Topological complexity sequences of groups

This paper introduces the topological complexity sequence of a group as an intrinsic refinement applicable to groups of infinite cohomological dimension, proving that this sequence is weakly increasing and unbounded for such groups while characterizing its growth and asymptotic behavior for finite groups of even order.

The Gist

The Big Picture: Navigating a Robot's World

Imagine you are programming a robot to move through a space. The robot needs to get from Point A to Point B.

• The Space ($X$): This is the environment the robot moves in.
• The Path: A line drawn from A to B is a possible motion.
• The Problem: Sometimes, the space is so twisty, tangled, or full of holes that you can't write a single, perfect set of instructions that works for every possible starting and ending point. You have to break the space into smaller zones. In each zone, you can write a simple, safe instruction. The Topological Complexity (TC) is simply a number that counts how many different "instruction zones" you need to cover the whole space.
  • If TC is low, the space is easy to navigate.
  • If TC is high, the space is chaotic and hard to navigate.
  • If TC is infinite, the space is so complex that no finite set of instructions can ever cover it.

The Problem with "Groups"

In mathematics, a Group is a set of rules for combining things (like rotating a shape or shuffling cards). Every group has a corresponding "shape" called a Classifying Space ($BG$). Mathematicians want to know the Topological Complexity of this shape to understand how "hard" it is to navigate the rules of that group.

The Catch:
For many interesting groups (specifically those with "infinite cohomological dimension"), the shape is so huge and complex that the Topological Complexity is infinite.

• Analogy: It's like asking, "How many instructions do I need to navigate an infinite universe?" The answer is "Infinity." While true, this isn't very helpful. It doesn't tell us how the complexity grows or if there are patterns. It just says "it's too big."

The Solution: The "Zoom-In" Sequence

The authors introduce a new way to look at these groups. Instead of looking at the whole infinite shape at once, they look at it in layers or stages.

Imagine the group's shape is a giant, infinite tower.

1. Stage 1 ($B_1G$): You look at just the bottom floor.
2. Stage 2 ($B_2G$): You look at the bottom two floors.
3. Stage $n$ ($B_nG$): You look at the first $n$ floors.

As you go up the tower (increasing $n$), you see more of the shape. The authors define a Topological Complexity Sequence: a list of numbers showing the complexity of the shape at each stage.

• $TC_1(G)$: Complexity of the first floor.
• $TC_2(G)$: Complexity of the first two floors.
• ...and so on.

Even if the whole tower is infinitely complex, each individual floor (or set of floors) has a finite complexity number. This allows mathematicians to study the growth of the complexity step-by-step.

Key Findings of the Paper

1. The "Staircase" Rule (Monotonicity)

The authors prove that for groups with infinite complexity, this sequence of numbers never goes down.

• Analogy: Imagine climbing a staircase where every step is at least as high as the one before it. You might stay on the same level for a bit, but you never go down.
• The Result: As you add more "floors" to your view of the group, the complexity either stays the same or gets harder. It never gets easier. Furthermore, because the group is infinitely complex, this number will eventually grow without bound.

2. How Fast Does It Grow? (The Growth Function)

The paper asks: "How quickly does the complexity rise?"
They define a "growth function" ($\alpha_G$). Think of this as a speedometer.

• If you ask, "How many stages ($n$) do I need to reach a complexity of 10?" the answer is a specific number.
• The authors found that for finite groups with an even number of elements (like the symmetries of a square or a cube), the complexity grows at a predictable rate.
• The Formula: As the numbers get huge, the complexity grows at roughly half the speed of the stage number.
  • Analogy: If you take 100 steps up the tower, the "difficulty meter" will have increased by about 50 points. It's a steady, predictable climb.

3. The Special Case of the Quaternion Group

The authors looked at a specific, tricky group called the Quaternion Group ($Q_8$).

• They used a specialized mathematical tool (called "sectional category weight") to get a more precise estimate for this specific group.
• The Result: For this specific group, their new, sharper tool showed that the complexity grows slightly slower than the general rule for even groups. It's like finding a specific type of staircase that has slightly shorter steps than the standard ones.

What They Didn't Solve (The Open Questions)

The paper ends by listing six puzzles they couldn't solve yet:

1. Does the "Staircase" rule apply to all groups? They proved it for infinite ones, but what about finite ones?
2. What about groups with an odd number of elements? They have a good rule for even groups, but odd groups are a mystery.
3. How "jumpy" is the growth? Does the complexity go up by 1 every time, or does it sometimes jump by 5?
4. What about "Sequential" complexity? (Imagine the robot has to stop at 3 intermediate points instead of just going straight from A to B). They defined this but didn't solve the growth rules for it yet.

Summary

This paper takes a mathematical concept that was previously "broken" (infinite complexity) and fixed it by looking at it in layers. They discovered that for many groups, the difficulty of navigating the group's rules increases steadily and predictably as you look deeper into the structure. They provided a formula for how fast this happens for even-sized groups and offered a sharper tool for specific, complex groups, while leaving several interesting mysteries for future mathematicians to solve.

Technical Summary

1. Problem Statement

The paper addresses a limitation in the study of Topological Complexity (TC), a numerical homotopy invariant introduced by Farber to quantify the instability of motion planning problems.

