Positive curvature and symmetry in small dimensions

Extending existing work in small dimensions, Dessai computed the Euler characteristic, signature, and elliptic genus for [Formula: see text]-manifolds of positive sectional curvature in the presence of torus symmetry. He also computes the diffeomorphism type by restricting his results to classes of manifolds known to admit non-negative curvature, such as biquotients. The first part of this paper extends Dessai’s calculations to even dimensions up to [Formula: see text]. In particular, we obtain a first characterization of the Cayley plane in such a setting. The second part studies a closely related family of manifolds called positively elliptic manifolds, and we prove a conjecture of Halperin in this context for dimensions up to [Formula: see text] or Euler characteristics up to [Formula: see text].

The diffeomorphism type of small hyperplane arrangements is combinatorially determined

It is known that there exist hyperplane arrangements with the same underlying matroid that admit non-homotopy equivalent complement manifolds. Here we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and the same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined, that is, they depend only on the underlying matroid. To prove this, we associate to every such matroid a topological space, that we call the reduced realization space; its connectedness, shown by means of symbolic computation, implies the desired result.

On Subcritically Stein Fillable 5-manifolds

We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case where the fundamental group is finite cyclic, and we show that on the 5-sphere, the standard contact structure is the unique subcritically ?llable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces

A surface ∑ endowed with a Poisson tensor π is known to admit a canonical integration, 𝒢(π), which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if π is not an area form on the 2-sphere, then 𝒢(π) is diffeomorphic to the cotangent bundle T*∑. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.

DOUBLE KNOT SURGERIES TO S4 AND S2 × S2

It is known that S4 is a union of two fishtails, and S2 × S2 is a union of two cusps (glued along their boundaries). Here we prove that for any choice of knots K, L ⊂ S3 the knot surgery operations [Formula: see text] and S2 × S2 ⇝ (S2 × S2)K, L along both of these fishtails and cusps, respectively, do not change the diffeomorphism type of these manifolds.