Fold maps and immersions from the viewpoint of cobordism

Abstract

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n + 1)-dimensional manifolds into an n-dimensional manifold Nn in terms of immersions with prescribed normal bundles. We compute that for Nn = Rn the cobordism group of simple fold maps is isomorphic to the direct sum of the (n - 1)-st stable homotopy group of spheres and the (n - 1)-st stable homotopy group of the space RP8. By using geometric invariants defined in the author�s earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group in terms of natural homomorphisms between cobordism groups of immersions. We also compute the ranks of the oriented bordism groups of simple fold maps.