• The Issue: For a discrete group $G$, the topological complexity is defined as $TC(G) = TC(BG)$, where $BG$ is the classifying space. However, if $G$ has infinite cohomological dimension (which includes all finite groups), $TC(G) = \infty$. This renders the invariant "meaningless" for a vast and important class of groups, as it fails to distinguish between different groups or capture their structural nuances.
• The Goal: The authors aim to define a refined invariant that remains finite and informative for groups of infinite cohomological dimension. They seek to understand the growth behavior of this new invariant, specifically for finite groups.

2. Methodology and Definitions

The authors introduce the Topological Complexity Sequence of a group $G$, denoted $\{TC_n(G)\}_{n \ge 1}$.

• Milnor Constructions: Instead of analyzing the infinite-dimensional classifying space $BG$ directly, the authors utilize the sequence of finite-dimensional approximations known as Milnor constructions. Let $E_n G$ be the $(n+1)$-fold join of $G$. The group $G$ acts freely on $E_n G$, and the quotient $B_n G = E_n G / G$ is the $n$-th Milnor construction.
• The Sequence: The sequence is defined as $TC_n(G) = TC(B_n G)$. Since $\dim(B_n G) = n$, it follows that $TC_n(G) \le 2n$, ensuring the values are always finite.
• Growth Function: To analyze the asymptotic behavior, they define a growth function $\alpha_G: \mathbb{N} \to \mathbb{Z}$:
  $$\alpha_G(n) = |\{k \ge 1 \mid TC_k(G) \le n\}|$$
  This function counts how many terms in the sequence are less than or equal to $n$.

Key Tools Used:

• Lusternik-Schnirelmann (LS) Category: Utilizing the inequality $cat(X) \le TC(X) \le 2 \cdot cat(X)$.
• Berstein Class: For groups of infinite cohomological dimension, a specific cohomology class $b \in H^1(BG; I)$ whose powers remain non-trivial, used to bound the LS category of $B_n G$.
• D-Topological Complexity ($TCD$): A variant of TC introduced by Farber et al., which relates to the sectional category of the diagonal map in the universal cover.
• Sectional Category Weight: A cohomological tool used to derive sharper lower bounds for specific groups (e.g., the Quaternion group).
• Blakers-Massey Theorem: Used to establish homotopy equivalences between $B_n G$ and $B_{n+1} G$.

3. Key Contributions and Results

A. Monotonicity and Unboundedness (Theorem 1.2)

The authors prove that for any group $G$ of infinite cohomological dimension, the sequence $\{TC_n(G)\}$ is:

1. Weakly Increasing: $TC_n(G) \le TC_{n+1}(G)$.
2. Unbounded: The sequence grows indefinitely.

• Significance: This establishes that $TC_n(G)$ acts as a valid approximation approaching the "infinite" value of $TC(G)$ as $n \to \infty$. This contrasts with groups of finite cohomological dimension (like $\mathbb{Z}$), where the sequence does not necessarily converge to $TC(G)$.

B. Growth Estimates for Finite Groups of Even Order (Theorem 1.4 & Corollary 1.5)

For a finite group $G$ of even order, the authors provide precise bounds on the growth function $\alpha_G(n)$:

• Lower Bound: $\lfloor n/2 \rfloor \le \alpha_G(n)$.
• Upper Bound: $\alpha_G(n) \le \beta(n)$, where $\beta(n)$ is a complex function involving the sum of digits in the dyadic expansion of $n$ (related to immersion dimensions of real projective spaces).
• Asymptotic Behavior: They prove that:
  $$\lim_{n \to \infty} \frac{\alpha_G(n)}{n} = \frac{1}{2}$$
  This indicates that for even-order finite groups, the topological complexity sequence grows linearly, with the "density" of values $\le n$ being exactly $1/2$.

C. Sharper Bounds for the Quaternion Group $Q_8$ (Proposition 4.4)

The authors demonstrate that the general upper bound $\beta(n)$ is not optimal for all groups.

• By applying sectional category weights to the cohomology of the Quaternion group $Q_8$, they derive a strictly sharper upper bound $\gamma(n)$ for $\alpha_{Q_8}(n)$.
• They show that $\beta(n) > \gamma(n)$ for infinitely many $n$, proving that the growth rate depends on the specific algebraic structure of the group, not just its order.

4. Significance

• Refinement of Invariants: The paper successfully resolves the issue of $TC(G) = \infty$ for infinite-dimensional groups by providing a sequence of finite invariants that capture the group's complexity.
• Connection to Geometry: The results link the topological complexity of Milnor constructions to the immersion dimension of real projective spaces ($RP^n$), leveraging deep results from geometric topology (Davis, Farber, Tabachnikov, Yuzvinsky).
• Algebraic Sensitivity: The work highlights that while the asymptotic growth rate ($1/2$) is universal for even-order finite groups, the specific growth function $\alpha_G(n)$ is sensitive to the group's internal structure (e.g., the difference between general even-order groups and $Q_8$).
• Foundational Questions: The paper concludes by posing six open problems, including whether monotonicity holds for all groups (not just infinite cohomological dimension), the behavior for groups of odd order, and the extension of these concepts to sequential topological complexity ($TC_r$).

5. Conclusion

Kishimoto and Minowa establish that the topological complexity sequence is a robust tool for studying groups with infinite cohomological dimension. They prove its monotonicity and unboundedness, determine its asymptotic growth rate for even-order finite groups, and demonstrate that finer algebraic structures yield sharper growth estimates. This work bridges homotopy theory, motion planning, and group cohomology, opening new avenues for analyzing the "complexity" of discrete groups.