The proposed project investigates local index theory and secondary invariants of foliations on a closed manifold.

Sara Azzali will focus on eta and torsion forms for foliations both in the Haefliger setting and in the noncommutative one. She will also work on applications of rho invariants and rho classes for foliations, and on the relation between analytically defined objects and constructions via K-theory exact sequences.

Sebastian Goette will focus on large time estimates for heat operators associated to Bismut superconnections. He will also continue to work on torsion invariants both for families and for foliations.

## Publications

We construct a Baum--Connes assembly map localised at the unit element of a discrete group $\Gamma$.

This morphism, called $\mu_\tau$, is defined in $KK$-theory with coefficients in $\mathbb{R}$ by means of the action of the projection $[\tau]\in KK_\mathbb{R}^\Gamma(\mathbb{C},\mathbb{C})$ canonically associated to the group trace of $\Gamma$.

We show that the corresponding $\tau$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture.

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

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

## Team Members

**Dr. Sara Azzali**

Project leader

Universität Potsdam

azzali(at)uni-potsdam.de

**Prof. Dr. Sebastian Goette**

Project leader

Albert-Ludwigs-Universität Freiburg im Breisgau

sebastian.goette(at)math.uni-freiburg.de