First Betti numbers of orbits of Morse functions on surfaces

Iryna Kuznietsova; Yuliia Soroka

REVIEW 3 major objections 6 minor 13 references

First Betti numbers of orbits of Morse functions on surfaces

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For groups in the class $\mathcal{G}$, the center and the abelianization are both isomorphic to $\mathbb{Z}^{\beta_1(\omega)}$, making the first Betti number of a surface Morse-function orbit equal to the number of $\mathbb{Z}$ symbols in…

desk verdict Solid new algebra computing centers and abelianizations of a natural wreath-product class; the geometric corollary is plausible but needs a path-component qualifier and a precise citation to Mak12. read the letter →

arxiv 1908.03014 v1 pith:GBZLRCRO submitted 2019-08-08 math.GR math.ATmath.GTmath.KT

Iryna Kuznietsova , Yuliia Soroka This is my paper · ORCID

classification math.GRmath.ATmath.GTmath.KT MSC 20E2220F1657T15

keywords wreathproductshomologygroupsMorsefunctionsfirstBettinumbercenterofagroupabelianizationorbitsclassG

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

The paper proves a structural fact about the class $\mathcal{G}$ of groups obtained from the trivial group by finitely many direct products and wreath products $A \wr_n \mathbb{Z}$, the semidirect product of $A^n$ with $\mathbb{Z}$ acting by cyclic shifts. For any group $G \in \mathcal{G}$, the center $Z(G)$ and the abelianization $G/[G,G]$ are isomorphic, and both are free abelian of rank equal to $\beta_1(\omega)$, the number of $\mathbb{Z}$ symbols in any presentation $\omega$ of $G$ in the alphabet $\{1,\mathbb{Z},(,),\times,\wr_2,\wr_3,\dots\}$. Because a quoted classification puts the fundamental groups of path components of orbits of Morse functions on connected compact oriented surfaces other than $S^2$ and $T^2$ inside $\mathcal{G}$, the paper obtains that the first homology of such an orbit is free abelian of that same rank. The result turns a syntactic count in a group word into a geometric invariant, the first Betti number of the orbit.

What carries the argument

The machinery is a pair of recursive descriptions of centers and commutator subgroups of wreath products $A \wr_n \mathbb{Z}$, the semidirect product of $A^n$ with $\mathbb{Z}$ acting by cyclic shifts. Theorem 2.1 computes the center of a wreath product $A\,\mathrm{Wr}_X B$ and its restricted version $A\,\mathrm{wr}_X B$ under a possibly non-effective action of $B$ on $X$: the center is a product or finite direct sum of copies of $Z(A)$ indexed by orbits, times $\ker\phi \cap Z(B)$. For the groups in $\mathcal{G}$, where $B=\mathbb{Z}$ acts by cyclic shifts on $X=\mathbb{Z}_n$, this gives $Z(A \wr_n \mathbb{Z}) \cong Z(A) \times \mathbb{Z}$. The second ingredient, Theorem 3.1, describes $[G \wr_n \mathbb{Z},\, G \wr_n \mathbb{Z}]$ as the tuples $(g_1,\dots,g_n,0)$ with $\prod g_i \in [G,G]$, and Theorem 3.2 turns this into $G \wr_n \mathbb{Z}/[G \wr_n \mathbb{Z},G \wr_n \mathbb{Z}] \cong G/[G,G] \times \mathbb{Z}$. Inducting over the length of a presentation using these two identities yields both halves of Theorem 1.2.

What would settle it

For a group in $\mathcal{G}$, present it as two different words in the alphabet $\{1,\mathbb{Z},(,),\times,\wr_2,\wr_3,\dots\}$ and compare the number of $\mathbb{Z}$ symbols; differing counts would disprove Theorem 1.2. Geometrically, take a non-generic Morse function on a genus-2 oriented surface with two critical points sharing one critical value and compute $\pi_1O_f(f)$; if this group is not in $\mathcal{G}$ or its abelianization is not free abelian of rank $\beta_1(\omega)$, the corollary fails.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.2, is that for every $G \in \mathcal{G}$ and every presentation $\omega$ of $G$, the three abelian groups $Z(G)$, $G/[G,G]$, and $\mathbb{Z}^{\beta_1(\omega)}$ are isomorphic; in particular $\beta_1(\omega)$ depends only on $G$, not on the presentation. The proof has two strands. Theorem 2.6 shows $Z(G) \cong \mathbb{Z}^{\beta_1(\omega)}$ by centering on a description, Theorem 2.1, of wreath-product centers for possibly non-effective actions. Theorem 3.3 shows $G/[G,G] \cong \mathbb{Z}^{\beta_1(\omega)}$ using Theorem 3.1, which identifies the commutator subgroup of $G \wr_n \mathbb{Z}$ as the tuples whose coordinate product lies in $[G,G]$. The geometric corollary then identifies $\beta_1(\omega)$ with the first Betti number of the path component of the orbit of a Morse function on a surface distinct from $S^2$ and $T^2$, via the Hurewicz theorem.

