Manuel Amann

On a generalized conjecture of Hopf with symmetry

A famous conjecture of Hopf is that the product of the two-dimensional sphere with itself does not admit a Riemannian metric with positive sectional curvature. More generally, one may conjecture that this holds for any nontrivial product. We provide evidence for this generalized conjecture in the presence of symmetry.

Positive curvature and rational ellipticity

Simply-connected manifolds of positive sectional curvature

Computational Complexity of Topological Invariants

M

PARTIAL CLASSIFICATION RESULTS FOR POSITIVE QUATERNION KÄHLER MANIFOLDS

are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include a small upper bound on the Euler characteristic and confirmations of famous conjectures by Hopf and Halperin under additional torus symmetry. We prove several cases (including all known even-dimensional examples of positively curved manifolds) of a conjecture by Wilhelm.

Topological properties of positively curved manifolds with symmetry

We answer the following question posed by Lechuga: Given a simply-connected space

Non-formal homogeneous spaces

X

Degrees of Self-Maps of Simply Connected Manifolds

with both

Mapping degrees of self-maps of simply-connected manifolds

H_*(X,\qq)

A note on the Hilali conjecture

and

Geometrically Formal Homogeneous Metrics of Positive Curvature

\pi_*(X)\otimes \qq