Families of Morse functions for manifolds with boundary

Maciej Borodzik; Weronika Buczy\'nska

REVIEW 2 major objections 5 minor 1 cited by

Families of Morse functions for manifolds with boundary

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Pith's one-line read Any path between two Morse functions on a manifold with boundary can be perturbed so that the family is Morse except at finitely many instants, each an interior birth/death, boundary birth/death, or collision.

desk verdict A genuinely new classification with a real, repairable gap: Proposition 3.6, the completeness step for Theorem 1.1, cites an undefined stratum and leaves most codimension counts as 'analogous.' read the letter →

arxiv 2507.15847 v1 pith:3SJTMDXW submitted 2025-07-21 math.GT

Maciej Borodzik , Weronika Buczyńska This is my paper · ORCID

classification math.GT MSC 57Q60

keywords MorsetheorymanifoldswithboundaryCerfequivariantjetsbirth-deathcollisionsplittinghandleslides

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

On a compact manifold with boundary, a generic one-parameter family of smooth functions is Morse at all but finitely many moments, and at each exceptional moment the family passes through one of exactly three local events: a birth or death of a pair of critical points in the interior, a birth or death of a pair of critical points on the boundary, or a collision of an interior critical point with a boundary critical point. The paper proves this list is complete by doubling the manifold and classifying codimension-one singularities of $\mathbb{Z}_2$-equivariant functions. The third event is the new one: unlike the previously studied 'boundary splitting' (which is codimension two), the collision is forced to appear in generic families and changes the stability type of the boundary critical point.

What carries the argument

The engine of the proof is the doubling construction: $N$ is glued to a copy of itself along its boundary to form a closed manifold $M$ with a $\mathbb{Z}_2$-involution $\tau$ whose fixed set is $\partial N$. A function on $N$ that is 'doublable' extends to a smooth $\tau$-invariant function on $M$, and the paper shows every boundary Morse function can be approximated by a doublable one without changing its critical points. In the space of $\mathbb{Z}_2$-equivariant functions, the equivariant jet transversality theorem of Wall (applied through Lemma 3.5) locates the codimension-one strata: $F^1_1$ for a degenerate interior critical point, $F^1_{2,1}$ for a degenerate boundary critical point whose kernel direction lies along the boundary, and $F^1_{2,2}$ for a boundary critical point whose kernel is the normal direction, with a nondegenerate quasihomogeneous quadratic form $B_\Phi$ controlling the higher terms. The quasihomogeneous form (weights $2$ on the tangential variables, $1$ on the normal variable) is the object that makes the $F^1_{2,2}$ case well-defined under coordinate changes, and the versal unfoldings of the three normal forms produce the three events.

What would settle it

Compute, for a concrete case such as $N$ the 2-disk (so $M$ is the 2-sphere), the codimensions of the higher strata $F^{\geq 2}_3$ (interior critical point with one-dimensional kernel and vanishing third derivative) and $F^{\geq 2}_5$ (boundary normal-kernel point with degenerate $B_\Phi$) inside the space of $\mathbb{Z}_2$-equivariant functions. If either codimension equals $1$, a generic one-parameter path must meet it, contradicting the completeness of Theorem 1.1; if both are $\geq 2$, the three-event list survives.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: if $F_0,F_1$ are two Morse functions on a smooth compact manifold $N$ with boundary and $F_\sigma$ is any path between them, then after perturbing the path relative to its endpoints one may assume that $F_\sigma$ is Morse except at finitely many parameters $\sigma_i$, and at each $\sigma_i$ there is exactly one non-Morse critical point whose transition is one of three types: birth/death of interior critical points, birth/death of boundary critical points, or a collision of a boundary and an interior critical point. The proof passes to the double $M = D(N)$ with the involution $\tau$, extends the functions to $\mathbb{Z}_2$-equivariant functions, and applies equivariant jet transversality to show that the only codimension-one strata of equivariant functions are $F^1_1$ (interior), $F^1_{2,1}$ (boundary, kernel tangent to the boundary), and $F^1_{2,2}$ (boundary, kernel normal to the boundary). Each stratum has a normal form, $y^3+\lambda y$, $x^2+y^3+\lambda y$, and $x^4+\lambda x^2$ (plus nondegenerate quadratic terms), whose versal deformations are exactly the three listed events.

Load-bearing premise

The proof requires that every degeneracy beyond the three listed strata occupies codimension at least two in the space of equivariant functions; the paper's dimension count for the higher strata is summarized as 'analogous' rather than carried out in detail, so an overlooked codimension-one stratum would add a fourth event to the list.