Load-bearing premise

The geometric conclusion rests on the quoted classification, not proved here, that the fundamental group of each Morse-function orbit component on a connected compact oriented surface other than $S^2$ and $T^2$ is built from trivial groups by direct products and cyclic wreath products; if that classification misses cases, the Betti-number corollary does not follow.

Editorial extensions

If this is right

The number $\beta_1(\omega)$ is a well-defined invariant of any group $G \in \mathcal{G}$: different presentations of the same group must contain the same number of $\mathbb{Z}$ symbols.
For a Morse function on a connected compact oriented surface other than $S^2$ and $T^2$, the first homology of the path component of its orbit is a free abelian group of rank $\beta_1(\omega)$.
Since $Z(G)$ and $G/[G,G]$ have the same rank for all $G \in \mathcal{G}$, the center's rank alone gives the first Betti number of the corresponding orbit component.
The abelianization of every group in $\mathcal{G}$ is torsion-free, so the first homology of each such orbit component has no torsion.

Reading between the lines

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

The proof suggests that $\beta_1(\omega)$ counts the infinite cyclic factors in the iterated wreath-product structure; one could try to show it agrees with the rank of the free abelian part of $G$ in any finite presentation, not only the special alphabet, which the paper does not address.
If the quoted classification result is extended to non-oriented surfaces or to Morse functions into $S^1$, the same algebraic theorem will assign first Betti numbers to orbit components in those settings; the extension is not proved here.
The equality $Z(G) \cong G/[G,G]$ for cyclic-shift wreath products invites testing whether other semidirect products $G^n \rtimes \mathbb{Z}$ with shift-like actions have the same property; the paper's argument uses the infinite order of the shift in an essential way.

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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

Colleague,

The algebraic core here is new and sound. For the class G generated from the trivial group by direct products and wreath products A ≀n Z, the paper proves that the center and the abelianization are both free abelian of the same rank, namely the number of Z symbols in any presentation. That is a genuine result. The key tools—Theorem 2.1 on centers of wreath products with non-effective actions and Theorem 3.2 on the abelianization of G ≀n Z—are correct, and the induction in Theorems 2.6 and 3.3 is clean. I do not see circularity or hidden parameters. The citation pattern is appropriate: the authors rely on Maksymenko and Kudryavtseva for the geometric side and on Meldrum for the standard wreath-product center; nothing is self-cited beyond reason.

The soft spots are on the geometric side. First, Corollary 1.4 states H1(O(f)) is free abelian of rank β1(ω), but the proof invokes Hurewicz for the path component Of(f). If the orbit has several path components, H1 of the whole orbit is a direct sum over components. The paper defines Of(f) earlier, so this is an easy fix: add the path-component qualifier to the statement, or prove all components have the same fundamental group and account for the number of components. As written, it is a real gap.

Second, Theorem 1.3 is load-bearing and is cited as “a direct consequence of results of [Mak12]” with no theorem number or page. The exact scope—all Morse functions or only generic ones, target R or S1, possibly non-oriented surfaces—is not pinned down. I do not doubt that [Mak12] contains something close to this, but a reader cannot verify it from the references as given. That is an addressable citation problem, not a fatal flaw.

The abstract also overstates the geometry by saying the rank is the first Betti number of the orbit, without the surface exclusions (S2 and T2) or the path-component qualifier. The exclusions are stated in Theorem 1.3 and Corollary 1.4, so this is presentation, but it should be fixed.

The converse part of Theorem 3.1 is a bit informal—the “same conjugacy class” induction is sketched—but Theorem 3.2 gives a direct homomorphism argument that establishes the same quotient, so the result is not in doubt.

Who is this for? People working on spaces of Morse functions, orbit fundamental groups, and wreath-product groups. The algebraic Lemma 2.2 and Theorem 2.1 are useful beyond the specific application.

I would send this to peer review. A serious referee can verify the Mak12 statement and push for the component fix. With those changes, the paper should be accepted. My own verdict is conditional: the mathematics I can check is correct, and the application is plausible but needs the stated corrections.

