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<p>To a topological groupoid endowed with an involution, we associate a topological groupoid of fixed points, generalizing the fixed-point subspace of a topological space with involution. We prove that when the topological groupoid with involution arises from a Deligne-Mumford stack over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \mathbb {R} \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>, this fixed locus coincides with the real locus of the stack. This provides a topological framework to study real algebraic stacks, and in particular real moduli spaces. Finally, we propose a Smith–Thom type conjecture in this setting, generalizing the Smith–Thom inequality for topological spaces endowed with an involution.</p>

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Topological groupoids with involution and real algebraic stacks

  • 期刊论文

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  • 发表日期：

    2026年6月26日

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  • Emiliano Ambrosi,
  • Olivier de Gaay Fortman

摘要

To a topological groupoid endowed with an involution, we associate a topological groupoid of fixed points, generalizing the fixed-point subspace of a topological space with involution. We prove that when the topological groupoid with involution arises from a Deligne-Mumford stack over \( \mathbb {R} \) R , this fixed locus coincides with the real locus of the stack. This provides a topological framework to study real algebraic stacks, and in particular real moduli spaces. Finally, we propose a Smith–Thom type conjecture in this setting, generalizing the Smith–Thom inequality for topological spaces endowed with an involution.