A Nielsen type periodic number for maps over $B$

Rafael Moreira de Souza; Weslem Liberato Silva

REVIEW 3 major objections 3 minor 14 references

A Nielsen type periodic number for maps over $B$

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper introduces a Nielsen-type periodic number $NBP_n(f)$ for fiber-preserving maps over a base space $B$, proves it is homotopy invariant over $B$, and computes it for $S^1$-bundles over $S^1$.

desk verdict A natural fiberwise Nielsen periodic number with a real, fixable counting error in the main theorem; worth refereeing. read the letter →

arxiv 2411.13233 v1 pith:N2YAZZL7 submitted 2024-11-20 math.AT

Weslem Liberato Silva , Rafael Moreira de Souza This is my paper · ORCID

classification math.AT MSC 55M2055R1037C25

keywords NielsenperiodicnumberReidemeisterclassesoverBfiber-preservingmapspointsfibrationshomotopyinvarianceminimalperiodstorusbundles

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

This paper extends the classical Nielsen periodic number from ordinary self-maps to fiber-preserving maps over a fixed base space $B$. Its central object is $NBP_n(f)$, defined as $n$ times the number of irreducible essential periodic orbits of Reidemeister classes over $B$, and the paper proves that this number is invariant under homotopies over $B$ and is a lower bound for the minimal number of periodic-point orbits among all maps homotopic over $B$ to $f$. When $B$ is a single point, $NBP_n(f)$ reduces to the classical Nielsen periodic number $NP_n(f)$. The paper also isolates a class of $n$-toral maps over $B$ for which $NBP_n(f)$ equals the inclusion-exclusion count $A_n(f)$, and computes $NBP_n(f)$ explicitly for $S^1$-bundles over $S^1$.

What carries the argument

The load-bearing mechanism is the fibrewise Reidemeister equivalence. For a point $x_0$ and a path $\omega$ in the fiber $p^{-1}(p(x_0))$ from $x_0$ to $f(x_0)$, the action $[c]*_B[\theta]=[\tilde H(1,s)*\omega^{-1}]$ is defined by lifting the base path $p(c)$ to a homotopy whose vertical sides are $\theta*\omega$ and $f(c)$; equivalence classes are the algebraic Reidemeister classes over $B$, and by Theorem 2.6 they are in bijection with the path-components of the space $E_B(f)$ of pairs $(x,\alpha)$ with $\alpha$ a path in the fiber from $x$ to $f(x)$. The periodic theory is carried by the induced map $[f_\omega]$ acting on classes of $f^n$, the maps $[\gamma_{m,n}]$ for $m|n$, and the notions of length, depth and irreducibility; $O_n(f)$ counts the irreducible essential orbits, and $NBP_n(f)$ multiplies this count by $n$. This setup is what makes homotopy invariance over $B$ and the inclusion-exclusion formula possible.

What would settle it

Take $B=S^1$ and let $E$ be the mapping torus of the transposition of a two-point fiber, so $E$ and $B$ are path-connected but the fiber is $S^0$. A fiber map over $B$ that swaps the two points of a fiber has no path $\omega$ in that fiber from $x_0$ to $f(x_0)$, so Definition 2.1, Theorem 2.6 and Proposition 4.2 cannot be formulated; this example marks exactly where the connected-fiber premise is load-bearing.

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Extended reading notes

Core claim

The central claim is that the Nielsen periodic count can be carried out fibrewise and behaves like its classical counterpart. For a fibration $Y\to E\to B$ and a fiber map $f:E\to E$ over $B$, the paper defines algebraic Reidemeister classes over $B$ for $f^n$ via a twisted action of $\pi_1(E,x_0)$ on $\pi_1(Y,x_0)$, using a path $\omega$ in the fiber from $x_0$ to $f(x_0)$. It shows these classes are in bijection with the path-components of the fiber space $E_B(f^n)$, so the geometric fixed-point classes over $B$ inject into them. The resulting number $NBP_n(f)=n\,O_n(f)$, where $O_n(f)$ counts irreducible essential periodic orbits of these classes, is proved to be a homotopy invariant over $B$ and to satisfy $NBP_n(f)\le\min\{\#\pi_0(P_n(g)):g\sim_B f\}$. Under the $n$-toral condition and the assumption that $0\neq NB(f^m)=RB(f^m)$ for every $m|n$, the paper proves $A_n(f)=NBP_n(f)$, and for $T=S^1\times S^1$ over $S^1$ with $f$ homotopic over $S^1$ to $f_{1,s}$, $s\neq0$, it yields $N_{S^1}P_n(f)=|s|\,p_1^{\alpha_1-1}\cdots p_l^{\alpha_l-1}(p_1-1)\cdots(p_l-1)$.

