On the classification of Smale–Barden manifolds with Sasakian structures

Smale–Barden manifolds [Formula: see text] are classified by their second homology [Formula: see text] and the Barden invariant [Formula: see text]. It is an important and difficult question to decide when [Formula: see text] admits a Sasakian structure in terms of these data. In this work, we show methods of doing this. In particular, we realize all [Formula: see text] with [Formula: see text] and [Formula: see text] provided that [Formula: see text], [Formula: see text], [Formula: see text] are pairwise coprime. We give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures. Our method allows us to completely solve the following problem of Boyer and Galicki in the class of semi-regular Sasakian structures: determine which simply connected rational homology 5-spheres admit negative Sasakian structures.

Tropically constructed Lagrangians in mirror quintic threefolds

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve. As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle–Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimension at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, including certain manifolds with non-trivial higher homotopy groups and certain products of aspherical manifolds.

LINEAR INDEPENDENCE IN THE RATIONAL HOMOLOGY COBORDISM GROUP

We give simple homological conditions for a rational homology 3-sphere $Y$ to have infinite order in the rational homology cobordism group $\unicode[STIX]{x1D6E9}_{\mathbb{Q}}^{3}$ , and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when $Y$ is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.

FREE FINITE GROUP ACTIONS ON RATIONAL HOMOLOGY 3-SPHERES

We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.