The solution on the geography-problem of non-formal compact (almost) contact manifolds

The paper demonstrates the existence of non-formal compact (almost) contact manifolds with specific first Betti numbers and, in certain dimensions, simple connectivity, thereby resolving a key problem in the geography of such manifolds.

The Gist

Imagine you are an architect trying to build a house. In the world of mathematics, specifically in a field called Topology, mathematicians build "houses" out of shapes called manifolds. These aren't houses you can live in; they are abstract spaces that can have any number of dimensions (3D, 5D, 100D, etc.).

This paper is about solving a specific puzzle regarding the "blueprints" of these mathematical houses. The puzzle is: Can we build a house that looks simple on the outside but is actually incredibly complex and twisted on the inside?

Here is the breakdown of the paper's story, using everyday analogies:

1. The Two Types of Blueprints: "Formal" vs. "Non-Formal"

Think of a Formal manifold like a perfectly symmetrical, Lego-built castle. If you take it apart and look at the individual bricks (the mathematical data), you can perfectly reconstruct the whole castle. It's predictable, orderly, and "honest."

A Non-Formal manifold is like a house built with a hidden, twisting staircase that connects the attic to the basement in a way that makes no sense if you just look at the rooms individually. The whole is greater than the sum of its parts. There are "ghosts" in the machine—hidden connections that only appear when you look at the whole structure.

Mathematicians call these hidden connections Massey Products. If a house has a non-zero Massey product, it's "non-formal" (twisted). If all Massey products are zero, it's "formal" (straightforward).

2. The "Contact" Constraint

The author isn't just building any house; he is building Contact Manifolds.

• The Analogy: Imagine a 3D room where every point has a specific "wind direction" blowing through it. This wind must swirl in a very specific, non-stop way (mathematically, it can't just stop or flatten out).
• In higher dimensions, this is called an Almost Contact structure. It's like a cosmic wind that flows through the entire shape of the universe.
• The paper asks: Can we build these "windy" houses that are also "twisted" (non-formal)?

3. The Geography Problem: "What Dimensions and Holes Are Possible?"

Mathematicians love to categorize things. They ask: "If I want a twisted, windy house, what sizes can I make it?"

• $m$: The number of dimensions (the size of the house).
• $b$: The number of "holes" in the house (like a donut has one hole, a figure-8 has two).

Before this paper, mathematicians knew the rules for most sizes, but there were some "forbidden zones" where they weren't sure if a twisted, windy house could exist. Specifically:

• Could you make a 5-dimensional windy house with exactly 1 hole?
• Could you make a 7-dimensional (or larger) windy house with 0 holes (a solid ball shape)?

4. The Author's Solution: "Yes, We Can!"

Christoph Bock, the author, says: "Yes, we can build them."

He uses a clever construction method involving Solvmanifolds.

• The Analogy: Think of a solvmanifold as a house built by taking a giant, infinite, twisting slide (a Lie group) and cutting it into a finite piece by wrapping it around a lattice (like wrapping a ribbon around a cylinder).
• He proves that if you pick the right "slide" (a specific type of mathematical group) and wrap it correctly, you get a house that:
  1. Has the "wind" flowing through it (it's a Contact manifold).
  2. Has the hidden, twisted connections (it's Non-Formal).
  3. Fits the specific dimensions and hole counts that were previously unknown.

5. The "Magic Trick" (The Boothby-Wang Fibration)

How did he prove the 5D and 7D cases? He used a mathematical "elevator."

• He started with a known "symplectic" house (a house with a special kind of magnetic field).
• He used a theorem (Boothby-Wang) to build a new house on top of it. Imagine taking a 2D sheet of paper and rolling it into a tube to make a 3D cylinder.
• This process adds one dimension. If the original sheet was "twisted" (non-formal), the new tube is also "twisted."
• By doing this, he turned a known 6D twisted shape into a 7D twisted windy shape, and a 4D shape into a 5D twisted windy shape.

