wolfgang Lück, David Rosenthal: On the K- and L-theory of hyperbolic and virtually finitely generated abelian groups
- Wegner, : The K-theoretic Farrell-Jones conjecture for CAT(0)-groups
- Bartels, Farrell, Tom: The Farrell-Jones Conjecture for cocompact lattices in virtually connected Lie groups
- Bartels, : The Farrell-Hsiang method revisited
- Winges, : A NOTE ON THE L-THEORY OF INFINITE PRODUCT CATEGORIES
- Pennig, : Twisted K-Theory with Coefficients in C*-Algebras
- Bartels, : On proofs of the Farrell-Jones Conjecture
- Bartels, Wolfgang Lueck, Holger Reich, Henrik Rueping: K- and L-theory of group rings over GL_n(Z)
- Mole, Tibor Macko, Philipp Kuehl: The total surgery obstruction revisited
- Mole, Henrik Rueping: Equivariant Refinements
- Mole, : Extending a metric on a simplicial complex
- Knopf, : Acylindrical Actions on Trees and the Farrell--Jones Conjecture
- Bartels, Christopher L. Douglas, Andre Henriques: Conformal nets IV: The 3-category
- Bartels, Christopher L. Douglas, Andre Henriques: Conformal nets II: conformal blocks
- Bartels, : Coarse flow spaces for relatively hyperbolic groups
- Bartels, Mladen Bestvina: The Farrell-Jones Conjecture for mapping class groups
- Barcenas Torres, : Equivariant Stable Homotopy Theory for Proper Actions of discrete Groups
- Barcenas Torres, : Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy
- Barcenas Torres, : SPACES OVER A CATEGORY AND BROWN REPRESENTABILITY
- Malhotra, : The Gromov-Lawson-Rosenberg conjecture for the Semi-dihedral group of order 16
- Joachim, Wolfgang Lück: Topological K-(co-)homology of classifying spaces of discrete groups
- Quick, : Continuous group actions on profinite spaces
- Quick, : Torsion algebraic cycles and etale cobordism
- Quick, : Some remarks on profinite completion of spaces
- Quick, : Profinite G-Spectra
- Quick, : Continuous Homotopy Fixed Points for Lubin-Tate Spectra
- Quick, Michael J. Hopkins: Hodge Filtered Complex Bordism
- Diarmuid Crowley, Clara Löh: Functorial semi-norms on singular homology and (in)flexible manifolds
- Albers, Will J. Mery: Translated points and Rabinowitz Floer homology
- Albers, Joel W. Fish, Urs Frauenfelder, Otto van Koert: The Conley-Zehnder indices of the rotating Kepler problem
- Albers, Urs Frauenfelder: A Gamma-structure on Lagrangian Grassmannians
- Fuchs, Lizhen Qin: A Geometric Proof of Removal of Boundary Singularities of Pseudo-Holomorphic Curves
- Albers, Will Merry: Orderability, contact non-squeezing, and Rabinowitz Floer homology
- Albers, Urs Frauenfelder: Exponential decay for sc-gradient flow lines
- Albers, Urs Frauenfelder: Square roots of Hamiltonian Diffeomorphisms
- Albers, Kai Cieliebak, Urs Frauenfelder: Symplectic Tate homology
- Albers, Urs Frauenfelder: Bubbles and Onis
- Albers, Kang, : Vanishing of Rabinowitz Floer homology on negative line bundles
- Albers, Urs Frauenfelder, Alexandru Oancea: Local systems on the free loop space and finiteness of the Hofer-Zehner capacity
- Albers, Fuchs, Will Merry: Positive loops and L^infinity-contact systolic inequalities
- Kang, : On reversible maps and symmetric periodic points
- Kang, Urs Frauenfelder: Real holomorphic curves and invariant global surfaces of section
- Kang, Jean Gutt: On the minimal number of periodic orbits on some hypersurfaces in R^{2n}
- Roettgen, Hansjörg Geiges, Kai Zehmisch: From a Reeb orbit trap to a Hamiltonian plug
- Roettgen, Hansjörg Geiges, Kai Zehmisch: Trapped Reeb orbits do not imply periodic ones
- Benedetti, Luca Asselle: Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level
- Benedetti, Luca Asselle: The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles
- Benedetti, Kai Zehmisch: On the existence of periodic orbits for magnetic systems on the two-sphere
- Benedetti, : Magnetic Katok Examples on the two-sphere
- Benedetti, Luca Asselle: Periodic orbits in oscillating magnetic fields on the two-torus
- Benedetti, : Lecture notes on closed orbits for twisted autonomous Tonelli Lagrangian flows
- Meiwes, Kathrin Naef: Translated points on hypertight contact manifolds