Editorial extensions

If this is right

A generic path between two boundary Morse functions can be assumed to avoid boundary splitting; the formerly codimension-two splitting event decomposes into a boundary birth/death followed by a collision.
The three events provide a complete set of moves for Cerf-type arguments on manifolds with boundary, so handle slides and pseudo-isotopy arguments for such manifolds can be built from them.
Every collision changes the stability (boundary stable or unstable) of the boundary critical point involved, so any generic transition that swaps boundary stability types must pass through a collision with an interior critical point.
Since every boundary Morse function can be approximated by a doublable one with the same critical points, the equivariant model captures all generic phenomena of boundary Morse families, not just a restricted subclass.
The classification is local and dimension-independent: the same three normal forms with added quadratic terms describe events in every dimension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

By the same equivariant transversality argument, the three-event list should also describe generic one-parameter families of Hajduk-style m-functions, after the translation explained in the paper; this would unify the two competing definitions of Morse functions with boundary.
The collision event, in which an interior and a boundary critical point merge and the boundary point flips stability, implies that invariants built from boundary Morse functions (such as boundary-relative chain complexes) must contain an explicit term tracking stability changes along homotopies, not just handles.
A testable numerical extension: for random one-parameter families on a low-dimensional bounded manifold (e.g., a disk or a ball), one should observe collisions with nonzero frequency, matching the claim that the collision is a codimension-one event.
The quasihomogeneous form $B_\Phi$ at the collision point may be the seed of a normal form for higher-codimension boundary singularities, suggesting an analogous classification of $k$-parameter families with events of codimension $k$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a genuinely new classification — the boundary-interior collision as a codimension-1 event is the sort of thing people will cite — but the paper as written does not complete the proof of its own Theorem 1.1. Proposition 3.6 is the load-bearing step, and its statement refers to F^{≥2}_4, which is never defined in Section 3.1. The only codimension counts actually carried out are for ~F^1_1 and ~F^1_2; the rest are declared analogous. Since Theorem 1.1 depends on ruling out every other codimension-1 stratum, this is a real gap, not a cosmetic one.

What's good: the three-way distinctions among boundary Morse notions are useful, the translation to doublable functions in Proposition 2.7 is careful, and the three normal forms plus standard unfoldings are explicit and checkable. The collision event — an interior point hitting a boundary point and changing boundary stability — is new to me, and the example in Section 4.2 showing how the codimension-2 boundary splitting decomposes into a birth plus a collision is a nice sanity check. The citation pattern looks fine; the authors' earlier work is used for context and examples, not as a load-bearing input.

Soft spots, in proportion: the main issue is the missing completeness proof. A referee should ask for the codimension computations for F^{≥2}_1, F^{≥2}_2, F^{≥2}_3, F^{≥2}_5 and a definition, or explicit deletion, of F^{≥2}_4. The proof of Proposition 3.15 also relies on versality in the equivariant setting; the argument is sketched rather than verified, and the remark about moduli honestly flags the subtlety. These are likely repairable, but they need to be in the text before the theorem is trustworthy.

Who this is for: topologists working with Morse functions on manifolds with boundary, especially people doing Cerf-theoretic or Floer-theoretic constructions. This deserves a serious referee; it is not a desk reject. Send it out, with the expectation of a revision that fills the completeness gap.

Referee Report

2 major / 5 minor

Summary. This paper studies one-parameter families of smooth functions on a compact manifold with boundary. The main result (Theorem 1.1) states that any path connecting two boundary Morse functions can be perturbed with endpoints fixed so that the family is Morse except at finitely many parameters, and at each exceptional parameter the local change is one of three types: birth/death of an interior critical point pair, birth/death of a boundary critical point pair, or a collision in which an interior critical point meets the boundary and changes the boundary point's stability. The proof doubles the manifold and phrases the problem in terms of Z2-equivariant functions on the double, stratifies the equivariant function space by jet conditions, and uses equivariant transversality (Wall) plus normal forms and versal unfoldings for the three codimension-one strata.

Significance. If the completeness of the list is established, this is a useful and nontrivial contribution: the boundary case has not previously been described in the Cerf-theoretic literature, and the collision phenomenon is genuinely new relative to the closed-manifold case. The paper's strengths are its clear equivariant framework, explicit normal forms in Propositions 3.9-3.11, the concrete standard unfoldings in Section 3.4, and the instructive example in Section 4.2 showing that boundary splitting is a codimension-two process decomposable into codimension-one events. The main caveat is that the central codimension count is not fully written out, so the completeness claim of Theorem 1.1 is not yet rigorously established in the text as it stands.

major comments (2)

