Let ξ be an oriented n-dimensional real vector bundle over an oriented closed m-manifold X. An r-field on ξ defined outside a finite subset of X has an index in the homotopy group πm−l(Vn,r) of the Stiefel manifold of r-frames in ℝn. The principal theorems of this paper relate the d and e-invariants of an associated ℝ/2-equivariant stable homotopy class, in certain cases, to computable cohomology characteristic numbers. Results of this type were first obtained by Atiyah and Dupont [5].

This paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie–Poisson systems in particular.

Using the KOℝ/2-theoretic obstruction theory developed in [4] and [5], necessary and sufficient conditions are derived for quaternionic projective spaces ℍPk and odd-dimensional complex projective spaces ℂP2k+1, of real dimension m say, to immerse in Euclidean space ℝ2m−1 in the range l ≦ 14. The results refine those obtained by Davis and Mahowald ([10, 11]) and earlier authors.

In this paper we study the fine geometric structure of a class of strongly continuous semigroups that satisfy the following property: the resolvent of the infinitesimal generator can be represented as the quotient of entire functions of finite exponential type. This class includes the solution map for functional differential equations and certain partial differential equations. In particular, we present necessary and sufficient conditions for one-to-oneness of the solution map and for completeness of the system of generalised eigenfunctions of the generator.