Topology of boundary special generic maps into Euclidean spaces

We introduce boundary special generic maps, a class of submersions from manifolds with boundary to Euclidean spaces whose restriction to the boundary has only boundary definite fold points as its singular points. We derive the differential-topological restrictions imposed by the existence of such maps on the global structure of the source manifolds. Furthermore, we apply our results to the non-singular extension problem, which asks when a map on a closed manifold extends to a non-singular map on a manifold with boundary, and obtain new results on non-singular extensions of special generic maps.

💡 Research Summary

The paper introduces a new class of smooth maps called boundary special generic maps. These are submersions from a compact, connected n‑dimensional manifold N with non‑empty boundary to Euclidean space ℝ^k (k < n) such that the interior of N has no singularities, while the restriction to the boundary ∂N has only boundary definite fold points as singularities. This notion extends the well‑studied special generic maps (all singularities are definite folds) to the setting where the source has a boundary, and it mirrors the Morse‑theoretic situation of a function with no interior critical points but only extrema on the boundary.

The authors’ main technical tool is the Reeb space W_F of a map F: N → ℝ^k, obtained by collapsing each connected component of a fiber to a point. They prove that for a boundary special generic map, W_F is a compact, connected, orientable k‑manifold with boundary, the Reeb map \bar F: W_F → ℝ^k is an immersion, and the quotient map q_F: N → W_F is smooth with a diffeomorphism between the singular set of F|∂N and ∂W_F.

Using this structure, the paper classifies the possible diffeomorphism types of N for various target dimensions:

1. Theorem 1.1 (k = 1). N admits a boundary special generic map into ℝ iff N is diffeomorphic to the n‑disk Dⁿ. The proof relies on the fact that the boundary map has exactly two critical points (a maximum and a minimum), making the Reeb space a closed interval, and then invoking classical results on the diffeomorphism groups of disks in all dimensions n ≥ 2.

2. Theorem 1.2 (k = 2, n ≥ 3). N admits such a map into ℝ² iff N is a boundary sum of finitely many D^{n‑1}‑bundles over S¹ (the case of zero summands being Dⁿ). The argument uses a handle decomposition of the 2‑dimensional Reeb space, which consists only of 0‑ and 1‑handles, and pulls this decomposition back to N via q_F.

3. Theorem 1.3 (k = 3, n ≥ 4, n ≠ 6,7). N admits a map into ℝ³ iff N is a boundary sum of finitely many D^{n‑2}‑bundles over S² (again, zero summands give Dⁿ). The proof mirrors that of Theorem 1.2 but in dimension three; the excluded dimensions require separate treatment because of exotic phenomena.

4. Theorem 1.4 (k ≥ 5, n − k = 1, n ≥ 6). If the Reeb space is contractible, then N must be diffeomorphic to Dⁿ. This result depends on classical facts about the diffeomorphism group of the (n‑1)‑disk and the fact that a contractible Reeb space forces the source to be a disk.

These theorems collectively show that the presence of only boundary definite fold points imposes very rigid global topological constraints on the source manifold.

The second part of the paper applies these classification results to the non‑singular extension problem: given a special generic map f : M → ℝ^k on a closed manifold M, does there exist a compact manifold N with ∂N = M and a boundary special generic map F : N → ℝ^k extending f? The authors obtain several striking corollaries:

• Corollary 1.5 (k = 1). A special generic map into ℝ extends non‑singularly iff M is a sphere S^m. Consequently, exotic spheres cannot admit such extensions.

• Corollary 1.6 (k = 2, m ≥ 7). Neither an exotic sphere Σ^m nor the connected sum (S¹ × S^{m‑1}) # Σ^m admits a non‑singular extension to a boundary special generic map into ℝ². The proof combines Theorem 1.2 with Schultze’s computation of the inertia group of S¹ × S^{m‑1}.

• Corollary 1.7 (k = 3, 3‑manifolds). A closed, irreducible 3‑manifold with non‑cyclic fundamental group cannot have a special generic map into ℝ³ that extends non‑singularly to a simply‑connected source. This follows from Theorem 1.3 and basic properties of lens spaces.

• Corollary 1.8 (k = m ≥ 7). An exotic sphere Σ^m does not admit a non‑singular extension to a boundary special generic map with contractible Reeb space into ℝ^m. The argument uses Theorem 1.4 and the fact that exotic spheres are stably parallelizable.

These corollaries demonstrate that the existence of a non‑singular extension imposes stronger constraints than the mere existence of a special generic map. In particular, they link extendability to subtle smooth‑structure phenomena (exotic spheres) and to algebraic invariants such as the fundamental group.

Overall, the paper makes three major contributions:

1. Conceptual Innovation: It defines and systematically studies boundary special generic maps, filling a gap in the literature on singularity theory for manifolds with boundary.

2. Classification Results: Through a blend of Reeb‑space analysis, handle decompositions, and classical diffeomorphism‑group facts, it provides complete diffeomorphism classifications for source manifolds in a wide range of target dimensions.

3. Applications to Extension Problems: By leveraging the classifications, it yields new non‑existence theorems for non‑singular extensions, revealing deep connections between singularity theory, differential topology, and smooth‑structure classification.

The work thus advances our understanding of how boundary singularities control global topology and opens avenues for further exploration of higher‑dimensional boundary phenomena and their impact on extension problems.