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 smooth fold maps f : Q^{n+1} → N^{n}, where Q^{n+1} and N^{n} are smooth manifolds of dimensions n + 1 and n respectively. A fold map is locally modeled by the normal form (x₁,…,x_{n‑1}, x_n² ± x_{n+1}²); the sign “+” gives a definite fold singularity, the sign “–” an indefinite one. The set of all singular points S_f is a smooth (n‑1)-dimensional submanifold of Q^{n+1}, and the restriction f|_{S_f} is a codimension‑one immersion into N^{n} when the map is in general position.

A simple fold map is defined as a fold map whose each connected component of any fiber contains at most one singular point. In the oriented setting this forces the indefinite singular fibers to be disjoint unions of a finite number of “figure‑eight” fibers and circles. This restriction dramatically simplifies the structure of the germ bundles associated with the singularities.

The central idea of the work is to encode the cobordism class of a simple fold map by two immersion data: the immersion of the indefinite singular set together with a prescribed normal bundle induced from the universal line bundle ε¹ → ℝP^∞, and the immersion of the definite singular set with normal bundle induced from the trivial line bundle over ℂP^∞. These give rise to homomorphisms  I_s : Cob^s(N) → Imm_{ε¹ℝP^∞}(n‑1, 1) and D_s : Cob^s(N) → Imm_{ε¹}(n‑1, 1), and analogous maps I_f, D_f for general fold maps.

The main theorem treats the case N = ℝⁿ. Using a Pontryagin–Thom construction, the authors show that the cobordism group of oriented simple fold maps  Cob^s(ℝⁿ) is naturally isomorphic to the direct sum of two stable homotopy groups:  Cob^s(ℝⁿ) ≅ π_{n‑1}^s(S⁰) ⊕ π_{n‑1}^s(ℝP^∞). Here π_{*}^s denotes the stable homotopy groups of spheres; the first summand corresponds to the immersion cobordism of the definite singular set (trivial normal bundle), while the second summand captures the immersion cobordism of the indefinite singular set with its ε¹‑twist over ℝP^∞. The proof proceeds by collapsing the one‑point compactification of ℝⁿ, mapping it into the Thom space of the universal line bundle, and then interpreting the resulting map in stable homotopy terms.

The paper also describes the natural inclusion  Cob^s(ℝⁿ) → Cob^f(ℝⁿ) from simple fold cobordism to full fold cobordism. This inclusion is realized on the immersion side by the homomorphism  Imm_{ε¹ℝP^∞}(n‑1, 1) → Imm_{det(γ₁×γ₁)}(n‑1, 1), induced by the diagonal inclusion ℝP^∞ → ℝP^∞ × ℝP^∞ of the classifying spaces of the line bundles. In other words, the extra structure present in a simple fold map (the Z₂‑reduction of the indefinite germ’s automorphism group) is forgotten when passing to general fold maps, and this loss is precisely encoded by the above bundle map.

Beyond the primary computation, the authors compute the ranks of the oriented (right‑left) bordism groups of simple fold maps, denoted Ω_n^{sf}. They show that these ranks coincide with the free part of the previously identified cobordism groups, confirming that simple fold maps impose strong topological constraints on the source manifolds, as observed in earlier works by Levine, Saeki, and others.

The paper also contains a detailed analysis of the germ bundles associated with definite and indefinite folds. For indefinite folds the universal germ bundle has structure group Z₂ ⊕ Z₂, which reduces to a diagonal Z₂ in the simple (or framed) case; this reduction yields the trivial line bundle over ℝP^∞ as the target of the universal bundle. For definite folds the target is the trivial line bundle over ℂP^∞, reflecting the fact that definite singularities have oriented normal bundles.

In low dimensions (e.g., n = 2, 3) the authors give explicit calculations, illustrating how the abstract stable homotopy groups translate into concrete cobordism invariants. They also discuss how the geometric invariants I_s, D_s, I_f, D_f, introduced in the author’s earlier works, provide a complete set of invariants for distinguishing cobordism classes.

Overall, the paper establishes a clear and computable bridge between the theory of fold maps and classical immersion cobordism, showing that for simple fold maps the cobordism classification reduces to well‑understood stable homotopy groups. This not only yields explicit algebraic invariants but also clarifies the relationship between simple and general fold cobordism via natural bundle homomorphisms. The results deepen our understanding of how singularity theory, immersion theory, and cobordism interact, and they provide tools that can be applied to further investigations of singular maps in higher codimensions.