Codimension one immersions and the Kervaire invariant one problem

Abstract

Let i: M↬ℝn+1 be a self-transverse immersion of a compact closed smooth n-dimensional manifold in (n + 1)-dimensional Euclidean space. A point of ℝn+1 is an r-fold intersection point of the immersion if it is the image under i of (at least) r distinct points of the manifold. The self-transversality of i implies that the set of r-fold intersection points is the image of an immersion of a manifold of dimension n+1-r (the empty set if r > n + 1). In particular, the set of (n + l)-fold intersection points is finite of order, say, θ(i). In this paper we are concerned with the set of values of θ(i) for (self-transverse) immersions of all (compact closed smooth) manifolds of given dimension n.