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Secondary invariants for foliations

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

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.

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.

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):
4Secondary 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):
4Secondary 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):
4Secondary 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.

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Dr. Sara Azzali