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Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants

This paper classifies finite groups of order 64 arising as specific quotients of the pure braid group on two strands and utilizes these algebraic results to construct families of double Kodaira fibrations that share identical biregular invariants and Betti numbers but possess distinct fundamental groups.

Original authors: Francesco Polizzi, Pietro Sabatino

Published 2026-04-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Francesco Polizzi, Pietro Sabatino

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to build a very specific, complex type of building called a Kodaira Fibration.

In the world of mathematics, these aren't buildings made of brick and mortar, but rather intricate, multi-dimensional shapes (surfaces) that have very strict rules about how they are constructed. Think of them as a "double-decker" structure where the building is simultaneously a stack of one type of floor plan and a stack of a different type of floor plan, all woven together perfectly.

For a long time, mathematicians have been trying to answer a tricky question: Can you build two of these structures that look exactly the same on the outside (same size, same shape, same number of rooms) but are actually made of completely different "materials" inside?

This paper by Francesco Polizzi and Pietro Sabatino says: Yes, and here is exactly how to do it.

Here is the breakdown of their discovery using simple analogies:

1. The Puzzle: The "DNA" of the Building

To build these shapes, the mathematicians use a secret code based on Groups (a concept from algebra that describes symmetry, like how a snowflake looks the same if you rotate it).

The Code: They needed to find a specific "group" (a set of rules for symmetry) that is just the right size to build these double-decker structures.
The Search: They had already checked small groups (sizes up to 32) and found only two that worked. They wondered: "What about bigger groups? Specifically, groups of size 64?"
The Filter: They realized that most groups of size 64 were like "bad blueprints." They had too much symmetry in the wrong places (mathematicians call these CCT groups), which made it impossible to build the required shape. They had to filter out the bad blueprints to find the rare, special ones.

2. The Discovery: The "Golden" Groups

After a massive amount of calculation (using a computer program called GAP, which is like a super-calculator for math), they found that out of hundreds of possible groups of size 64, only seven specific groups could actually build these structures.

Think of it like finding that out of 64 different types of Lego bricks, only 7 specific colors can be used to build a working double-decker tower.

3. The Twist: The "Look-Alike" Twins

Here is the most exciting part. The authors took two of these special groups (let's call them Group A and Group B) and built their double-decker structures.

The Exterior: When you look at the outside of the buildings made from Group A and Group B, they are identical.
- They have the same number of floors.
- They have the same total area.
- They have the same number of windows (Betti numbers).
- They have the same "weight" (topological invariants).
The Interior: But if you look inside the "walls" (the fundamental group), they are completely different.
- The "DNA" of the building made from Group A is different from Group B.
- Even more surprisingly, for one specific group, there were two different ways to arrange the internal wiring, resulting in two buildings that look the same but have different "torsion" (a specific type of internal twist).

4. The Analogy: The Identical Houses

Imagine you have two houses that look exactly the same from the street. They have the same roof, the same paint, the same number of windows, and the same square footage. A real estate appraiser would say they are worth the exact same amount.

However, if you walk inside:

House A has a secret tunnel system that connects the kitchen to the attic in a specific way.
House B has a completely different tunnel system that connects the kitchen to the basement.

To the outside world, they are twins. But to someone who knows how to navigate the inside, they are totally different worlds.

5. Why This Matters

Before this paper, mathematicians knew that such "twins" existed, but they were rare, isolated examples.

This paper is a breakthrough because:

It found a whole family: They didn't just find one pair; they found two entire families (3-dimensional families) of these buildings. You can tweak the construction slightly and get a new building that still has the same "look-alike" properties.
It solved a mystery: It proved that you can have surfaces that are indistinguishable by almost every standard measurement (size, shape, curvature) but are fundamentally different in their core structure.

Summary

Polizzi and Sabatino acted like master detectives. They sifted through thousands of mathematical "blueprints" (groups of order 64), filtered out the ones that wouldn't work, and found the few that could build these complex shapes. They then proved that you can build two versions of these shapes that are perfectly identical on the outside but completely different on the inside, creating a new class of mathematical "twins" that have never been seen before.

It's a bit like discovering that you can bake two cakes that taste, smell, and look exactly the same, but one is made with a secret ingredient that changes its molecular structure entirely.

1. Problem Statement

The paper addresses the classification and construction of double Kodaira fibrations, which are compact complex surfaces $S$ admitting two distinct Kodaira fibrations $f_1: S \to B_1$ and $f_2: S \to B_2$ (where the fibers are not all biholomorphic).

The specific problem tackled is the existence and classification of diagonal double Kodaira fibrations. These arise from finite Galois covers $f: S \to \Sigma_b \times \Sigma_b$ branched over the diagonal $\Delta$ , where $\Sigma_b$ is a Riemann surface of genus $b$ . The authors aim to:

Classify finite groups $G$ of order up to 64 that admit a diagonal double Kodaira structure (an algebraic encoding of the covering map).
Investigate whether surfaces constructed from these groups can share identical biregular invariants (e.g., $K_S^2$ , $c_2(S)$ , $q(S)$ ) and Betti numbers but possess non-isomorphic fundamental groups (and thus be non-homeomorphic).

Previous work by the authors ([PolSab22]) had solved this for groups of order $\le 32$ , finding that only two extra-special groups of order 32 admit such structures. This paper extends the analysis to groups of order 64.

2. Methodology

The authors employ a "detopologization" strategy, transforming the geometric problem of constructing branched covers into a purely algebraic and combinatorial problem involving finite groups.

