arxiv: 2605.07015 · v1 · submitted 2026-05-07 · 🧮 math.GN

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Nielsen coincidence theory of (n,m)-valued pairs of maps

Alan \.Zeromski, Grzegorz Graff, P. Christopher Staecker

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Pith reviewed 2026-05-11 00:49 UTC · model grok-4.3

classification 🧮 math.GN

keywords nielsen coincidence theoryn-valued mapsm-valued mapscoincidence pointsgraph intersectionshomotopy invariantscircle mapsmulti-valued maps

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The pith

A corrected Nielsen invariant for (n,m)-valued map pairs, defined by graph intersections, lower-bounds coincidence points and is sharp on the circle.

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

Pairs of an n-valued map and an m-valued map have coincidence points wherever their images overlap. An earlier attempt to build a Nielsen-style invariant that lower-bounds the number of such points in a homotopy class had technical flaws. The authors replace the construction with one that counts intersections between the graphs of the two maps in the product space. The resulting number N(f:g) is a homotopy invariant that works for any connected finite polyhedron and gives the minimal number of coincidences across all homotopic pairs. When the domain is the circle, this bound is achieved exactly by some pairs, and the authors compute its precise value.

Core claim

The authors establish a modified invariant N(f:g) for pairs consisting of an n-valued map f and an m-valued map g. It is defined in terms of the intersection points of the graphs of f and g and is shown to be a homotopy invariant that lower-bounds the number of coincidence points for all homotopic pairs on connected finite polyhedra. For maps defined on the circle the invariant is sharp, and its exact numerical value can be determined directly from the given maps.

What carries the argument

The intersection of the graphs of the n-valued map f and the m-valued map g, used to define the homotopy invariant N(f:g) that counts essential coincidence classes.

If this is right

Any pair homotopic to a given (f,g) must have at least N(f:g) coincidence points.
On the circle the bound N(f:g) is attained by some homotopic pair.
For circle maps the exact minimal number can be computed from the degrees of the component single-valued maps.
The construction applies uniformly to all connected finite polyhedra.

Where Pith is reading between the lines

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

The graph-intersection method may extend to other one-dimensional domains where intersections can be counted explicitly.
If the invariant remains sharp in higher dimensions, it would give exact minimal coincidence counts for multi-valued maps on surfaces.
The approach could be used to bound coincidences in dynamical systems whose iterates are multi-valued.

Load-bearing premise

That redefining the invariant through graph intersections removes the earlier flaws and produces a true homotopy invariant that lower-bounds the actual number of coincidence points.

What would settle it

A pair of homotopic (n,m)-valued maps on a polyhedron whose number of coincidence points is strictly smaller than the value of N(f:g) computed for that homotopy class.

Figures

Figures reproduced from arXiv: 2605.07015 by Alan \.Zeromski, Grzegorz Graff, P. Christopher Staecker.

**Figure 1.** Figure 1: Two multimaps of the circle, having 5 graph intersection points (circled), but only 3 domain coincidence points. To briefly outline the problems in [2], consider [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Three identical vertical blocks of ϕ6,2. Proof. Let us notice that am = bn is equivalent to a n = b m , so the slopes of the lines of these power maps are equal. This means that for every s ∈ [0, 1) ϕn,a(s) = a n s + 0 n = b m s + 0 m = ϕm,b(s), and so CoinX(ϕn,a : ϕm,b) = S 1 . For the proof of the second part of the theorem, assume without loss of generality that n ≥ m. Consider the translation φ(x) := ϕ… view at source ↗

**Figure 3.** Figure 3: Domain coincidences of (ϕ2,1, ϕ3,−1). 0 1 x1 x3 x4 x5 1 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

read the original abstract

We consider pairs of maps $(f,g)$, where $f$ is an $n$-valued map and $g$ is an $m$-valued map, defined on connected finite polyhedra. A point $x$ such that $f(x)\cap g(x)\neq \emptyset$ is called a coincidence point of $f$ and $g$. A useful device for studying coincidence points would be a Nielsen-type invariant which provides a lower bound for the number of coincidence points of all $(n, m)$-valued pairs of maps homotopic to $(f,g)$. The construction of such an invariant $N(f:g)$ was proposed in [J. Fixed Point Theory Appl. 14, 309--324 (2013)]. Unfortunately, this approach has some flaws. In this paper, we present a modified construction that yields a corrected form of the invariant, defined in terms of the intersection points of the graphs of $f$ and $g$. In the case of $(n, m)$-valued pairs of maps of the circle our invariant provides a sharp lower bound, which we precisely determine.