Load-bearing premise

Every fiber $Y=p^{-1}(b)$ of the fibration is assumed path-connected, because Definition 2.1 needs a path $\omega$ lying entirely inside a single fiber from $x_0$ to $f(x_0)$; if a fiber is disconnected, such a path need not exist and the Reidemeister classes over $B$ are undefined.

Editorial extensions

If this is right

If $NBP_n(f)\neq 0$, then $n$ belongs to the homotopy-invariant period set $HBPer(f)$, so the invariant is a tool for detecting periods that persist under homotopies over $B$.
When $B$ is a point, $NBP_n(f)$ collapses to the classical Nielsen periodic number $NP_n(f)$, recovering the ordinary theory as a special case.
For $S^1$-bundles over $S^1$, the closed formula $N_{S^1}P_n(f)=|s|\,p_1^{\alpha_1-1}\cdots(p_l-1)$ turns the invariant into an explicit count in terms of one integer $s$.
The inequality $NBP_n(f)\le\min\{\#\pi_0(P_n(g)):g\sim_B f\}$ converts the algebraic count into a lower bound on periodic-point orbits for every map homotopic over $B$ to $f$.

Reading between the lines

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

The same depth-and-length machinery may yield closed formulas for other fiber bundles, such as nilmanifold or infranilmanifold bundles, where the fundamental-group action on the fiber is more complicated than in the $S^1$ case.
If a sharpening theorem over $B$ were established, the inequality in Proposition 4.2 would become an equality; the paper does not prove such a result.
The conditions in Theorem 5.3 might be relaxed to a weaker essentiality assumption, which would make $NBP_n(f)$ computable for a larger class of fiber maps.

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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

Short version: this is a competent, incremental extension of Heath–Piccinini–You periodic Nielsen theory to maps over a base, using the Gonçalves–Koschorke fiberwise Nielsen theory. The new invariant NBP_n(f) is natural, homotopy invariant over B, and does what a Nielsen periodic number should. But Theorem 5.3, the main computational result, has a proof gap: the counting sum runs over the wrong set of divisors. As printed, the equality is false. That needs fixing.

Credit where due: the paper defines algebraic periodic Reidemeister classes over B, gives them a depth and length, and packages them into NBP_n(f). The classical number is recovered when B is a point, which is the right sanity check. The homotopy invariance argument in Proposition 4.2(i) is mostly careful — the path homotopy over B is worked out in some detail. The torus computation in Proposition 5.8 gives a clean closed formula, the kind of concrete result that makes the theory usable. The n-toral and NB=RB hypotheses are stated plainly as assumptions, not smuggled in.

Soft spots, in proportion. The Theorem 5.3 proof: Definition 5.2 sets A_m(f) = NB(f^m) − Σ_{k|m,k

Second, the standing assumption that all spaces are path-connected is ambiguous about the fiber. Definition 2.1 needs a path ω inside the fiber; if fibers are not path-connected, the construction collapses. Path-connected E and B do not force path-connected fibers. The authors should say explicitly that the fiber is path-connected (or adjust the definition).

Minor: the lower-bound proof in Proposition 4.2(ii) is a bit compressed but the idea is standard and likely fine.

Bottom line: the paper is a solid incremental contribution, not a breakthrough. The right audience is people in Nielsen theory or fiberwise fixed point theory. It deserves a serious referee; the main theorem is plausible but unproved as written. Send it out, ask for the divisor-sum correction and a clarification of the fiber assumption.

Referee Report

3 major / 3 minor

Summary. The paper introduces a Nielsen-type periodic number NBP_n(f) for fiber-preserving self-maps f:E→E over a base B, extending the classical Nielsen periodic number NP_n. It defines algebraic and geometric Reidemeister classes over B, studies orbits and depths of periodic classes, and proposes a lower bound NBP_n(f) ≤ min{#π0(P_n(g)) | g ∼_B f}. Under an n-toral hypothesis and the assumption NB(f^m)=RB(f^m)≠0 for all m|n, it claims A_n(f)=NBP_n(f), where A_n is defined by Möbius inversion over the divisors of n. The paper then applies this to S^1-bundles over S^1 and derives an Euler-phi-like formula in Proposition 5.8.

