We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rational homology spheres which admit almost complex structures, and infinitely many which do not. We then show that if a closed almost complex manifold has sum of Betti numbers three, then its dimension must be a power of two.

No abstract

No abstract

For compact complex surfaces (M 4 , J) of Kähler type, it was previously shown [31] that the sign of the Yamabe invariant Y (M ) only depends on the Kodaira dimension Kod(M, J). In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we also reprove a result from [2] that explains why the exclusion of class VII is essential here.

The question of which manifolds are spin or spin c has a simple and complete answer. In this paper we address the same question for spin h manifolds, which are less studied but have appeared in geometry and physics in recent decades. We determine that the first obstruction to being spin h is the fifth integral Stiefel-Whitney class W 5 . Moreover, we show that every orientable manifold of dimension 7 and lower is spin h , and that there are orientable manifolds which are not spin h in all higher dimensions. We are then led to consider an infinite sequence of generalised spin structures. In doing so, we determine an answer to the following question: is there an integer k such that every manifold embeds in a spin manifold with codimension k?