Referee Report

3 major / 6 minor

Summary. The paper defines a class G of groups built from the trivial group by direct products and wreath products A ≀n Z, where A ≀n Z is the semidirect product A^n ⋊ Z with the cyclic shift action. Theorem 1.2 asserts that for any G in G and any presentation ω of G in the corresponding alphabet, the center Z(G) and the abelianization G/[G,G] are both free abelian of rank equal to the number of Z symbols β1(ω) in the presentation. The proof computes the center of a wreath product with a non-effective action (Theorem 2.1), the commutator subgroup of A ≀n Z (Theorem 3.1), and then runs an induction on presentations (Theorems 2.6 and 3.3). The final part applies the algebraic result to fundamental groups of orbits of Morse functions on oriented surfaces, invoking [Mak12] to state that these groups lie in G, and deduces a formula for the first Betti number of the orbit.

Significance. The algebraic core is clean and self-contained: the center computation for non-effective actions (Theorem 2.1) is a useful extension of Meldrum's theorem, and the induction in Theorems 2.6 and 3.3 is elegant and does not assume the geometric conclusion. If correct, Theorem 1.2 gives a syntactic invariant of the class G. The geometric application is potentially interesting but is not established in the paper: it rests entirely on an externally quoted classification and contains a path-component error. The paper is honest in separating the algebraic derivation from the geometric input, which is a strength; there is no circularity.

major comments (3)

[Corollary 1.4 and its proof] The statement concludes H1(O(f),Z) ≅ Z^{β1(ω)}, but the proof applies Hurewicz to O(f) as if it were path-connected, while the paper earlier defines Of(f) as the path component and states G = π1Of(f). If O(f) has several path components, H1(O(f),Z) is the direct sum of the homology of its components, and there is no reason for its rank to equal β1(ω). This is load-bearing because the advertised geometric interpretation is the first Betti number of the orbit. The corollary should be restated for Of(f), or the connectedness of O(f) must be established or cited.
[Theorem 1.3] Theorem 1.3 is quoted as a 'direct consequence of results of [Mak12]' without a theorem number, page, or precise statement. It is unclear whether [Mak12] covers all Morse functions or only generic ones, whether orientedness of the surface and the choice P = R vs P = S^1 affect the conclusion, and whether the finite-group quotient cases from [Kud12a] and [Kud12b] are included. Since Corollary 1.4 depends entirely on this external classification, the authors should provide a precise reference and, ideally, the exact statement of the result used.
[Theorem 3.1] The proof of the characterization of [G ≀n Z, G ≀n Z] is incomplete. The displayed elements h2 and f have too many coordinates for n = 2; the argument appears to assume n ≥ 3, and the cases n = 1, 2 are not treated. Moreover, the verification that the element f is a commutator with the given c and d is not carried out, and a direct computation with the stated definitions does not yield the claimed f. The proof also uses the phrase 'same conjugacy class' where 'same coset modulo the commutator subgroup' is meant. Since Theorem 3.2 and Theorem 3.3 rely on this description, the proof must be repaired by a separate treatment of small n and an explicit commutator computation, or by citing a standard reference.

minor comments (6)

[Definition 1.1] In the definition of A ≀n Z, the clause 'n ∈ Z' should be 'n ≥ 1'; as written, negative n is undefined.
[Theorem 1.3 / Corollary 1.4] The notation π1O(f) and π1Of(f) is used inconsistently; since Of(f) is defined as the path component, the theorem and corollary should consistently refer to π1Of(f) or explicitly state the abuse of notation.
[Lemma 2.2] In the case φ_l(y) ≠ y, the displayed equation f(y)g_{y,c}(y) = f(y) omits the dependence on p; one must set p = e to obtain the contradiction. This step should be clarified.
[Theorem 3.1] The element f displayed in the proof has an inconsistent number of coordinates; it should contain exactly n group entries plus the final integer coordinate. Also, 'same conjugacy class' should be 'same coset modulo the commutator subgroup'.
[Corollary 2.5] The notation 'D(A) ×_n Z' is nonstandard and unclear; the proof suggests a direct product with the subgroup nZ. Please use standard notation.
[Abstract and introduction] The abstract states that 'this rank is a first Betti number of the orbit of Morse function'; if the intended statement concerns the path component, the wording should be adjusted to avoid the path-component issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the algebraic derivations are self-contained, and the geometric corollary's dependence on [Mak12] is an external input rather than a circular reduction.