[Section 3.2, Proposition 3.6] The proof of Proposition 3.6 is the load-bearing completeness argument for Theorem 1.1, but it contains only the codimension computations for ~F^1_1 and ~F^1_2. For F^{≥2}_3, F^{≥2}_4, F^{≥2}_5 and for all s≥2 multijet strata, the text says the counting is 'analogous' without giving the count. In addition, F^{≥2}_4 is listed in Proposition 3.6 but never defined in Section 3.1, while F^{≥2}_3 is used for two different subspaces (one with p0 outside M^{Z2}, one with p0 on M^{Z2}). Since an unaccounted stratum of codimension one would add an event to the generic list, this gap must be closed by explicit counts or by a systematic stratification argument that covers all named and multijet strata.
[Section 3.5, Proposition 3.15] The versality proof is still a sketch. The paper cites [AGZV12] and states that infinitesimal versality 'can be readily verified,' but the actual verification is replaced by the claim that transversality of the jet map to W is equivalent to transversality of the parameter map T to the RL-orbit. This equivalence is not automatic: the jet map has additional source directions ∂/∂z_i that are not present in T, and one must prove that these directions lie in the tangent space of the orbit or are otherwise harmless. The argument should be written as a lemma, with the equivariant adaptation of the versality theorem made explicit. Because Proposition 3.15 is what turns a transverse crossing of a stratum into one of the three standard unfoldings, this is load-bearing.

minor comments (5)

[Section 3.1] The symbol F^{≥2}_3 is defined twice for different strata, and the text around ~F^1_{2,1} writes F^2_3, probably meaning F^{≥2}_3. Please disambiguate the notation and define F^{≥2}_4 if it is intended to appear in Proposition 3.6.
[Section 3.3, Proposition 3.11] The proof refers to 'Proposition 3.11' where Proposition 3.10 is meant, and the displayed normal form and intermediate formulas write sum_{i=1}^n y_i^2 although the local coordinates contain only n−1 y-variables alongside x.
[Lemma 3.5] The dimension statements 'dim \tilde M = 1 + s dim M' and 'dim \tilde M^s = s + s dim M' are unclear; it should be stated that the relevant domain is \tilde M^s_0 of dimension 1+s dim M, and the intersection computation should be made with respect to this subspace.
[Theorem 1.1 and Section 2.1] The term 'Morse function' in Theorem 1.1 should be tied explicitly to one of the three variants discussed in Section 2.1, presumably 'boundary Morse', to avoid ambiguity about the space of functions to which the perturbation is applied.
[Throughout] There are small typos, including 'critical pint' in the introduction and 'F^1_{2,1} ∪ F^2_3' in Section 3.1, which should be 'F^1_{2,1} ∪ F^{≥2}_3'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the main theorem is derived from external transversality, equivariant Morse theory, and versality results; the proof gap in Proposition 3.6 is a completeness issue, not a circular reduction.

full rationale

Theorem 1.1 is obtained by (i) approximating boundary Morse functions by doublable functions (Proposition 2.7, proved in the paper), (ii) passing to Z2-equivariant functions on the double, (iii) applying Wall's equivariant multijet transversality theorem (Theorem 2.14) to control codimension-one strata, and (iv) invoking AGZV versality to obtain the three normal forms (Proposition 3.15). None of these steps defines the target result in terms of itself. No parameter is fitted and no conclusion of Theorem 1.1 is assumed in the stratification of Section 3.1. The authors' earlier work [BNR16, BM25, BP16] is used for context, terminology, and the boundary Morse lemma; the boundary Morse lemma is not the theorem being proved and does not force the generic-event list. The cited results of Wall, Wassermann, Milnor, and AGZV are external mathematical theorems with stated assumptions that do not include the conclusion of Theorem 1.1. The substantive weakness in the manuscript is a proof gap: Proposition 3.6 asserts the codimension count for several higher strata by analogy, and F^{≥2}_4 is not explicitly defined, so the exhaustiveness of the F^1 decomposition is not fully demonstrated in the text. That is a correctness/completeness risk, not circularity, because the asserted stratification is independent of the theorem's conclusion and is not introduced as a renamed version of the desired bifurcation list. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard external results: Wall's equivariant jet transversality theorem, Wassermann's equivariant Morse theory, and the Arnold-Gusein-Zade-Varchenko versality theorem. No fitted parameters are introduced. The paper's own new content, including the stratification, normal forms, and unfoldings, is derived rather than postulated. Earlier work by the same authors is used only for context and examples.

assumptions (5)

standard math Equivariant jet transversality theorem of Wall (1985, Theorem 2.1) for Z2-manifolds
Used in Subsection 2.5 and Section 3.2 to establish that equivariant Morse functions are residual and that generic paths hit only the codimension-1 strata.
standard math Wassermann's equivariant Morse theory, Lemmas 4.1 and 4.8
Provides the equivariant Morse lemma and the open-density of equivariant Morse functions, cited in Subsection 2.4.
standard math Versality theorem of Arnold, Gusein-Zade and Varchenko, Theorem 8.3
Used in Proposition 3.15 to show that the standard unfoldings of Subsection 3.4 are versal; the paper notes the equivariant adaptation is routine.
domain assumption N is a smooth compact manifold with boundary and F_sigma is a smooth path of functions with boundary Morse endpoints
This is the setting of Theorem 1.1; compactness is used for finiteness of singular parameters and for the collar and bump-function arguments in Proposition 2.7.
standard math The space of smooth equivariant functions on the double is path-connected
Used in the proof of Theorem 1.1 to join any two equivariant Morse functions by an equivariant path that can be perturbed rel endpoints; it holds because the function space is an affine space.

