On the cohomology algebra of some classes of geometrically formal manifolds

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is eit...

We prove the formality and the evenness of odd-degree Betti numbers for compact K\"ahler orbifolds, by adapting the classical proofs for K\"ahler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any K\"ahler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sasakian manifold with Betti numbers $b_1=0$ and $b_2\,> 1$.

We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2<2$. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number $b_2\geq 2$, which is formal. Therefore, such an example does not admit any 3-Sasakian structure. Examples of 7-dimensional simply connected compact formal Sasakian manifolds, with $b_2\geq 2$, are also given.

We discuss the question of geometric formality for rationally elliptic manifolds of dimension $6$ and $7$. We prove that a geometrically formal six-dimensional biquotient with $b_{2}=3$ has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with $b_2\leq 2$ and $b_3=0$ can not be geometrically formal. As it follows from their real homotopy classification, the seven-dimensional geometrically formal rationally elliptic m...

We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle, and are related to symplectic geometry and Seiberg-Witten theory. We also prove that a manifold admits a metric with harmonic forms whose product is not harmonic if and only if it is not a rational homology sphere.

We review topological properties of K\"ahler and symplectic manifolds, and of their odd-dimensional counterparts, coK\"ahler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the K\"ahler/symplectic situation) and the $b_1=1$ case (in the coK\"ahler/cosymplectic situation).

We classify manifolds of small dimension that admit both, a Riemannian metric of non-negative scalar curvature, and a -- a priori different -- metric for which all wedge products of harmonic forms are harmonic. For manifolds whose first Betti numbers are sufficiently large, this classification extends to higher dimensions.

By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from K\"ahler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we explicitly compute the cohomology rings for several cases of interest and give a formula for homotopy equivalence in one particular 7-dimensional case. We also...

In this thesis, we study cohomological properties of non-K\"ahler manifolds. In particular, we are concerned in investigating the cohomology of compact (almost-)complex manifolds, and of manifolds endowed with special structures, e.g., symplectic structures, D-complex structures in the sense of F. R. Harvey and H. B. Lawson, exhaustion functions satisfying positivity conditions.

Let $(M,I,J,K,\Omega)$ be a compact HKT manifold and denote with $\partial$ the conjugate Dolbeault operator with respect to $I$, $\partial_J:=J^{-1}\overline\partial J$, $\partial^\Lambda:=[\partial,\Lambda]$ where $\Lambda$ is the adjoint of $L:=\Omega\wedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{\bullet,0},\partial,\partial_J)$ and $(A^{\bullet,0},\partial,\partial^\Lambda)$ showing a similar behavior to K\"ahler manifolds. In particula...