6. Why Does This Matter?

You might ask, "Who cares about twisted 5D houses?"

• The Big Picture: In physics and geometry, we often want to know if certain structures (like those describing the universe or quantum fields) can exist.
• The Surprise: There was a belief that if a house is "Contact" (has that specific wind flow), it might be forced to be "Formal" (simple). This paper proves that no, you can have a Contact house that is incredibly complex and twisted.
• Sasakian Structures: The paper notes that even if a house admits a "Sasakian" structure (a very nice, rigid type of geometry), it doesn't mean the house has to be simple. You can have a rigid, beautiful structure that still has hidden, twisted secrets inside.

Summary

Christoph Bock solved a puzzle by showing that complexity and "windy" geometry can coexist in dimensions 5, 7, and higher. He proved that you can build mathematical "houses" with specific numbers of holes that are both:

1. Windy (Contact manifolds).
2. Twisted (Non-formal).

He did this by constructing them out of "slides" (Lie groups) and using a "elevator" (fibration) to lift them into higher dimensions, proving that the universe of these shapes is much richer and more twisted than previously thought.

Technical Summary

Here is a detailed technical summary of the paper "The solution on the geography-problem of non-formal compact (almost) contact manifolds" by Christoph Bock.

1. Problem Statement

The paper addresses the geography problem for non-formal compact manifolds equipped with contact structures. Specifically, it seeks to determine the existence of non-formal compact contact manifolds of dimension $m$ (where $m$ is odd) with a specific first Betti number $b_1 = b$.

• Context: A manifold is formal if its rational homotopy type is determined entirely by its cohomology ring (specifically, if its minimal model is quasi-isomorphic to its cohomology algebra). Non-formal manifolds possess non-trivial higher-order cohomology operations, such as Massey products.
• Known Results:
  • For general oriented compact manifolds, Fernández and Muñoz established the existence of non-formal examples for $m \ge 3, b \ge 2$; $m \ge 5, b=1$; and $m \ge 7, b=0$.
  • For compact symplectic manifolds (even dimensions), similar results exist.
  • For contact manifolds (odd dimensions), it was previously known that non-formal examples exist for $m$ odd, $b \ge 2$.
• The Gap: The paper aims to resolve the remaining cases for contact manifolds:
  1. The case $m=5$ with $b_1=1$.
  2. The case $m \ge 7$ with $b_1=1$.
  3. The case $m \ge 7$ with $b_1=0$ (simply-connected).

2. Methodology

The author employs a combination of Lie group theory, solvmanifold constructions, and rational homotopy theory.

A. Solvmanifolds and Contact Structures

The core construction relies on solvmanifolds (quotients $\Gamma \backslash G$ where $G$ is a simply-connected solvable Lie group and $\Gamma$ is a lattice).

• Proposition 2.1: If a connected Lie group $H$ admits an almost complex structure, then any quotient of $\mathbb{R} \ltimes H$ by a lattice admits a contact structure.
• Theorem 2.2: The author classifies 5-dimensional solvable Lie groups $G$ that admit lattices. It is shown that for most groups (excluding specific exceptions like $G_{5.2}$), the resulting solvmanifold is contact. This reduces the problem to finding specific Lie algebras that yield non-formal cohomology.

B. Rational Homotopy Theory and Massey Products

To prove non-formality, the paper utilizes Massey products.

• Lemma 3.1 & 3.2: If a manifold is formal, all Massey products (and $a$-Massey products) in its cohomology must vanish.
• Strategy: The author constructs manifolds where specific Massey products are non-zero, thereby proving non-formality.
• Boothby-Wang Fibration: To move from symplectic manifolds (even dimension) to contact manifolds (odd dimension), the paper uses the Boothby-Wang construction. Given a symplectic manifold $(M, \omega)$ with an integral cohomology class, there exists a principal $S^1$-bundle $E \to M$ where $E$ is a contact manifold. The Gysin sequence is used to relate the cohomology of $E$ to $M$.