Algebraic Formulation: The existence of a diagonal double Kodaira fibration is equivalent to the existence of an admissible epimorphism $\phi: P_2(\Sigma_b) \to G$ , where $P_2(\Sigma_b)$ is the pure braid group on two strands. This corresponds to finding a diagonal double Kodaira structure on $G$ : an ordered set of $4b+1$ generators satisfying specific surface and conjugacy relations, where the generator $z$ (corresponding to the loop around the diagonal) has order $n \ge 2$ .
Group-Theoretic Filtering:
- CCT-Groups: The authors utilize the property of Center Commutative-Transitive (CCT) groups. They prove that CCT groups cannot admit prestructures (and thus no diagonal double Kodaira structures). They systematically classify non-CCT groups for orders $33 \le |G| \le 63$ .
- Monolithic vs. Non-Monolithic: They distinguish between monolithic groups (having a unique minimal normal subgroup) and non-monolithic groups. They prove that structures on non-monolithic groups are always lifts of structures from proper quotients.
Computational Algebra: Due to the large number of groups of order 64 (267 groups total, 124 non-abelian non-CCT), the authors used the GAP4 computer algebra system. They developed a custom script (CheckStructures) to:
1. Generate all possible structures modulo the automorphism group $\text{Aut}(G)$ .
2. Verify lifting properties.
3. Compute the first homology group $H_1(S, \mathbb{Z})$ of the resulting surface $S$ using the orbifold braid group presentation.

3. Key Contributions and Results

A. Classification of Groups of Order 64

The paper provides a complete classification of groups of order 64 admitting a diagonal double Kodaira structure (specifically of type $(b, n) = (2, 2)$ ).

Non-Existence for $33 \le |G| \le 63$ : The authors prove that no finite group of order between 33 and 63 admits a diagonal double Kodaira structure.
Existence for $|G| = 64$ : Exactly 7 isomorphism classes of groups of order 64 admit such structures. They are categorized as follows:
- Non-Monolithic Groups (5 cases): $G(64, 199), G(64, 200), G(64, 201), G(64, 264), G(64, 265)$ $G (64, 199), G (64, 200), G (64, 201), G (64, 264), G (64, 265)$ .
  - These structures are all lifts from the two extra-special groups of order 32 ( $G(32, 49)$ and $G(32, 50)$ ).
  - For $G(64, 264)$ and $G(64, 265)$ , the structures are non-strong (the covering is not a direct product).
- Monolithic Groups (2 cases): $G(64, 249)$ $G (64, 249)$ and $G(64, 266)$ $G (64, 266)$ .
  - These structures are "new"; they do not arise as lifts from any quotient of order 32.
  - $G(64, 249)$ admits strong structures.
  - $G(64, 266)$ admits non-strong structures.

B. Geometric Invariants and Topological Distinction

The authors compute the topological and biregular invariants for the surfaces $S$ associated with these groups.

Shared Invariants: Surfaces associated with different groups (e.g., $G(64, 199)$ $G (64, 199)$ vs. $G(64, 266)$ $G (64, 266)$ ) can share:
- Base genera ( $b_1, b_2$ ) and fiber genera ( $g_1, g_2$ ).
- Biregular invariants: Geometric genus $p_g(S)$ , irregularity $q(S)$ , and canonical square $K_S^2$ .
- Betti numbers: $b_0, b_1, b_2, b_3, b_4$ .
Topological Distinction (The Main Breakthrough):
- While the Betti numbers are identical, the torsion part of the first homology group $H_1(S, \mathbb{Z})$ differs.
- Specifically, for the group $G(64, 266)$ $G (64, 266)$ , there are two distinct families of structures yielding surfaces with:
  1. $H_1(S, \mathbb{Z}) \cong \mathbb{Z}_{12} \oplus (\mathbb{Z}_2)^3$
  2. $H_1(S, \mathbb{Z}) \cong \mathbb{Z}_{12} \oplus (\mathbb{Z}_2)^2 \oplus \mathbb{Z}_4$
- Since the fundamental group $\pi_1(S)$ is determined by the structure, and the homology groups differ, these surfaces are not homotopy equivalent, and thus not homeomorphic.

C. Construction of Families

The construction depends on the choice of the base curve $\Sigma_2$ (genus 2), which varies in a 3-dimensional moduli space. This leads to the existence of two 3-dimensional families of Kodaira doubly fibred surfaces ( $\mathcal{F}_1$ and $\mathcal{F}_2$ ) with:

$b_1 = b_2 = 3$ , $g_1 = g_2 = 41$ , $\sigma(S) = 32$ .
Identical biregular invariants ( $p_g=93, q=6, K^2=736$ ) and Betti numbers.
Different fundamental groups (distinguished by the torsion of $H_1$ ).

4. Significance

This work represents a major advancement in the study of Kodaira fibrations and the topology of complex surfaces:

Counterexamples to Uniqueness: It provides the first explicit examples of positive-dimensional families of doubly fibred Kodaira surfaces that share the same base/fiber genera, biregular invariants, and Betti numbers, yet are not homotopy equivalent. This challenges the intuition that these invariants might determine the topology of such surfaces.
Algebraic-Geometric Bridge: The paper successfully demonstrates how deep algebraic properties of finite groups (specifically the structure of order 64 groups and their braid group representations) directly dictate the topological classification of complex surfaces.
Computational Rigor: It showcases the necessity of computational group theory (GAP4) in modern algebraic geometry, handling cases where manual calculation is infeasible due to the sheer number of group isomorphism classes.
New Structures: The identification of "monolithic" structures that are not lifts from smaller groups expands the known landscape of diagonal double Kodaira fibrations beyond the previously known order 32 cases.

In conclusion, Polizzi and Sabatino have resolved the classification for order 64 and utilized these results to construct a novel class of surfaces that are indistinguishable by standard invariants but distinct topologically, opening new avenues for research in the moduli of complex surfaces.