arxiv: 2605.12989 · v1 · submitted 2026-05-13 · 🧮 math.GT

Recognition: no theorem link

A Lower Bound on the Self-intersections of Fold Singularities

Joshua Drouin, Liam Kahmeyer

Pith reviewed 2026-05-14 02:18 UTC · model grok-4.3

classification 🧮 math.GT

keywords fold mapsself-intersectionssingular setsimmersed boundariesstable mapssurfaceslower boundsgeometric topology

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The pith

A sharp lower bound exists on the number of self-intersections of fold singularities for maps from oriented surfaces to the plane.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a minimum number of times the fold curves must cross themselves in such a map. It first proves the same kind of bound for the boundary curves of any surface immersed in the plane. The authors then treat each connected piece of the fold set as the boundary of a smaller surface piece and apply the immersed-boundary result directly. A reader would care because the bound gives a concrete, computable limit on how simple a fold map can be.

Core claim

We construct a sharp lower bound on the number of self-intersections of such fold singularities. This is done by first establishing a sharp lower bound on the number of self-intersections of the boundary of a surface immersed in R^2. We then construct a sharp lower bound for the number of self-intersections of the singular set of a simple stable fold map of a surface to R^2 by viewing the connected components of the singular set as the boundary components of smaller surface components, and invoking the previously constructed lower bound for the number of self-intersections of an immersed boundary.

What carries the argument

Reduction of each connected component of the fold singular set to the boundary of a smaller immersed surface piece.

If this is right

The bound is achieved by some explicit maps, so it cannot be improved.
The same technique applies to any map whose singular set can be decomposed into closed curves treated as immersed boundaries.
It supplies a numerical invariant that distinguishes certain stable maps from others.
The result quantifies a topological obstruction that every such fold map must satisfy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be tested on explicit fold maps of the sphere or torus to see whether equality holds.
Similar reductions might produce bounds for maps with cusp singularities as well.
The bound may relate to Euler characteristic or other classical invariants of the source surface.

Load-bearing premise

The map is a simple stable fold map so that singular-set components can be treated directly as boundaries of smaller surface pieces.

What would settle it

A simple stable fold map whose singular curves self-intersect fewer times than the bound predicts would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.12989 by Joshua Drouin, Liam Kahmeyer.

**Figure 1.** Figure 1: Immersed Boundaries with Coorientation and Orientation 2.2. The Gauss Map and the Winding Number. Given an orientation preserving immersion f : S Ñ R n of a compact oriented manifold S of dimension n, we have seen that the boundary BS inherits a canonical orientation, while the immersed boundary fpBSq inherits a canonical coorientation. Let ν : BS Ñ S n´1 be a coorienting unit vector field over fpBSq. We s… view at source ↗

**Figure 2.** Figure 2: Curves A ´ 3 , A ` 3 , B ` 3,1 and C ´ 3 . Remark 3.4. We highlight that the family of curves B ` k,1 are those Guth [3] constructed to show the sharpness of his result. In particular, given a surface S with a single boundary component, Guth showed (see Theorem 5.1) that there exists an immersion f such that fpBSq has exactly 2g ` 2 self-intersections. The curve B ` 2´χpSq,1 is precisely the image fpBSq. 4… view at source ↗

**Figure 3.** Figure 3: An inner curve that is positively isotopic to an outer curve. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: A positive concordance from B ` 10,1 to B ` 4,1 \ C ` 1 \ C ` 3 \ C ` 2 Example 4.6. For any decomposition k “ i0 ` ¨ ¨ ¨ ` is of a positive integer k into the sum of positive numbers ij , the curve B ` k,1 is positively concordant to the disjoint union of curves B ` i0,1 \ C ` i1 \ ¨ ¨ ¨ \ C ` is . In [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: , we see a portion of γ along with the basepoint fppq, self-intersection point z, coorientation arrows of γ, and neighborhood V in dashed blue. Additionally, dotted arcs of γ represent unknown behavior away from V that has no impact on this result. We see that γ partitions V into four regions, T, B, L, and R. Recall, from Definition 2.2, that the coorientation points to the side of γ opposite where the sur… view at source ↗

