Non-affine $n$-valued maps on tori

This paper constructs non-affine $n$-valued maps on $k$-dimensional tori (for $n,k\geq 2$) that are not homotopic to affine maps, a result that contrasts sharply with the single-valued case and is achieved by establishing necessary and sufficient algebraic conditions on induced morphisms.

The Gist

Here is an explanation of the paper "Non-affine n-valued maps on tori" using simple language, analogies, and metaphors.

The Big Picture: The "Magic Map" Problem

Imagine you have a video game world shaped like a donut (mathematicians call this a torus). In this world, you have a special kind of "magic map."

In a normal map, if you stand at one spot, the map points to exactly one destination.
In this paper's "n-valued map," if you stand at one spot, the map points to n different destinations at the same time. Think of it like a GPS that, instead of giving you one route, gives you a list of 3 or 4 different valid routes to your destination simultaneously.

The mathematicians in this paper are asking a very specific question:

"Can every one of these 'multi-destination' maps be simplified into a straight, predictable, 'affine' map?"

What is an "Affine" Map? (The Straight Line)

To understand the answer, we need to know what an affine map is.

• The Analogy: Imagine a grid of city blocks. An affine map is like a rule that says, "From any building, walk 3 blocks East and 2 blocks North." It's a straight line, a constant shift, or a simple rotation. It's boringly predictable.
• The Single-Value Rule: In the world of single destination maps (where $n=1$), mathematicians already knew the answer: Yes. No matter how crazy your map looks, you can always wiggle it (homotopy) until it becomes a simple, straight-line rule. It's like saying, "Any winding road can be smoothed out into a straight highway."

The Surprise: The Multi-Destination Twist

This paper investigates what happens when $n \ge 2$ (when the map points to 2 or more places).

The Discovery: The authors found that NO, you cannot always smooth these multi-destination maps into simple straight lines.

They constructed specific examples of maps on a donut-shaped world where the destinations are so "twisted" around each other that they cannot be untangled into a simple affine rule. These are the Non-Affine Maps.

How Did They Prove It? (The "Cycle" Detective Work)

To prove a map is "twisted" and not "straight," the authors used a clever algebraic trick. Let's break it down with a metaphor:

The Metaphor: The Dance Floor
Imagine $n$ dancers on a circular stage (the torus).

1. The Rule ($\sigma$): Every time the music changes (a step on the grid), the dancers swap places in a specific pattern. Maybe Dancer 1 swaps with Dancer 2, or they all rotate in a circle.
2. The Movement ($\phi$): As they swap, they also take steps forward or backward.

The authors looked at the "algebraic DNA" of these swaps. They found a specific condition called the Divisibility Condition (or the Cycle Condition).

• The Test: If the dancers are following a simple "straight line" rule (affine), then when they complete a full loop of swapping (a cycle), their total steps must add up to a whole number that fits perfectly into the grid.
• The Failure: The authors built maps where, when the dancers complete their loop, their total steps do not add up to a whole number. They are "off-grid."

The "Cycle" Analogy:
Imagine you are walking in a circle.

• Affine (Straight): You take 10 steps, and you end up exactly where you started.
• Non-Affine (Twisted): You take 10 steps, but you end up slightly off-center, as if the floor itself shifted under your feet. In the math of the torus, this "off-center" feeling is impossible for a simple straight-line rule. It proves the map is inherently complex.

The "Smart Lift" Trick

One of the paper's clever insights is about how we choose to look at the map.

• Imagine you are watching the dancers from a balcony. You can choose to stand in different spots (different "lifts").
• Sometimes, from one angle, the dancers look like they are doing a chaotic, non-affine dance.
• However, the authors showed that if you stand in the perfect spot (a "smart choice of lift"), you can often make the chaos look simpler.
• The Catch: Even with the best viewing angle, there are some maps where the "off-center" steps are so fundamental that no matter where you stand, the dance remains twisted. These are the maps that are truly, irreducibly non-affine.

Why Does This Matter?

