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Topological groupoids with involution and real algebraic stacks.

Emiliano Ambrosi1, Olivier de Gaay Fortman2

  • 1Institut de Recherche Mathématique Avancée (IRMA), 7 Rue René Descartes, Strasbourg, 67084 France.

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|June 29, 2026
PubMed
Summary
This summary is machine-generated.

We introduce a topological groupoid of fixed points, generalizing fixed-point subspaces. For Deligne-Mumford stacks over real numbers, this fixed locus matches the real locus, offering a framework for studying real algebraic stacks.

Keywords:
14A2014D2314P2522A22

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Area of Science:

  • Algebraic Geometry
  • Topology
  • Differential Geometry

Background:

  • Topological groupoids and involutions are fundamental in various mathematical fields.
  • Understanding fixed points is crucial in topology and geometry.
  • Real algebraic stacks and moduli spaces are active research areas.

Purpose of the Study:

  • To generalize the concept of fixed-point subspaces to topological groupoids.
  • To establish a connection between the fixed locus of a topological groupoid and the real locus of a Deligne-Mumford stack.
  • To develop a topological framework for studying real algebraic stacks and propose a new conjecture.

Main Methods:

  • Association of a topological groupoid of fixed points to a given topological groupoid with involution.
  • Demonstration of the coincidence of the fixed locus and real locus for stacks over the real numbers.
  • Formulation of a Smith-Thom type conjecture in the context of topological groupoids with involutions.

Main Results:

  • A new construction of a topological groupoid of fixed points is presented.
  • It is proven that for Deligne-Mumford stacks over R, the fixed locus equals the real locus.
  • A novel topological framework for the study of real algebraic stacks is established.

Conclusions:

  • The introduced framework provides powerful tools for analyzing real algebraic stacks and moduli spaces.
  • The generalization of the fixed-point subspace concept deepens the understanding of involutions in groupoids.
  • The proposed Smith-Thom type conjecture opens new avenues for research in algebraic topology and geometry.