We give a bordism‐theoretic characterization of those closed almost contact (2q+1)‐manifolds (with q⩾2) that admit a Stein fillable contact structure. Our method is to apply Eliashberg's h‐principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7‐manifold with torsion‐free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.

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Let X be a closed m-dimensional spin manifold which admits a metric of positive scalar curvature and let R + (X) be the space of all such metrics. For any g ∈ R + (X), Hitchin used the KO-valued α-invariant to define a homomorphism An−1 : πn−1(R + (X), g) → KOm+n. He then showed that A0 = 0 if m = 8k or 8k + 1 and that A1 = 0 if m = 8k − 1 or 8k.In this paper we use Hitchin's methods and extend these results by proving that A8j+1−m = 0 whenever m ≥ 7 and 8j − m ≥ 0. The new input are elements with non-trivial α-invariant deep down in the Gromoll filtration of the group Γ n+1 = π0(Diff(D n , ∂)). We show that α(Γ 8j+2 8j−5 ) = {0} for j ≥ 1. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.

We define a Z 48 -valued homotopy invariant ν(ϕ) of a G 2 -structure ϕ on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. For manifolds of holonomy G 2 obtained by the twisted connected sum construction, the associated torsion-free G 2 -structure always has ν(ϕ) = 24. Some holonomy G 2 examples constructed by Joyce by desingularising orbifolds have odd ν.We define a further homotopy invariant ξ(ϕ) such that if M is 2-connected then the pair (ν, ξ) determines a G 2 -structure up to homotopy and diffeomorphism. The class of a G 2 -structure is determined by ν on its own when the greatest divisor of p 1 (M ) modulo torsion divides 224; this sufficient condition holds for many twisted connected sum G 2 -manifolds.We also prove that the parametric h-principle holds for coclosed G 2 -structures. Proposition 1.10. Let f : M ∼ = M be a spin diffeomorphism with mapping torus T f . Then D(ϕ, f * ϕ) = 24 A(T f ) ∈ Z.

ABSTRACT. A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds.In this paper, we use information about the degrees of maps between manifolds to construct new functorial semi-norms with interesting properties. In particular, we answer a question of Gromov by providing a functorial semi-norm that takes finite positive values on homology classes of certain simply connected spaces. Our construction relies on the existence of simply connected manifolds that are inflexible in the sense that all their self-maps have degree −1, 0, or 1. The existence of such manifolds was first established by Arkowitz and Lupton; we extend their methods to produce a wide variety of such manifolds.