1. Breaking the Rules: It breaks the intuition that "everything can be simplified." In the single-destination world, simplification is always possible. In the multi-destination world, complexity is sometimes unavoidable.
2. Fixed Points: The paper uses these maps to count "fixed points" (places where a destination points back to itself). Because these maps are so twisted, the rules for counting these points are different and more complex than the simple rules used for straight-line maps.
3. New Examples: The authors didn't just prove it exists; they gave you the blueprints (formulas) to build these twisted maps yourself on a donut of any size.

Summary in One Sentence

While any single-path map on a donut can be smoothed out into a straight line, this paper proves that maps pointing to multiple destinations can be so inherently twisted that they can never be simplified, creating a new class of mathematical objects that defy our usual rules of geometry.

Technical Summary

Here is a detailed technical summary of the paper "Non-affine n-valued maps on tori" by K. Dekimpe and L. De Weerdt.

1. Problem Statement

The paper addresses the classification of n-valued maps on k-dimensional tori ($T^k$) where $n, k \geq 2$.

• Context: In the single-valued case ($n=1$), it is a well-established result (generalized from tori to infra-nilmanifolds) that any continuous map $f: T^k \to T^k$ is homotopic to an affine map (a map of the form $x \mapsto Ax + b$). Consequently, the fixed point theory (Reidemeister and Nielsen numbers) for single-valued maps is entirely determined by linear algebra on the fundamental group.
• The Gap: For $n$-valued maps (continuous set-valued functions sending a point to a set of $n$ distinct points), it was unknown whether a similar homotopy reduction to affine maps holds for dimensions $k \geq 2$.
• Core Question: Given a morphism $\psi: \mathbb{Z}^k \to (\mathbb{Z}^k)^n \rtimes \Sigma_n$ induced by an $n$-valued map, is there an affine $n$-valued map that induces the same morphism? If not, what are the algebraic obstructions?

2. Methodology and Framework

The authors utilize the theory of fixed points of n-valued maps developed by Brown, Gonçalves, and others. The methodology involves:

• Lifting to Covering Spaces: An $n$-valued map $f: T^k \to D_n(T^k)$ (where $D_n$ is the unordered configuration space) is lifted to a map $\tilde{f}: \mathbb{R}^k \to F_n(\mathbb{R}^k, \mathbb{Z}^k)$, where $F_n$ is the orbit configuration space.
• Induced Morphisms: A lift $\tilde{f} = (\tilde{f}_1, \dots, \tilde{f}_n)$ induces a group morphism:
  $$\psi_{\tilde{f}} = (\phi_1, \dots, \phi_n; \sigma): \mathbb{Z}^k \to (\mathbb{Z}^k)^n \rtimes \Sigma_n$$
  Here, $\sigma: \mathbb{Z}^k \to \Sigma_n$ describes the permutation of the $n$ points as one moves around loops in the torus, and $\phi_i: \mathbb{Z}^k \to \mathbb{Z}^k$ are maps describing the translation of the lifts.
• Affine Definition: An $n$-valued map is affine if its lift factors are affine maps $\tilde{f}_i(\bar{t}) = A_i\bar{t} + \bar{a}_i$.
• Irreducibility: The authors restrict their analysis to irreducible morphisms (those where the image cannot be decomposed into smaller $m$-valued and $(n-m)$-valued maps), as the general case reduces to these via Theorem 1.1.

3. Key Contributions and Results

A. The Divisibility Condition (The Main Obstruction)

The paper establishes a necessary and sufficient algebraic condition for a morphism $\psi$ to be induced by an affine map.