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 paper fixes the flawed 2013 invariant for (n,m)-valued map coincidences by counting graph intersections and gives an explicit sharp lower bound on the circle.

read the letter

The main point is that the authors repair the 2013 construction of a Nielsen coincidence invariant for pairs of n-valued and m-valued maps. They replace the earlier version with a count of intersection points between the graphs of the two maps in the product space, prove that this count is a homotopy invariant, and show it lower-bounds the number of actual coincidence points. On the circle they compute the invariant in closed form and exhibit maps where the bound is achieved, so it is sharp there.

Referee Report

0 major / 3 minor

Summary. The paper corrects flaws in a 2013 construction of a Nielsen-type coincidence invariant N(f:g) for pairs of n-valued and m-valued maps on connected finite polyhedra. It redefines the invariant directly from the intersection points of the graphs of f and g, proves that the resulting count is a homotopy invariant, and shows that it lower-bounds the number of coincidence points. For the special case of maps on the circle the authors compute the invariant in closed form and establish that the bound is sharp.

Significance. If the invariance and lower-bound arguments hold, the work supplies a usable, geometrically defined tool in multi-valued coincidence theory that avoids the defects of the prior approach. The explicit, realizable formula on the circle is a concrete advance that can serve as a benchmark for extensions to higher-dimensional polyhedra. The construction is parameter-free and rests on standard graph-intersection techniques, which strengthens its potential utility.

minor comments (3)

[Introduction] The introduction should state the explicit closed-form expression for the circle case in a theorem immediately after the definition, rather than deferring it to the final section, to make the sharpness claim easier to locate.
[Section 2] Notation for the new invariant N(f:g) should be contrasted more clearly with the 2013 version (e.g., by a short table or sentence) so readers can see precisely which quantities have been altered.
[Section 3] The proof that the graph-intersection count is homotopy invariant would benefit from an explicit reference to the relevant lemma or proposition number when the homotopy is applied to the graphs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. We are pleased that the geometric construction of the corrected invariant N(f:g) and its sharpness on the circle are viewed as useful advances in multi-valued coincidence theory.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces a corrected invariant N(f:g) explicitly defined via intersection points of the graphs of the n-valued map f and m-valued map g. It proves homotopy invariance of this count and establishes the lower-bound property for coincidence points directly from the definition and standard Nielsen theory techniques on polyhedra. The 2013 reference is invoked only to note prior flaws that the new graph-based construction avoids; the current proofs and the explicit sharp formula for circle maps do not reduce to any fitted parameter, self-citation chain, or input by construction. The derivation remains self-contained against external topological benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard domain assumptions of Nielsen theory together with the claim that graph intersections yield a homotopy-invariant count.

axioms (2)

domain assumption The maps are defined on connected finite polyhedra
Explicitly stated as the setting for the pairs (f,g).
domain assumption Homotopy classes of (n,m)-valued maps are well-defined and the invariant is constant on them
Required for any Nielsen-type lower bound to be useful.

pith-pipeline@v0.9.0 · 5502 in / 1221 out tokens · 68013 ms · 2026-05-11T00:49:57.388799+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean (and Cost/FunctionalEquation.lean) reality_from_one_distinction; washburn_uniqueness_aczel unclear
We present a modified construction that yields a corrected form of the invariant, defined in terms of the intersection points of the graphs of f and g. ... for an (n,m)-valued pair of self-maps of S¹ of degrees a and b, the minimal number of graph intersection points is equal to |am−bn|.

Reference graph

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