3. Key Contributions and Results

Theorem 1.4: The 5-Dimensional Case ($m=5, b_1=1$)

• Construction: The author utilizes the solvmanifold constructed in a previous work ([3, Proposition 7.2.9]) as a quotient of the almost abelian Lie group $G_{5.15}^{-1}$ by a lattice.
• Result: This manifold has $b_1=1$. By Theorem 2.2, it admits a contact structure. By analyzing its cohomology, the author confirms it possesses a non-vanishing Massey product.
• Conclusion: There exists a non-formal compact contact 5-manifold with $b_1=1$.

Theorem 1.5: The Higher Odd-Dimensional Case ($m \ge 7, b_1=1$)

• Construction:
  1. Start with the completely solvable Lie group $G = G_{6.15}^{-1}$ and a lattice $\Gamma$.
  2. Construct the solvmanifold $M = \Gamma \backslash G$.
  3. Define a symplectic form $\tilde{\omega}$ on $M$ using the basis of left-invariant 1-forms.
  4. Apply the Boothby-Wang construction to obtain a principal $S^1$-bundle $E \to M$, where $E$ is a contact manifold of dimension $2n+1 = 7$.
• Proof of Non-formality:
  • The cohomology of $M$ is computed explicitly using Hattori's Theorem.
  • A triple Massey product $\langle [x_6], [x_6], [\tilde{\omega}] \rangle$ is calculated in $H^*(M)$. It is shown to be non-zero (specifically, it equals $[x_{456}]$).
  • Using the Gysin sequence, the author shows that the pullback of this Massey product to the total space $E$ remains non-vanishing.
• Generalization: For $m > 7$ (odd), the author takes the product $M \times (S^2)^k$. By Lemma 3.4, the product of a non-formal manifold with a formal one (like $S^2$) remains non-formal.

Theorem 1.3: The Simply-Connected Case ($m \ge 7, b_1=0$)

• Result: The paper confirms the existence of simply-connected non-formal compact contact manifolds for $m \ge 7$.
• Method:
  • For $m=7$, the construction in Theorem 1.5 can be adapted or referenced.
  • For $m \ge 13$, the author provides a shorter proof idea based on Cavalcanti's work ([6]), which constructed a simply-connected non-formal symplectic 12-manifold. By applying the Boothby-Wang construction to this 12-manifold, a simply-connected non-formal contact 13-manifold is obtained. Higher dimensions follow via products with $S^2$.

4. Significance

• Completing the Classification: This paper fully solves the geography problem for non-formal compact contact manifolds. It establishes that non-formal contact manifolds exist for all odd dimensions $m \ge 5$ and all admissible Betti numbers ($b_1 \ge 1$ for $m=5$; $b_1 \ge 0$ for $m \ge 7$).
• Sasakian Structures: The paper addresses a subtle point regarding Sasakian geometry. While Kähler manifolds are always formal, the existence of a Sasakian structure (a contact analogue of Kähler) does not imply formality. The constructed examples are contact manifolds that are non-formal, demonstrating that formality is not a topological obstruction to the existence of Sasakian structures.
• Methodological Advancement: The work effectively bridges the gap between symplectic geometry (even dimensions) and contact geometry (odd dimensions) using solvmanifolds and the Boothby-Wang fibration, providing a robust toolkit for generating counterexamples to formality in odd dimensions.

Summary of Main Theorems Proven

1. Theorem 1.4: Existence of non-formal compact contact 5-manifolds with $b_1=1$.
2. Theorem 1.5: Existence of non-formal compact contact $m$-manifolds ($m \ge 7$, odd) with $b_1=1$.
3. Theorem 1.3: Existence of simply-connected non-formal compact contact $m$-manifolds ($m \ge 7$, odd).

These results, combined with previous work, confirm that the conditions for the existence of non-formal compact contact manifolds are:

• $m \ge 5, b_1 \ge 1$
• $m \ge 7, b_1 = 0$