full rationale

The central algebraic result Theorem 1.2 is proved by an explicit induction on the number of letters in a presentation of a group in the class G. Theorem 2.1 and Corollary 2.5 compute Z(A ≀n Z) directly from the semidirect product definition, and Lemma 2.2 and Lemma 2.3 verify the center via centralizer calculations; no step assumes that a group's center rank equals the count of Z-symbols. The induction in Theorem 2.6 decomposes a presentation either as a direct product of two shorter presentations or as ω1 ≀n Z, using only the already-proved center formula plus the evident additivity of β1 under direct and wreath products. Theorem 3.3 repeats the same independent induction for G/[G,G], using Theorem 3.2, which constructs an explicit quotient homomorphism. No parameter is fitted, and no conclusion is used as an input. The geometric corollary invokes Theorem 1.3, quoted from [Mak12], as an external classification saying π1 O(f) ∈ G; this is an imported premise rather than a re-derivation of the target conclusion, so any concern about its scope or proof is a verification and correctness issue, not circularity. There are no load-bearing self-citations by the present authors. Accordingly the score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters and no new entities. Its algebraic proof is self-contained apart from standard group theory; the geometric interpretation rests on one external classification from [Mak12].

assumptions (3)

domain assumption Theorem 1.3 from [Mak12]: for a connected compact oriented surface M distinct from S^2 and T^2 and a Morse function f, pi_1 O(f) belongs to the class G.
External classification result quoted without proof in Section 1; it connects the algebraic theorem to orbits of Morse functions.
standard math Hurewicz theorem: for path-connected X, H_1(X,Z) is isomorphic to pi_1 X / [pi_1 X, pi_1 X].
Used in the proof of Corollary 1.4 to pass from the fundamental group to first homology.
standard math Standard group theory and set-theoretic foundations (ZFC), properties of semidirect products, centers, and commutator subgroups.
Underlying framework for all sections; no special axioms introduced.

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Cite this review

Pith. "Pith review of First Betti numbers of orbits of Morse functions on surfaces." pith.science (2026). https://pith.science/paper/GBZLRCRO

@misc{pith2026190803014,
  author       = {Pith},
  title        = {Pith review of: First Betti numbers of orbits of Morse functions on surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GBZLRCRO}},
  note         = {Machine review of arXiv:1908.03014}
}

abstract

In this article we study algebraic properties of the specific class of groups $\mathcal{G}$ generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. We prove that for any group $G\in\mathcal{G}$ the ranks of the center $Z(G)$ and the quotient by commutator subgroup $G/[G,G]$ coincide. Moreover, this rank is a first Betti number of the orbit of Morse function.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

[1]

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[2]

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, On the homotopy type of spaces of M orse functions on surfaces , Sb. Math. 204 (2013), no. 1, 75--113

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, Topology of spaces of functions with prescribed singularities on the surfaces, Dokl. Akad. Nauk 93 (2016), no. 3, 264--266. 3527003

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[9]

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, Homotopy types of right stabilizers and orbits of smooth functions on surfaces, Ukrainian Math. Journal 64 (2012), no. 9, 1186--1203 (Russian)

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Meldrum, J. D. P., Wreath products of groups and semigroups, Pitman Monographs and Surveys in Pure and Applied Mathematics, 74 (1995), 324

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S. Maksymenko. Homotopy types of right stabilizers and orbits of smooth functions on surfaces. Ukrainian Math. Journal , 64(9):1186--1203, 2012

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Maksymenko

S. Maksymenko. Homotopy types of stabilizers and orbits of M orse functions on surfaces. Ann. Global Anal. Geom. , 29(3):241--285, 2006

2006

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

B. G. Feshchenko, Deformation of smooth functions on 2 -torus whose K ronrod- R eeb graphs is a tree , Topology of maps of low-dimensional manifolds, vol. 12, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., no. 6, Nats\= onal. Akad. Nauk Ukra\" ni, \=Inst. Mat., Kiev, 2015, pp. 204--219

work page 2015

[2] [2]

Bohdan Feshchenko, Actions of finite groups and smooth functions on surfaces, Methods Funct. Anal. Topology 22 (2016), no. 3, 210--219. 3554649

work page 2016

[3] [3]

Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. 1867354

2002

[4] [4]

E. A. Kudryavtseva, Special framed M orse functions on surfaces , Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2012), no. 4, 14--20. 3026876

work page 2012

[5] [5]

Notes 92 (2012), no

, The topology of spaces of M orse functions on surfaces , Math. Notes 92 (2012), no. 1-2, 219--236, Translation of Mat. Zametki 9 2 (2012), no. 2, 241--261. 3201559

work page 2012

[6] [6]

, On the homotopy type of spaces of M orse functions on surfaces , Sb. Math. 204 (2013), no. 1, 75--113

work page 2013

[7] [7]

, Topology of spaces of functions with prescribed singularities on the surfaces, Dokl. Akad. Nauk 93 (2016), no. 3, 264--266. 3527003

work page 2016

[8] [8]

S. I. Maksymenko, Homotopy types of stabilizers and orbits of M orse functions on surfaces , Ann. Global Anal. Geom. 29 (2006), no. 3, 241--285. MR2248072 (2007k:57067)

work page 2006