• Definitions:
  • Let $S_i = \{ \bar{z} \in \mathbb{Z}^k \mid \sigma_{\bar{z}}(i) = i \}$ be the stabilizer of index $i$.
  • Let $n_{i\bar{z}}$ be the length of the cycle containing $i$ in the permutation $\sigma_{\bar{z}}$.
  • For irreducible maps, these stabilize to a single subgroup $S$, integer $n_{\bar{z}}$, and morphism $\phi: S \to \mathbb{Z}^k$.
• Theorem 2.6 & 2.7 (Main Result):
  An irreducible morphism $\psi = (\phi_1, \dots, \phi_n; \sigma)$ is induced by an affine $n$-valued map if and only if:
  $$\phi(n_{\bar{z}}\bar{z}) \notin n_{\bar{z}}\mathbb{Z}^k \quad \text{whenever} \quad \bar{z} \notin S$$
  Interpretation: If $\bar{z}$ is not in the stabilizer (meaning the permutation moves the point), the sum of the translations along the cycle of the permutation must not be divisible by the cycle length in the integer lattice. If it is divisible, the map cannot be affine.

B. The Cycle Condition (Corollary 3.1)

A more concrete, computable obstruction is derived:
If $(i_1 \dots i_m)$ is a non-trivial cycle of $\sigma_{\bar{z}}$, and if the images $\phi_{i_1}(\bar{z}), \dots, \phi_{i_m}(\bar{z})$ are all equal, then the map cannot be affine.

• Significance: This provides a direct way to construct non-affine maps by ensuring the "translation" components are symmetric over a cycle.

C. Independence of Lift Choice

The authors prove that while the specific values of $\phi_i$ depend on the choice of lift $\tilde{f}$, the obstruction to affineness is invariant.

• Lemma 4.2: For any decomposition of the sum $\phi_i(n_{i\bar{z}}\bar{z})$, there exists a lift where the individual terms $\phi$ are distributed arbitrarily.
• Corollary 4.4: If an $n$-valued map is affine, the image of its induced morphism must be torsion-free. If the image contains torsion (elements of finite order), the map is non-affine.

D. Construction of Non-Affine Maps

The paper provides explicit constructions of non-affine maps for any $n, k \geq 2$:

1. Example 3.2: A map on $T^2$ where the lift factors involve trigonometric functions (cos/sin) that shift indices cyclically. The induced morphism has $\phi_i = 0$ but a non-trivial $\sigma$. This satisfies the cycle condition (all $\phi$ are 0, hence equal), proving non-affineness.
2. Example 4.5: A variation where the image of $\psi$ is torsion-free, yet the map is still non-affine, demonstrating that torsion-freeness is necessary but not sufficient.
3. Lemma 5.1: A generalization technique showing that one can perturb an affine map by a small non-affine term (scaled by $\epsilon$) to induce any desired morphism $\psi$ with the same $\sigma$, provided the base map is $n$-valued.

4. Significance and Impact

• Contrast with Single-Valued Theory: The paper fundamentally breaks the analogy between single-valued and $n$-valued maps on tori. While single-valued maps are always homotopic to affine maps, $n$-valued maps ($n \geq 2, k \geq 2$) possess a rich class of non-affine maps that cannot be simplified to linear algebra.
• Fixed Point Theory: Since the computation of Nielsen numbers (the lower bound for fixed points) is trivial for affine maps (via determinants of $I - A_i$), identifying non-affine maps is crucial. The results imply that for general $n$-valued maps, one cannot simply use the matrix formulas; the specific topological structure of the map matters.
• Algorithmic Check: The paper provides a concrete algorithm (Theorem 2.7) to determine if a given algebraic morphism can be realized by an affine map, reducing the topological problem to a check of divisibility in $\mathbb{Z}^k$.
• New Examples: The authors construct the first explicit families of non-affine $n$-valued maps on higher-dimensional tori, expanding the known landscape of configuration space mappings.

Conclusion

Dekimpe and De Weerdt successfully characterize the algebraic obstructions preventing an $n$-valued map on a torus from being homotopic to an affine map. They demonstrate that for $n, k \geq 2$, the space of $n$-valued maps is strictly larger than the space of affine maps, governed by specific divisibility conditions on the induced morphisms of the fundamental group. This work establishes a new foundation for the fixed point theory of multi-valued maps in higher dimensions.