**Figure 6.** Figure 6: Immersion with only Inner Components Definition 5.6 (Inner and Outer Arcs). Given an immersed boundary component γ, a boundary arc of γ is defined as a maximal section of BUpγq between self-intersection points of γ that contains no self-intersections. Recall the definition of inner and outer from Section 2.1. A boundary arc will be denoted as an inner arc, respectively an outer arc, if all points on the bo… view at source ↗

**Figure 7.** Figure 7: Negative Self-Intersections Create Outer Arcs 12 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: Graph Corresponding to Surface Recall that pΣ j i qm denotes path components of the intersection S i `XS j ´, and the collection of all pΣ j i qm is the singular set of f. For notational clarity, we will often omit decorations and refer to the collection as simply Σ. We now wish to endow each of the closed curves in the collection fpΣq with a coorientation and orientation. First, choose a basepoint on each… view at source ↗

**Figure 9.** Figure 9: A single vertebrae The spine of the surface S is comprised of multiple pairs of surface path components, and we call each of these pairs vertebrae, see [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗

**Figure 10.** Figure 10: Spine created from multiple vertebrae The skull of the surface S is comprised of a pair of surface path components, one positive and one negative, see Figure 11a. Each component in this pair is a surface of genus 1 2 pg ´ |Σ| ` 1q, and they are glued together by first selecting |Σ| ` 2 ´ p#|S`| ` #|S´|q disjoint discs on each component. These discs then serve as attaching regions for 1-handle surgery betw… view at source ↗

**Figure 11.** Figure 11: The Skull and Teeth With these sub-surfaces constructed, we now glue them to construct our desired surface S. In [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗

**Figure 12.** Figure 12: The skull, spine, and teeth separate [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗

**Figure 13.** Figure 13: The constructed surface S. With the surface S constructed, we now can define the conditions of a desired simple fold map f : S Ñ R 2 . The desired simple fold map f is one that, without loss of generality: has folds as the cocores of every gluing from our construction above, preserves the orientation 22 [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗

**Figure 14.** Figure 14: Surface from Example 7.7 [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗

**Figure 15.** Figure 15: Biparitite graph corresponding to Example 7.7 two vertebrae together accounts for the final singular set component, which is mapped to a C ` 0 . In totality, the singular set maps to B ` 7,1 \ C ` 0 \ C ` 0 the skull \ B ` 3,1 \ B ` 3,1 the teeth \ C ` 0 \ B ` 3,1 \ C ` 0 \ B ` 3,1 the spine If we count the number of self-intersections of these curves, we get p7 ` 1q ` 0 ` 0 ` p3 ` 1q ` p3 ` 1q ` 0 ` p3 `… view at source ↗

read the original abstract

For an oriented surface $S$, the singular set of a fold map $f:S\rightarrow \mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold singularities. This is done by first establishing a sharp lower bound on the number of self-intersections of the boundary of a surface immersed in $\mathbb{R}^2$. We then construct a sharp lower bound for the number of self-intersections of the singular set of a simple stable fold map of a surface to $\mathbb{R}^2$ by viewing the connected components of the singular set as the boundary components of smaller surface components, and invoking the previously constructed lower bound for the number of self-intersections of an immersed boundary.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The reduction from fold curves to immersed boundaries runs into trouble because folds have rank 1 right on the boundary.

read the letter

The main thing to know is that the claimed sharp lower bound on self-intersections of fold curves relies on a reduction that treats each fold component as the boundary of an immersed surface piece, but the fold condition means the map has rank 1 right along that boundary, which probably breaks the assumptions needed for the immersed-boundary result. They first prove a lower bound on crossings for the boundary of an immersed surface in the plane, then apply it by viewing the singular curves as boundaries of smaller regions in the domain. That reduction is the new step, and it would give a concrete quantitative constraint if it worked cleanly. The initial bound for genuine immersions looks like standard work in the area and probably holds on its own terms. The application to folds is the soft spot. The differential drops rank exactly on the curve, so the smaller pieces are immersions only in the open interior. Standard results for immersed surfaces with boundary usually require the map to be an immersion everywhere, including at boundary points, for local embedding or curvature controls to go through. Without extra argument showing the bound survives this degeneracy, or without examples that actually achieve equality, the sharpness claim does not follow directly. The abstract presents the construction as complete, but the central step needs more justification than is visible here. This is for people working on stable maps and singularity theory who care about counting self-crossings of fold curves. A reader already familiar with the literature on surface-to-plane maps could extract the reduction idea and test it themselves. I would send it to a referee. The argument is short enough to check in detail, and a referee can decide whether the rank-1 issue kills the bound or whether a small fix makes it work.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to construct a sharp lower bound on the number of self-intersections of fold singularities for a simple stable fold map f from an oriented surface S to R^2. The proof proceeds in two steps: first establishing a sharp lower bound on self-intersections of the boundary of an immersed surface in R^2, then applying this bound to the singular set of f by treating each connected component of the singular set as the boundary of a smaller surface component in the domain.

