A compact, connected, combinatorial 4-maifold is embeddable in R^7 if its twisted normal Stiefel-Whitney class in dimension 3 is trivial.

Introduction. LetXbe a submanifold ofY, in either the topological, smooth, or piecewise linear ( =PL) categories. Anormal cell bundleonXinYis a bundle ξ = (p, E, X) in the category whose fibre is a closed cell, and such thatEis a neighbourhood ofXinYandp: E → Xis a retraction. The triple (Y, X, ξ) is atubular neighbourhood, or briefly, atube. For convenience we may refer to a tube by its cell bundle.

Let X be an analytic vector field on a real or complex 2-manifold, and K a compact set of zeros of X whose fixed point index is not zero. Let A denote the Lie algebra of analytic vector fields Y on M such that at every point of M the values of X and [X,Y] are linearly dependent. Then the vector fields in A have a common zero in K. Application: Let G be a connected Lie group having a 1-dimensional normal subgroup. Then every action of G on M has a fixed point.

Abstract. A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The principal result is that limit sets of such systems cannot be more complicated than invariant sets of systems of one lower dimension. In fact orthogonal projection along any positive direction maps a limit set homeomorphically and equivariantly onto an invariant set of a Lipschitz vector field in a hyperplane. Limit sets are nowhere dense, unknotted and unlinked. In dimension 2 every trajectory is eventually monotone.In dimension 3 a compact limit set which does not contain an equilibrium is a closed orbit or a cylinder of closed orbits.

Immersions of an m-manifold in an n-manifold, n>m, are classified up to regular homotopy by the homotopy classes of sections of a vector bundle E associated to the tangent bundle of M.  When N = Rⁿ , the fibre of E is the Stiefel manifold of m-frames in n-space.