## Prof. Dr. Sebastian Goette

### Project leader

Albert-Ludwigs-Universität Freiburg im Breisgau

E-mail: sebastian.goette(at)math.uni-freiburg.de

Telephone: +49 761 203 5571

Homepage: http://home.mathematik.uni-freiburg.de/g…

## Project

**4**Secondary invariants for foliations

## Publications within SPP2026

In a previous work, a six-parameter family of highly connected 7-manifolds which admit an $\mathrm{SO}(3)$-invariant metric of non-negative sectional curvature was constructed. Each member of this family is the total space of a Seifert fibration with generic fibre $\mathbb S^3$ and, in particular, has the cohomology of an $\mathbb S^3$-bundle over $\mathbb S^4$. In the present article, the linking form of these manifolds is computed and used to demonstrate that the family contains infinitely many manifolds which are not even homotopy equivalent to an $\mathbb S^3$-bundle over $\mathbb S^4$, the first time that any such spaces have been shown to admit non-negative sectional curvature.

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

In this short note we observe the existence of free, isometric actions of finite cyclic groups on a family of 2-connected 7-manifolds with non-negative sectional curvature. This yields many new examples including fake, and possible exotic, lens spaces.

**Related project(s):****4**Secondary invariants for foliations**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