## Dr. Martin Kerin

### Project leader

WWU Münster

E-mail: m.kerin(at)uni-muenster.de

Telephone: +49 251 83-32731

Homepage: https://www.uni-muenster.de/IVV5WS/WebHo...

## Publications within SPP2026

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjecture in the presence of a slice-maximal torus action is proved.

Journal | Int. Math. Res. Not. IMRN |

Volume | 18 |

Pages | 5786--5822 |

Link to preprint version | |

Link to published version |

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds**15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected 5-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension 4 are connected sums of *\(S^3\)*-bundles over \(S^2\). Furthermore, the Betti numbers of the 5-manifolds and of the quotient 4-manifolds are related by a simple formula involving the number of fixed-point components. We also investigate semi-free \(S^3\) actions on simply connected 8-manifolds with quotient a 5-manifold and show, in particular, that the Pontrjagin classes, the \(\hat A\) -genus and the signature of the 8-manifold must all necessarily vanish.

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds

In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.

**Related project(s):****4**Secondary invariants for foliations**11**Topological and equivariant rigidity in the presence of lower curvature bounds

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds

Let \(M^n, n \in \{4,5,6\}\), be a compact, simply connected *n*-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on \(M^n\) by a torus \(T^{n-2}\) is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

Journal | Math. Z. |

Volume | 276 |

Pages | 133--152 |

Link to preprint version | |

Link to published version |

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds