Averaging formulas for the Reidemeister trace, Lefschetz and Nielsen numbers of $n$-valued maps

This paper establishes averaging formulas for the Reidemeister trace, Lefschetz, and Nielsen numbers of $n$-valued maps on closed manifolds by expressing them through single-valued maps on finite orientable covering spaces, with explicit results derived for infra-nilmanifolds.

The Gist

Imagine you are standing in a room with a magical, multi-headed mirror. When you look into it, you don't just see one reflection of yourself; you see $n$ different reflections scattered around the room. In mathematics, this is called an $n$-valued map. Instead of pointing to a single spot, the map points to a set of spots.

The big question mathematicians ask about these mirrors is: "Do any of these reflections actually land on top of the real you?" In other words, is there a point $x$ such that $x$ is one of the $n$ spots the map points to?

This paper by Karel Dekimpe and Lore De Weerdt is a guidebook on how to count these "self-touching" points (fixed points) and understand the geometry of these multi-headed mirrors, specifically for complex shapes like donuts, Klein bottles, and higher-dimensional versions of them.

Here is the breakdown of their discovery using simple analogies.

1. The Problem: The "Too Complicated" Mirror

In the old days, mathematicians studied mirrors that showed only one reflection (single-valued maps). They had a perfect toolkit to count how many times the reflection touched the real person. They used something called the Lefschetz number (a quick count) and the Nielsen number (a more precise count of distinct groups of touches).

But when you have $n$ reflections, the old tools break.

• The Issue: You can't just look at the whole mirror and say, "Okay, let's lift this up to a bigger room to see it clearly." In the single-reflection world, you could always zoom out to a "universal cover" (a bigger, simpler version of the room) to do the math.
• The Failure: With $n$ reflections, the reflections might get tangled up in a way that prevents you from zooming out cleanly. The "lift" doesn't work anymore. It's like trying to untangle a knot of $n$ strings by pulling on just one end; the whole thing just gets messier.

2. The Solution: The "Split" Strategy

The authors' brilliant idea is to stop trying to handle the $n$ reflections as one giant, messy blob. Instead, they split the problem.

Imagine your multi-headed mirror is actually just $n$ separate, single-headed mirrors glued together, but they are dancing in a synchronized routine.

• The Trick: They prove that even though the $n$ reflections are linked, you can analyze them by looking at coincidences between single reflections and the "floor" of the room.
• The Analogy: Instead of asking, "Does the multi-headed monster touch me?", they ask, "Does reflection #1 touch me? Does reflection #2 touch me? Does reflection #3 touch me?"
• They then use a technique called Averaging. Imagine you have a blurry photo of a crowd. Instead of trying to count everyone in the blur, you take many clear, zoomed-in snapshots of small groups, count them, and then average the results to get the total.

3. The "Averaging Formula"

This is the core of the paper. They derived a formula that says:

The total number of fixed points for the $n$-valued map = The average of the fixed points of all the "lifted" single maps.

Think of it like this:

• You have a complex machine with $n$ gears turning.
• You can't easily count the total rotations of the whole machine.
• But, if you take the machine apart, look at each gear individually while it's spinning on a different, simpler track (a "covering space"), count the rotations there, and then take the average of all those counts... you get the exact answer for the whole machine.

This formula works for three different types of "counts":

1. The Lefschetz Number: A quick algebraic sum (like a rough estimate).
2. The Nielsen Number: A precise count of distinct groups of fixed points (the "real" answer).
3. The Reidemeister Trace: A detailed map showing exactly where and how these points are connected (the "blueprint").

4. The Special Case: The "Infra-Nilmanifold"

The paper gets even more practical when applied to a specific type of shape called an infra-nilmanifold.

• What is it? Think of a flat torus (a donut) or a Klein bottle (a bottle with no inside or outside). These are shapes that are "flat" but loop back on themselves.
• Why is it special? For these shapes, the "splitting" trick works perfectly every time. The authors show that for these shapes, you don't just get a vague estimate; you get an exact, calculable formula.
• The Result: You can plug in some numbers (matrices representing how the map stretches and twists the space) and immediately calculate the exact number of fixed points without having to do any complex geometry.

5. Why Does This Matter?

Before this paper, if you had a map with 3 or 4 reflections on a complex shape, you were stuck. You couldn't use the old formulas, and you couldn't easily compute the answer.

This paper provides a universal translator. It translates the hard problem of "multi-valued maps" into the easy problem of "single-valued coincidences."

• For Mathematicians: It opens the door to solving fixed-point problems for complex systems (like fluid dynamics or network flows) where multiple outcomes are possible at once.
• For the Rest of Us: It's a reminder that when a problem seems too tangled to solve as a whole, sometimes the best way forward is to break it into smaller, simpler pieces, solve those, and then average the results.

In a nutshell: The authors found a way to count the "ghosts" in a multi-reflection mirror by breaking the mirror into pieces, counting the ghosts in each piece on a simpler stage, and then averaging the counts to get the truth.

Technical Summary

Here is a detailed technical summary of the paper "Averaging Formulas for the Reidemeister Trace, Lefschetz and Nielsen Numbers of n-valued Maps" by Karel Dekimpe and Lore De Weerdt.

1. Problem Statement and Context

The Problem:
The paper addresses the computation of fixed point invariants (Lefschetz number $L(f)$, Nielsen number $N(f)$, and Reidemeister trace $RT(f)$) for $n$-valued maps. An $n$-valued map $f: X \to D_n(X)$ assigns a set of $n$ distinct points to each $x \in X$, where $D_n(X)$ is the unordered configuration space.

While averaging formulas exist for single-valued maps (relating invariants on a manifold $X$ to lifts on a finite covering space $\bar{X}$), these formulas do not directly generalize to the $n$-valued setting.

• Limitation of Previous Approaches: A direct generalization of the single-valued averaging formula (Theorem 2.5 in the paper) requires the $n$-valued map to lift to a finite cover that is a nilmanifold. The authors demonstrate (via Example 2.6) that $n$-valued maps on infra-nilmanifolds often do not admit such lifts, rendering the existing averaging formulas inapplicable.
• Goal: To establish a new averaging formula for $n$-valued maps that works even when the map cannot be lifted to a nilmanifold in the traditional sense, specifically by linking $n$-valued fixed points to coincidences of single-valued maps between covering spaces.

2. Methodology

The authors develop a framework connecting the fixed point theory of $n$-valued maps to the coincidence theory of single-valued maps through a "split lift" construction.

A. The Split Lift Construction

1. Setup: Let $X$ be a closed manifold with universal cover $\tilde{X}$ and fundamental group $\pi$. Let $\Gamma \subset \pi$ be a finite index normal subgroup, and $\bar{X} = \Gamma \backslash \tilde{X}$ be the corresponding finite cover.
2. Invariant Subgroup: The authors introduce an $f$-$\Gamma$-invariant subgroup $S \subset \pi$. This subgroup must be contained in $\Gamma$, contained in the kernel of the permutation part of the induced morphism, and map into $\Gamma$ under the component morphisms.
3. Intermediate Cover: They construct an intermediate cover $\hat{X} = S \backslash \tilde{X}$.
4. Decomposition: An $n$-valued map $f: X \to D_n(X)$ lifts to a map $\tilde{f}: \tilde{X} \to F_n(\tilde{X}, \pi)$. Using the subgroup $S$, this lift induces a split lift $\bar{f} = (\bar{f}_1, \dots, \bar{f}_n)$, where each $\bar{f}_i$ is a single-valued map from $\hat{X}$ to $\bar{X}$.
5. Fixed Point Relation: A point $x \in X$ is a fixed point of the $n$-valued map $f$ if and only if it is a coincidence point of some $\bar{\alpha} \bar{f}_i$ and the projection map $q: \hat{X} \to \bar{X}$ for some $\bar{\alpha} \in \pi/\Gamma$ and $i \in \{1, \dots, n\}$.

B. Algebraic Decomposition

The core technical challenge is decomposing the sum over the Reidemeister classes of the $n$-valued map (which are complex twisted conjugacy classes in $\pi \times \{1, \dots, n\}$) into sums over the simpler Reidemeister coincidence classes of the single-valued components.

• The authors utilize the Orbit-Stabilizer Theorem and properties of coincidence subgroups ($coin(\phi, \psi)$) to relate the indices of fixed point classes of $f$ to the coincidence indices of the pairs $(\bar{\alpha}\bar{f}_i, q)$.
• They prove that the index of a fixed point class of the $n$-valued map is proportional to the index of the corresponding coincidence class of the single-valued maps, scaled by the cardinality of a specific coincidence subgroup.

3. Key Contributions and Results

The paper derives three main averaging formulas:

1. Averaging Formula for the Reidemeister Trace (Theorem 4.1)

The Reidemeister trace of the $n$-valued map $f$ is expressed as an average of the coincidence Reidemeister traces of the associated single-valued maps:
$$RT(f, \tilde{f}) = \frac{1}{[\pi : S]} \sum_{i=1}^n \sum_{\bar{\alpha} \in \pi/\Gamma} \hat{r}_{\alpha}^i \left( RT(\bar{\alpha}\bar{f}_i, \alpha\tilde{f}_i; q, \text{id}) \right)$$
Here, $\hat{r}_{\alpha}^i$ is a natural map between the free abelian groups generated by the respective Reidemeister classes. This formula successfully bypasses the need for the $n$-valued map to lift to a nilmanifold.

2. Averaging Formula for the Lefschetz Number (Theorem 5.1)

By summing the coefficients of the trace formula, the authors derive:
$$L(f) = \frac{1}{[\pi : S]} \sum_{i=1}^n \sum_{\bar{\alpha} \in \pi/\Gamma} L(\bar{\alpha}\bar{f}_i, q)$$
This expresses the Lefschetz number of the $n$-valued map as the average of the Lefschetz coincidence numbers of the single-valued components.

3. Averaging Formula for the Nielsen Number (Theorem 6.1)

For the Nielsen number, the authors obtain an inequality:
$$N(f) \geq \frac{1}{[\pi : S]} \sum_{i=1}^n \sum_{\bar{\alpha} \in \pi/\Gamma} N(\bar{\alpha}\bar{f}_i, q)$$
Equality holds if and only if a specific condition regarding the coincidence subgroup is met (specifically, the coincidence subgroup must be contained in $S$ whenever the coincidence class is essential).

4. Explicit Formulas for Infra-nilmanifolds (Section 7)

The paper applies these general results to infra-nilmanifolds (manifolds finitely covered by nilmanifolds).

• Context: On nilmanifolds, Lefschetz and Nielsen numbers can be computed using determinants of Lie algebra morphisms.
• Result (Theorem 7.3): For an $n$-valued map on an infra-nilmanifold, the Nielsen number is given by an explicit determinant formula:
  $$N(f) = \frac{1}{[\pi : \Gamma]} \sum_{i=1}^n \sum_{\bar{\alpha} \in \pi/\Gamma} \left| \det(I - (\tau_{\alpha}\phi'_i)_*) \right|$$
  where $(\tau_{\alpha}\phi'_i)_*$ is the Lie algebra morphism induced by the unique extension of the component maps to the Lie group.
• Significance: This provides a computable, practical method for determining fixed point invariants for $n$-valued maps on a broad class of manifolds where previous methods failed.

4. Significance and Impact

• Overcoming Lifting Obstructions: The primary contribution is solving the problem that $n$-valued maps often do not lift to nilmanifolds. By shifting the perspective from lifting the $n$-valued map to analyzing coincidences of single-valued maps on intermediate covers, the authors create a robust framework applicable to a wider range of topological spaces.
• Unification of Theories: The work effectively unifies fixed point theory for set-valued maps with coincidence theory for single-valued maps, demonstrating that the complex invariants of $n$-valued maps can be decomposed into standard invariants of single-valued maps.
• Computational Utility: The explicit formulas for infra-nilmanifolds (Theorems 7.1 and 7.3) allow for the calculation of these invariants using linear algebra (determinants of matrices), making the theory applicable to concrete examples (as demonstrated in Example 7.7 with the Klein bottle).
• Generalization: The results generalize the classical averaging formulas of fixed point theory, extending their validity to the multi-valued setting while maintaining the structural elegance of the single-valued case.

In summary, Dekimpe and De Weerdt provide a rigorous theoretical bridge between $n$-valued fixed point theory and single-valued coincidence theory, yielding powerful new tools for calculating fixed point invariants on manifolds, particularly infra-nilmanifolds.