Significance. If the central definition and theorems were correct, the paper would contribute a natural Nielsen periodic number over a base space, with a homotopy-invariant lower bound and explicit computations for S^1-bundles. The algebraic framework of Reidemeister classes over B and the use of standard fibration and index theory from [3] and [14] are sensible, and the paper has no fitted parameters or circular dependencies. However, the main invariant as defined does not satisfy the advertised lower bound, and the proof of the key equality in Theorem 5.3 has a false divisor summation. These are load-bearing issues that currently prevent acceptance.

major comments (3)

[Definition 4.1 and Proposition 4.2(ii)] The definition NBP_n(f)=n×O_n(f), where O_n(f) counts orbits of irreducible essential Reidemeister classes, is not compatible with the claimed lower bound. Consider the paper's own example setup with T=S^1×S^1, p(x,y)=y, and f(x,y)=(xy,y); this is f_{1,1} in Proposition 5.8. For n=2, the paper's computation gives R_{S^1}(f^2)=2 and R_{S^1}(f)=1, so there is exactly one irreducible essential class of f^2. Because the fiber map is trivial on π1 of the fiber at the fixed fiber y=1, the induced action [f^ω] on R_{S^1}(f^2) is the identity, so this irreducible class forms a singleton orbit. Hence O_2(f)=1 and Definition 4.1 gives NBP_2(f)=2. But P_2(f)=S^1×{-1}, which is connected, so #π0(P_2(f))=1. This contradicts Proposition 4.2(ii). The proof of Proposition 3.3, which asserts that x and f^l(x) lie in different path components of Fix(f^d), is also false in this example: x and f(x) lie in the same connected fiber S^1×{-1}. The invariant needs to be redefined—likely by counting irreducible essential classes rather than multiplying the number of class-orbits by n.
[Theorem 5.3, proof] The displayed equality in the proof of Theorem 5.3 is false as written: the proof states that the number of Reidemeister classes of f^n with depth m<n is NB(f^m)−Σ_{j|n, j<n}A_j(f)=A_m(f). Definition 5.2 defines A_m(f)=NB(f^m)−Σ_{k|m, k<m}A_k(f), so the summation must be over the proper divisors of m, not of n. For n=6 and m=2, the displayed right-hand side NB(f^2)−A_1(f)−A_2(f)−A_3(f) is not equal to A_2(f). The subsequent conclusion NBP_n(f)=NB(f^n)−Σ_{k|n,k<n}A_k(f)=A_n(f) depends on this incorrect counting step. A corrected argument can likely be supplied by considering the fixed-point set of [f^ω]^d on R_B(f^n) and applying Möbius inversion, but that argument is not present in the manuscript.
[Proposition 5.7] The proof of Proposition 5.7 establishes only the injectivity of [γ_{k,n}], not condition (i) of Definition 5.1, which requires d(<[α]_m>)=l(<[α]_m>) for every m|n. This is not a minor omission: for the map f_{1,1} of the previous comment, condition (i) fails for m=2, since the unique irreducible class of f^2 has depth 2 but orbit length 1. Thus Proposition 5.7's conclusion that such f is n-toral is false without an additional hypothesis, and Proposition 5.8 cannot invoke Theorem 5.3 for f_{1,s} merely from the equality NB(f^m)=RB(f^m).

minor comments (3)

[Section 2, opening] The standing assumption that 'all spaces are path-connected' should explicitly state that every fiber of p is path-connected, not only E and B. The path ω in Definition 2.1 exists only if x0 and f(x0) lie in the same path component of the fiber.
[References] Reference [14] is listed as 'Springer-Verlag, 1918'; the intended year is presumably 1978.
[Proposition 3.3] The final sentence of Proposition 3.3 claims that the points x, f^l(x), ... belong to different path components of Fix(f^d). This is not established by the cited Proposition 2.8 and is false for the f_{1,1} example discussed above; the proof needs to be reworked or the statement corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant is definitional and the main theorem depends on explicit hypotheses and external Nielsen theory.

full rationale

The derivation chain is self-contained in the relevant sense. NBP_n(f) is introduced in Definition 4.1 as n times the number of irreducible essential Reidemeister orbits; this is a definition, not a disguised fit or a renamed input. Homotopy invariance (Proposition 4.2(i)) is proved by constructing the fiber homotopy equivalence from [12] and using the Reidemeister-class machinery from [3] and [14]; these are external, independently checkable results, and the proof does the categorical work rather than citing the conclusion. Theorem 5.3 does not obtain A_n(f)=NBP_n(f) by renaming: A_n(f) is defined recursively in Definition 5.2 from the Nielsen numbers NB(f^m), and the proof invokes the n-toral hypotheses (injectivity of [γ_m,n] and equality NB=RB) to count irreducible essential classes. Theorem 5.4 is a purely algebraic inclusion-exclusion identity, and Proposition 5.8 substitutes the independently established values N_S1(f^m)=m|s| from [3]. No parameter is fitted to the target quantity, and no load-bearing claim is justified exclusively by the authors' own prior work; the only same-author reference [13] is not cited in the body. The apparent wrong divisor sum in the proof of Theorem 5.3 is a correctness concern rather than a circularity: it does not equate the target formula with its inputs by construction, and the equality would need a repaired index set rather than a removal of a derivation. The path-connectedness assumptions are explicit standing hypotheses, not hidden circular inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters appear in this pure mathematics paper. The central claim rests on standard fibration theory, the path-connectedness of the fiber, the imported index theory of [3], and the explicit n-toral and rank hypotheses used in the computation. No new physical or geometric entities are postulated.

