Topological groupoids with involution and real algebraic stacks

Technical Summary: Topological Groupoids with Involution and Real Algebraic Stacks

Problem Statement
The paper addresses the challenge of studying the topology of real moduli spaces associated with real algebraic stacks. While the topology of the real locus $X(\mathbb{R})$ of a real algebraic variety $X$ is well-understood via the Smith–Thom inequality (relating the cohomology of $X(\mathbb{R})$ to $X(\mathbb{C})$), this framework fails for general moduli problems. In many cases, such as moduli of abelian varieties or curves, the real moduli space does not coincide with the real locus of the coarse moduli space. Consequently, standard topological methods for spaces with involution are insufficient. The authors aim to develop a topological framework capable of handling the intrinsic stacky structure of these moduli problems, specifically by generalizing the notion of fixed points from topological spaces to topological groupoids.

Methodology
The authors adopt a topological perspective, translating the algebraic geometry of real Deligne–Mumford (DM) stacks into the language of topological groupoids equipped with an involution.

1. Topological G-Groupoids: The authors define a topological G-groupoid (where $G \cong \mathbb{Z}/2$) as a topological groupoid $\mathcal{X} = [R \rightrightarrows U]$ equipped with an involution $\sigma$. This structure generalizes topological spaces with involution.
2. Fixed Locus Construction: A central methodological contribution is the definition of the fixed locus $|\mathcal{X}^G|$ for such a groupoid. Unlike the naive fixed point set of the coarse space, $|\mathcal{X}^G|$ is defined as the set of isomorphism classes of a specific "fixed-point groupoid" $\mathcal{X}^G$. The objects of $\mathcal{X}^G$ are pairs $(x, \phi)$ where $x \in U$ and $\phi: x \xrightarrow{\sim} \sigma(x)$ is an isomorphism satisfying $\sigma(\phi) \circ \phi = \text{id}$.
3. Comparison with Real Stacks: The authors establish a correspondence between real DM stacks and topological G-groupoids. For a real DM stack $\mathcal{X}$, they construct a topological groupoid $\mathcal{X}_{\mathbb{C}} = [R(\mathbb{C}) \rightrightarrows U(\mathbb{C})]$ with a natural anti-holomorphic involution. They prove that the fixed locus of this groupoid is canonically equivalent to the groupoid of real points $\mathcal{X}(\mathbb{R})$.
4. Topological Analysis: The paper utilizes properties of "open equivalences" of topological groupoids to show that topological invariants (like the coarse space topology) are independent of the choice of étale presentation. This allows for local analysis of the real locus.
5. Smith–Thom Generalization: The authors investigate whether the Smith–Thom inequality, which bounds the cohomology of the fixed locus by that of the total space, holds for groupoids. They demonstrate that the naive inequality fails for the coarse space $|\mathcal{X}|$ because it ignores automorphisms. They propose a corrected conjecture involving the inertia groupoid $\mathcal{I}\mathcal{X}$, which encodes automorphisms.

Key Contributions and Results

• Theorem 1.4 (Homeomorphism of Topologies): The authors prove that for a separated Deligne–Mumford stack $\mathcal{X}$ of finite type over $\mathbb{R}$, the bijection between the fixed locus of the associated topological groupoid $|\mathcal{X}^G|$ and the set of isomorphism classes of real points $|\mathcal{X}(\mathbb{R})|$ is a homeomorphism. Here, $|\mathcal{X}^G|$ carries the quotient topology from the fixed-point groupoid, and $|\mathcal{X}(\mathbb{R})|$ carries the real analytic topology. This result validates the use of topological groupoids as a rigorous framework for the topology of real stacks.
• Theorem 1.5 (Structure of Real Loci Maps): For a separated DM stack $\mathcal{X}$ where the automorphism group order is constant, the map from the real locus $|\mathcal{X}(\mathbb{R})|$ to the real locus of the coarse moduli space $M(\mathbb{R})$ is shown to be an open map and a topological covering over its image. This result relies on working "Euclidean locally" via the topological groupoid structure, analogous to working analytically locally for varieties.
• Conjecture 1.6 (Smith–Thom for Groupoids): The paper formulates a generalized Smith–Thom inequality. It conjectures that for a topological G-groupoid $\mathcal{X}$, the dimension of the cohomology of the fixed locus is bounded by the dimension of the cohomology of the inertia coarse space:
  $$\dim H^*(|\mathcal{X}^G|, \mathbb{Z}/2) \leq \dim H^*(|\mathcal{I}\mathcal{X}|, \mathbb{Z}/2)$$
  The authors note that the naive bound using $|\mathcal{X}|$ fails (e.g., for classifying stacks $B\Gamma$).
• Proposition 1.8 (Verification for Classifying Stacks): The conjecture is verified for the classifying groupoid $B\Gamma$ of a finite group $\Gamma$ with an involution. The proof reduces the topological inequality to a group-theoretic statement: $\#H^1(G, \Gamma) \leq \#(\Gamma/\Gamma)$, where the RHS is the number of conjugacy classes. This inequality is proven using orbit-stabilizer arguments.
• Groupoid Cohomology Variants: The paper explores a complementary approach using groupoid cohomology. It notes that while groupoid cohomology is often infinite-dimensional, the ratio of partial sums of Betti numbers for $B\Gamma$ and its fixed locus converges to a rational value derived from Poincaré series. This leads to Question 5.6 regarding the existence of a uniform bound for these ratios.

Significance
The paper claims significance primarily as the first part of a two-part series establishing a topological framework for real algebraic stacks.

• Framework Validation: It demonstrates that topological groupoids with involution provide a natural and rigorous setting to study real moduli spaces, overcoming the limitations of coarse moduli spaces which often fail to capture the correct topology of the real locus.
• Foundational Results: By proving that the real analytic topology coincides with the quotient topology of the fixed-point groupoid (Theorem 1.4), the authors provide the necessary topological groundwork for future computations.
• New Inequalities: The formulation of the Smith–Thom type conjecture for groupoids (Conjecture 1.6) offers a new direction in real algebraic geometry, suggesting that the "size" of the real locus is controlled not just by the complex locus, but by the complex inertia stack.
• Verification: The verification of the conjecture for classifying stacks (Proposition 1.8) serves as a crucial test case, confirming that the proposed inequality holds in non-trivial stacky situations where the naive inequality fails.

The authors explicitly state that the development of techniques to compute the topology of $|\mathcal{X}(\mathbb{R})|$ for general stacks and the full verification of the conjecture in the algebraic setting will be the subject of the subsequent paper [AGF25].