The Kervaire invariant of immersions

Abstract

where f2. G is an appropriate cobordism theory of immersed submanifolds of Euclidean space. By generalizing techniques of Browder [2] we shall give necessary and sufficient Adams spectral sequence conditions for an element in t2. G to have nonzero Kervaire invariant. Also we will prove that for every j > 1 there exists a closed, differentiable manifold of dimension 2 ) § together with an immersion in R2s§ which has nonzero Kervaire invariant. Before we state our results more precisely, we first recall some preliminaries about the Kervaire invariant. The classical Kervaire invariant of stably framed corbodism, K: f24kf + 2---'Z/2, is the obstruction to a framed cobordism class containing a framed homotopy sphere. The question of the existence of smooth, stably framed, closed manifolds with nonzero Kervaire invariant has intrigued topologists since the early 1960's. By using the Thom-Pontr jagin construction which equates framed cobordism, f2 fr, with the stable homotopy groups of spheres, n,(S~ W. Browder gave necessary and sufficient conditions on the mod 2 Adams spectral sequence f r s 0 for there to exist elements in f 2 . r c . ( S ) which have Kervaire invariant one. In [2] he proved the following.