Topology of Galois conjugate character varieties

Technical Summary: Topology of Galois Conjugate Character Varieties

Problem Statement
The paper investigates the topological dependence of character varieties $M_{n,d}$ on the degree parameter $d$. Let $C$ be a nonsingular projective curve of genus $g \geq 2$, and let $n \geq 2$ be an integer coprime to $d$. The character variety $M_{n,d}$ is defined as the Betti moduli space of representations of $\pi_1(C)$ into $GL_n(\mathbb{C})$ with determinant $\xi_n^d$, where $\xi_n$ is a primitive $n$-th root of unity.

A central question, posed by Hausel in 2005 (Question 0.1), asks whether $M_{n,d}$ and $M_{n,d'}$ are homeomorphic (or homotopy equivalent) for any $d, d'$ coprime to $n$. While it is known that these varieties are Galois conjugate and share identical Betti numbers and rational cohomology rings, the existence of a homeomorphism preserving the unitary locus (which distinguishes $d$ from $d'$ via the Harder–Narasimhan correspondence) remains an open problem. Previous topological invariants, including fundamental groups and higher homotopy groups, fail to distinguish these spaces. The authors aim to determine if the integral cohomology ring provides a finer invariant capable of distinguishing the homotopy types of Galois conjugate character varieties.

Methodology
The authors develop a framework combining integral structures, automorphism groups of cohomology rings, and tautological relations to detect topological differences.

1. Cohomological Automorphisms and Integral Structures:
   The paper establishes that the rational cohomology ring $H^*(M_{n,d}, \mathbb{Q})$ is isomorphic to a graded $\mathbb{Q}$-algebra $R_{g,n}$ tensored with the cohomology of a torus, where $R_{g,n}$ is independent of $d$ (Theorem 0.3). The authors define $G_{g,n}$ as the group of graded automorphisms of $R_{g,n}$. They propose that if the integral cohomology rings of $M_{n,d}$ and $M_{n,d'}$ were isomorphic, there would exist an automorphism in $G_{g,n}$ that preserves the integral lattices induced by different degrees $d$ and $d'$.

2. Weakly Diagonal Automorphisms:
   A key structural reduction involves the subgroup $G^{wd}_{g,n} \subset G_{g,n}$ of "weakly diagonal" automorphisms, which preserve specific lines in low-degree cohomology components ($\mathbb{Q}\alpha_3$ and $\mathbb{Q}\beta_2$). Theorem 0.5 states that if $G^{wd}_{g,n} = G_{g,n}$ (i.e., all automorphisms are weakly diagonal), then the integral cohomology rings of $M_{n,d}$ and $M_{n,d'}$ are isomorphic if and only if $d \equiv \pm d' \pmod n$.

3. Tautological Relations:
   To verify the condition $G^{wd}_{g,n} = G_{g,n}$, the authors analyze the ideal of relations among tautological generators ($\alpha_i, \beta_i, \psi_{i,j}$). They utilize:

   • The closed formulas for Poincaré polynomials (Hausel–Rodriguez-Villegas, Schiffmann, Mellit) to determine the lowest degree in which a non-trivial relation must exist.
   • The relations for stable vector bundles (Earl–Jeffrey–Kirwan) to identify candidates for these relations in the character variety setting.
   • A "distinguished" tautological relation $\Upsilon_{g,n}$ in the lowest possible cohomological degree $r_{g,n} = 4g(n-1) - 2n$.

4. Computational Verification:
   For the specific case $(g, n) = (2, 5)$, the authors perform explicit linear algebra calculations to:

   • Determine the unique homogeneous relation $\Upsilon_{2,5}$ in degree 22 (Theorem 3.16).
   • Compute the dimension of the Lie algebra of the stabilizer of this relation within the automorphism group.
   • Verify that the dimension of the full automorphism group is small enough to force the "weakly diagonal" condition.

Key Contributions and Results

• Theorem 0.2 (Main Result): For a curve of genus $g=2$, the topological spaces $M_{5,1}$ and $M_{5,2}$ are not homotopy equivalent. This provides the first counterexample to Hausel's 2005 question, demonstrating that Galois conjugate character varieties can have distinct homotopy types.
• Theorem 0.5: Establishes a criterion for the non-existence of isomorphisms between integral cohomology rings based on the structure of the automorphism group $G_{g,n}$. Specifically, if all automorphisms are weakly diagonal, the integral structures distinguish the degrees.
• Theorem 0.6: Proves that for $(g, n) = (2, 5)$, the group of automorphisms $G_{2,5}$ coincides with the subgroup of weakly diagonal automorphisms ($G^{wd}_{2,5} = G_{2,5}$). This is achieved by showing the dimension of the Lie algebra of the stabilizer of the relation $\Upsilon_{2,5}$ is 12, which is strictly less than the dimension required for non-diagonal automorphisms to exist.
• Theorem 0.7: Identifies a unique distinguished tautological relation $\Upsilon_{g,n} = 0$ in degree $r_{g,n}$ for $g \geq 2, n \geq 3$. For $(2, 5)$, this relation is explicitly computed (Theorem 3.16) and shown to be symplectic-invariant.
• Proposition 1.3: Confirms that higher homotopy groups $\pi_k(M_{n,d})$ for $k > 1$ are independent of $d$, reinforcing that homotopy groups are insufficient to distinguish these spaces.

Significance and Claims
The paper claims to answer negatively Hausel's 2005 question regarding the homeomorphism of Galois conjugate character varieties. The authors assert that the integral cohomology ring is the first known topological invariant sensitive to the degree $d$ of the character variety, revealing topological differences that are "hidden deeper" than rational cohomology or Betti numbers.

The work highlights that while Galois conjugate varieties often share rational cohomology, their integral structures can differ significantly. The method relies on the interplay between the Chern grading (splitting the weight filtration), the tautological relations, and the rigidity of the automorphism group of the cohomology ring. The authors note that their approach is computationally intensive; the proof for $(2, 5)$ required solving a large linear algebra problem involving the stabilizer of a specific relation. They suggest that stronger computational resources could yield further counterexamples for other pairs $(g, n)$.

The paper does not claim a general classification for all $g, n$, but rather provides a structural framework (Conjecture 0.4) and a concrete method to disprove homotopy equivalence in specific cases where the automorphism group is sufficiently constrained by tautological relations.