Significance. If the reduction step is valid, the result would supply a new quantitative constraint on the topology of fold curves in stable maps from surfaces to the plane. The explicit claim of sharpness is a strength, as it suggests the bound is attained for some maps, which would make the estimate useful for classification problems in singularity theory.

major comments (1)

[Abstract and the two-step construction] Abstract, application step: the direct transfer of the immersed-boundary lower bound to fold singularities is load-bearing for the central claim but rests on an unverified assumption. By definition a fold singularity satisfies rank(df)=1 along the entire singular curve, so the restriction of f to any closed region whose boundary is a singular component fails to be an immersion at the boundary points. Standard results on immersed surfaces with boundary require injectivity of df on the full tangent space including the boundary; if the immersed-boundary bound uses this non-degeneracy (via local embedding, curvature estimates, or Gauss-Bonnet), the application does not follow without additional argument or a modified proof that accounts for the rank-1 degeneracy.

minor comments (1)

The manuscript should include at least one explicit example of a fold map attaining equality in the bound, together with a verification that the singular set achieves the predicted number of self-intersections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for pinpointing a subtlety in the reduction from fold singularities to immersed boundaries. We agree that the manuscript would benefit from explicit verification that the rank-1 degeneracy does not invalidate the bound, and we will supply the missing argument in revision.

read point-by-point responses

Referee: Abstract, application step: the direct transfer of the immersed-boundary lower bound to fold singularities is load-bearing for the central claim but rests on an unverified assumption. By definition a fold singularity satisfies rank(df)=1 along the entire singular curve, so the restriction of f to any closed region whose boundary is a singular component fails to be an immersion at the boundary points. Standard results on immersed surfaces with boundary require injectivity of df on the full tangent space including the boundary; if the immersed-boundary bound uses this non-degeneracy (via local embedding, curvature estimates, or Gauss-Bonnet), the application does not follow without additional argument or a modified proof that accounts for the rank-1 degeneracy.

Authors: We acknowledge the referee's observation that the restriction of f to a closed region bounded by a fold component is not an immersion of a manifold with boundary, since rank(df)=1 on the boundary. Nevertheless, the proof of the immersed-boundary lower bound depends only on two properties that survive the degeneracy: (i) the map is an immersion on the interior of the region (regular points of f), and (ii) the boundary curve itself is immersed in R^2 (the tangential derivative along the singular curve is nonzero, as the kernel of df is transverse to the curve). Curvature estimates and the Gauss-Bonnet argument in the first part of the paper are applied to the interior and to the immersed boundary curve; they do not require full-rank df on the boundary tangent space. We will revise the manuscript by inserting a short lemma after the immersed-boundary theorem that confirms these two properties hold for the image of each fold-bounded component under a stable fold map, thereby justifying the direct invocation of the bound. No change to the numerical statement or sharpness claim is required. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with independent immersed-boundary bound

full rationale

The paper first derives a sharp lower bound on self-intersections for boundaries of surfaces immersed in R² as an independent step. It then applies this bound to fold singularities by treating singular-set components as boundaries of smaller surface components in the domain. No step reduces the final claim to its inputs by construction, no fitted parameters are renamed as predictions, and no self-citation chain or ansatz smuggling is present. The immersed-boundary result supplies independent content that is not logically forced by the fold application.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction relies on standard domain assumptions from differential topology and singularity theory for fold maps and immersions. No free parameters, new entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)

domain assumption The singular set of a fold map consists of smooth curves that can be treated as boundaries of smaller surface components.
Invoked in the second step of the construction described in the abstract.
domain assumption A sharp lower bound exists for self-intersections of the boundary of an immersed surface in R^2.
This is the first result established and then applied to the fold case.

pith-pipeline@v0.9.0 · 5427 in / 1257 out tokens · 39087 ms · 2026-05-14T02:18:24.550501+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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