## Dr. Sara Azzali

### Project leader

Universität Potsdam

E-mail: azzali(at)uni-potsdam.de

Telephone: +49 331 977-1629

Homepage: http://www.math.uni-potsdam.de/~azzali/

## Project

**4**Secondary invariants for foliations

## Publications within SPP2026

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 $\Gamma$. The right hand side of $\mu_\tau$ is functorial with respect to the group $\Gamma$.

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

We construct $\eta$- and $\rho$-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

Journal | Math. Proc. Camb. Philos. Soc. |

Publisher | Cambridge University Press |

Volume | August 2018 |

Link to preprint version | |

Link to published version |

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

The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral $\zeta$-invariants using lifted defect formulae which express discrepancies of $\zeta$-regularised traces in terms of Wodzicki residues. We derive Atiyah's $L^2$-index theorem as an instance of the $\mathbb{Z}_2$-graded generalisation of the canonical lift of spectral $\zeta$-invariants and we show that certain lifted spectral $\zeta$-invariants for geometric operators are integrals of Pontryagin and Chern forms.

Journal | Trans. Amer. Math. Soc |

Volume | to appear |

Link to preprint version |

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