assumptions (4)

standard math Covering homotopy property for fibrations (Whitehead, Elements of Homotopy Theory, I.7.16 and I.7.18)
Used to lift homotopies in Definition 2.1, Lemma 2.3, Theorem 2.6, and throughout the periodic class arguments; the lifts are the backbone of the Reidemeister action over B.
domain assumption E, B, and the fiber Y are compact, path-connected manifolds without boundary
Stated at the start of Section 2. It guarantees a path omega inside the fiber from x0 to f(x0), so loop classes in pi_1(Y,x0) exist and the Reidemeister construction is well defined. Without path-connected fibers the construction is partial.
standard math Nielsen classes over B carry an index as in Goncalves-Koschorke [3, Section 5]
The definition of essential Reidemeister classes in Definition 3.14 and the lower-bound proof of Proposition 4.2 depend on this imported index theory. The paper does not reprove it.
ad hoc to paper f is n-toral over B and NB(f^m)=RB(f^m) != 0 for every m|n
These are the hypotheses of Theorem 5.3 and Corollary 5.5. The rank condition ensures every Reidemeister class is essential; the n-toral condition ensures irreducible orbits and the counting formula match.

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Cite this review

Pith. "Pith review of A Nielsen type periodic number for maps over $B$." pith.science (2026). https://pith.science/paper/N2YAZZL7

@misc{pith2026241113233,
  author       = {Pith},
  title        = {Pith review of: A Nielsen type periodic number for maps over $B$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N2YAZZL7}},
  note         = {Machine review of arXiv:2411.13233}
}

abstract

Let $Y \to E \stackrel{p}{\to} B$ be a fibration and let $f: E \to E$ be a fiber map over $B$. In this work, we study the geometric and algebraic Reidemeister classes of the iterates of $f$ and introduce a Nielsen-type periodic number over $B$, denoted by $N_B P_n(f)$. When $B$ is a point, then $N_B P_n(f)$ coincides with the classical Nielsen periodic number.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [3]

D. L. Gon¸ calves, U. Koschorke, Nielsen coincidence theory of ﬁber-preserving maps and Dol d’s ﬁxed point index , Topol. Methods Nonlinear Anal. 33 (1), 85 - 103, 2009

work page 2009

[2] [14]

G. W. Whitehead; Elements of Homotopy Theory , Springer-Verlag, 1918

work page 1918

[3] [1]

Alsed` a, S

L. Alsed` a, S. Baldwin, J. Llibre, R. Swanson, W. Szlenk; Minimal sets of periods for torus maps via Nielsen numbers , Paciﬁc J. Math. vol 169, no 1, 1995

work page 1995

[4] [2]

D. L. Gon¸ calves, Fixed points of S1-ﬁbrations, Paciﬁc J. Math. vol 129, no 2, 1987

work page 1987

[5] [4]

D. L. Gon¸ calves, A. K. M. Libardi, D. Vendr´ uscolo and J. P. Vieira, On minimal ﬁxed point set of ﬁber preserving maps of S1-bundles over S1 , to appear in Topology and its Applications

work page

[6] [5]

Heath, R

P. Heath, R. Piccinini, and C. You, Nielsen type numbers for periodic points I , Springer Lecture Notes in Math., vol. 1411, pg 88-106, 1989

work page 1989

[7] [6]

Jezierski The Nielsen coincidence numbers of maps into tori , Quaestiones Mathematicae, 24:2, 217 - 223, 2001

J. Jezierski The Nielsen coincidence numbers of maps into tori , Quaestiones Mathematicae, 24:2, 217 - 223, 2001

work page 2001

[8] [7]

Jezierski, W

J. Jezierski, W. Marzantowicz; Homotopy Methods in Topological Fixed and Periodic Point The ory, Topological Fixed Point Theory and Its Applications, Sprin ger, vol. 3, 2006

work page 2006