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 explicitly calculated for codimension two immersions of (4k − 1)dimensional homotopy spheres. This group is proved to determine the corresponding groups in all codimensions. The results show in particular that in codimensions smaller than 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.