Fold maps and immersions from the viewpoint of cobordism

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. 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 by natural homomorphisms between cobordism groups of immersions. We also compute the ranks of the oriented (right-left) bordism groups of simple fold maps.

💡 Research Summary

The paper investigates cobordism classes of oriented simple fold maps (f\colon M^{n+1}\to N^{n}) and shows that they can be completely described by immersion data with prescribed normal bundles. A simple fold map is a smooth map whose only singularities are fold points, locally modeled by ((x_{1},\dots ,x_{n-1},x_{n}^{2}\pm x_{n+1}^{2})). The singular set (\Sigma_{f}) is a smooth ((n-1))-dimensional submanifold of the source, and its normal bundle in (M) carries essential information about the cobordism class of (f).

The author introduces two geometric invariants. First, the normal bundle of (\Sigma_{f}) can be identified either with the normal bundle of the standard sphere (S^{n-1}) or with the normal bundle of the infinite‑dimensional real projective space (\mathbb{R}P^{\infty}). For each identification one obtains an immersion \