THE GROUP OF IMMERSIONS OF HOMOTOPY $(4k-1)$-SPHERES

Abstract

Regular homotopy classes of immersions of homotopy $n$-spheres into $(n+q)$-space form an Abelian group under connected summation. The subgroup of immersions of those homotopy spheres which bound parallelizable manifolds is computed for $n=4k-1$ and arbitrary codimension $q$. A consequence of this computation is that in codimensions $q<2k+1$, the group of immersed homotopy $(4k-1)$-spheres is not the direct sum of the group of immersed standard spheres and the group of homotopy spheres. Also, our results shed light on Brieskorn's famous equations: for any exotic sphere (bounding a parallelizable manifold), a countable number of distinct equations give rise to codimension-two embeddings of this exotic sphere. It follows from our results that these embeddings lie in pairwise distinct regular homotopy classes, and that any regular homotopy class containing embeddings is representable by a map arising from such an equation.