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Pith. "Pith review of Families of Morse functions for manifolds with boundary." pith.science (2026). https://pith.science/paper/3SJTMDXW

@misc{pith2026250715847,
  author       = {Pith},
  title        = {Pith review of: Families of Morse functions for manifolds with boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SJTMDXW}},
  note         = {Machine review of arXiv:2507.15847}
}

read the original abstract

We study 1-parameter families of Morse functions for manifolds with boundary. We list all degeneracies that may occur in generic 1-parameter families.

Figures

Figures reproduced from arXiv: 2507.15847 by the authors.

**Figure 1.** Boundary stable (left) and boundary unstable (right) critical points. We have sketched the flow of the gradient of the corresponding Morse function. Remark 2.4. Failure to openness and density of boundary Morse functions makes it hard to define precisely what is a generic path of boundary Morse functions, Formally one would have to consider e.g. generic functions withing the closure of boundary Morse functions in th… view at source ↗

**Figure 2.** Standard unfolding of a F 1 1 singularity. Left: λ < 0. Middle: λ = 0. Right: λ > 0. We have drawn exemplary gradient flow of Gλ. The dotted part on each picture is the symmetric copy, indicating the τ -action. 3.4. Unfoldings of F 1 1 , F 1 2,1 and F 1 2,2 . Having defined a normal form of each of the three singularities, we pass to describing local unfoldings. Later, in Subsection 3.5, we will show that this unfol… view at source ↗

**Figure 3.** Standard unfolding of a F 1 2,1 singularity. Left: λ < 0. Middle: λ = 0. Right: λ > 0. We present the flow of ∇Gλ [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Standard unfolding of a F 1 2,2 singularity. Left: λ < 0. Right: λ > 0. We present the flow of ∇Gλ. Case λ = 0 differs from λ > 0 only by the speed of convergence to the critical point in the x-direction. This deformation is not as standard as the previous deformations…

**Figure 5.** Figure 5: The behavior of Φλ,µ. The derivative BΦλ,µ Bµ vanishes on x = 0 and on one of the horizontal lines, whose height is µ. The derivative BΦλ,µ Bλ vanishes on one of the hyperbolas (or a degenerate hyperbola) depending on the parameter λ. Critical points of Φλ,µ are read o…

**Figure 6.** Figure 6: Bifurcation of Φλ,µ. The discriminant set {λ = 0} ∪ {λ = −3µ 2 } is drawn. For each of the four regions, we sketch the gradient vector field near the critical locus of Φλ,µ. From this discussion it follows that in the space of parameters we can specify the following fo…

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Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

auser Classics. Birkh\

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko. Singularities of differentiable maps. V olume 1 . Modern Birkh\"auser Classics. Birkh\"auser/Springer, New York, 2012. Classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous based on a previous translation by Mark Reynolds, Reprint of the 1985 edition

work page 2012

[2] [2]

Merging boundary critical points of a M orse function

Maciej Borodzik and Marcin Mielniczuk. Merging boundary critical points of a M orse function. Proc. Amer. Math. Soc. Ser. B , 12:64--77, 2025

work page 2025

[3] [3]

Morse theory for manifolds with boundary

Maciej Borodzik, Andr\'as N\'emethi, and Andrew Ranicki. Morse theory for manifolds with boundary. Algebr. Geom. Topol. , 16(2):971--1023, 2016

work page 2016

[4] [4]

Embedded M orse T heory and R elative S plitting of C obordisms of M anifolds

Maciej Borodzik and Mark Powell. Embedded M orse T heory and R elative S plitting of C obordisms of M anifolds. J. Geom. Anal. , 26(1):57--87, 2016

work page 2016

[5] [5]

J. Cerf. La stratification naturelle des espaces de fonctions diff\'erentiables r\'eelles et le th\'eor\`eme de la pseudo-isotopie. Inst. Hautes \'Etudes Sci. Publ. Math. , (39):5--173, 1970

work page 1970

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B. Hajduk. Minimal m -functions. Fund. Math. , 111(3):179--200, 1981

work page 1981

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Jankowski and R

A. Jankowski and R. Rubinsztein. Functions with non-degenerate critical points on manifolds with boundary. Comment. Math. Prace Mat. , 16:99--112, 1972

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