Fold cobordisms and stable homotopy groups

We prove that for n ≧ 1 and q > 0 the (oriented) cobordism group of fold maps of (oriented) ( n + q )-dimensional manifolds into ℝ n contains the direct sum of ⌊ q + 1)/2⌋ copies of the ( n − 1)th stable homotopy group of spheres as a direct summand. We also prove that for k ≧ 1 and q = 2 k −1 the cobordism group of fold maps of unoriented ( n + q )-dimensional manifolds into ℝ n also contains the n th stable homotopy group of the space ℝ P∞ as a direct summand. We have the analogous results about bordism groups of fold maps as well.

The Kahn–Priddy theorem

Let ΦSr(X) be the stable homotopy groupwhere SnX means the n-fold suspension of X. For example, the groups ΦSr(S0) are the stable homotopy groups of spheres. Letbe the ‘infinite-dimensional’ orthogonal group. Then topologists are familiar with the ‘stable J-homomorphism’G. W. Whitehead observed that J factors through an ‘even more stable’ J-homomorphismhe conjectured that J′ is epi (for r > 0).