Topological groupoids with involution and real algebraic stacks

Abstract 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 $$ \mathbb {R} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> , 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.