Research Seminar on Topology (Oberseminar Topologie)
Mondays 17-18 in 0.05 (PER 23)

On the homology of the free loop space of a closed manifold $M$ there exists the so-called Chas-Sullivan product. It is a product defined via the concatenation of loops and can, for example, be used to study closed geodesics of Riemannian or Finsler metrics on $M$. In this talk I will outline how one can use the geometry of symmetric spaces to partially compute the Chas-Sullican product. In particular, we will see that the powers of certain non-nilpotent classes correspond to the iteration of closed geodesics in a symmetric metric. If time permits, some results on the Goresky-Hingston cohomology product will also be mentioned. This talk is based on joint work with Maximilian Stegemeyer

In this talk I will provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres. The main tool will be so-called twisted suspensions, which are manifold analogues of the classical suspension operation on a topological space. As application we obtain a complete classification of closed, simply-connected manifolds with a free torus action of cohomogeneity four and new examples of manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions. This is joint work with Fernando Galaz-García.

The classification of compact surfaces is a standard result which is presented in almost all first courses in topology; less common is the classification of noncompact surfaces. This talk will cover some properties of noncompact surfaces, before developing the invariants which classify them. In addition to invariants familiar from the compact classification theorem, we must include the space of ends, and some of its notable subspaces, of the surface. Earlier considerations will also justify additional invariants regarding the boundary components of the surface.

The surgery theorem of Gromov and Lawson is a powerful tool to construct Riemannian manifolds of positive scalar curvature. For the stricter condition of positive Ricci curvature, however, it is not known if a surgery theorem in the same generality holds. If one imposes additional restrictions on the metric and the dimensions involved, then there exist surgery results for positive Ricci curvature by Sha-Yang and Wraith. In this talk we review these results and present a generalization of the surgery theorem of Wraith.

In this talk we use the results of the previous talk to construct metrics of positive Ricci curvature on manifolds obtained by plumbing. As application we show that one obtains infinite families of new examples of simply-connected 6-manifolds with a metric of positive Ricci curvature.

Equivariant cohomology as introduced by A. Borel is an algebraic invariant associated to continuous Lie group actions that, roughly speaking, captures information about the fixed point set of the action, very much like ordinary cohomology encodes the presence of higher-dimensional "holes" on a topological space. Determining the equivariant cohomology of an action is, in general, a challenging task. The purpose of this talk is to introduce a class of actions for which it actually is possible to compute - and to visualize - their equivariant cohomology, the so-called GKM actions. We will briefly review some necessary background on equivariant cohomology, then explain how to associate a graph to GKM actions and how to recover their equivariant cohomology from this graph. Afterwards, we will discuss recent results about or involving GKM actions.

An almost 100 year old conjecture of Hopf states that even-dimensional positively curved manifolds have positive Euler characteristic. While this seems out of reach in full generality, some progress has recently been made assuming in addition that the positively curved metric has some symmetries. If a torus acts on a closed manifold, the Euler characteristic of the manifold is the same as that of the fixed point set, and the problem is reduced to understanding which fixed point sets may occur. We prove that if the metric admits an isometric T⁴-action, then all fixed point components have the rational cohomology ring of spheres or complex or quaternionic projective spaces, in particular have positive Euler characteristic, proving the Hopf conjecture for manifolds with T⁴-symmetry. This builds upon work of Kennard, Wiemeler and Wilking, who proved the same for manifolds admitting a T⁵-action in 2021.

Let $G$ be a Lie group and $BG$ a classifying space. To any $G$-space one can associate its Borel equivariant cohomology, which naturally comes equipped with the structure of a module over the cohomology ring of $BG$. If the equivariant cohomology, considered as a module in this way, is a free module, then the action is said to be equivariantly formal. This condition imposes severe restrictions on the possible topology of the space. For example, if $G$ is a torus acting with finitely many fixed points, then the ordinary cohomology of the space must necessarily vanish in odd degrees. Nonetheless, there are many interesting classes of actions which are equivariantly formal, such as Hamiltonian torus actions on symplectic manifolds. We will show that the isotropy action on certain $\Gamma$-symmetric spaces is equivariantly formal, which also implies that these spaces are formal in the sense of Rational Homotopy Theory.

I present splitting theorems for mean convex subsets in RCD spaces. This extends results for Riemannian manifolds with boundary by Kasue, Croke and Kleiner to a non-smooth setting. A corollary is a Frankel-type theorem. I also show that the notion of mean curvature bounded from below for the boundary of an open subset is stable w.r.t. to uniform convergence of the corresponding boundary distance function.