Group actions on stacks and applications to equivariant string topology for stacks

Abstract

This paper is a continuations of the project initiated in the book string topology for stacks. We construct string operations on the SO(2)-equivariant homology of the (free) loop space $L(X)$ of an oriented differentiable stack $X$ and show that $H_{\ast +dim(X)-2}^{SO(2)}(L(X))$ is a graded Lie algebra. In the particular case where $X$ is a 2-dimensional orbifold we give a Goldman-type description for the string bracket. To prove these results, we develop a machinery of (weak) group actions on topological stacks which should be of independent interest. We explicitly construct the quotient stack of a group acting on a stack and show that it is a topological stack. Then use its homotopy type to define equivariant (co)homology for stacks, transfer maps, and so on.

Several new examples added, 57 pages

References in corpus (4)

• Foundations of Topological Stacks I
• Higher Hochschild cohomology, Brane topology and centralizers of $E_n$-algebra maps
• On string topology of classifying spaces
• String topology of classifying spaces

Cited by in corpus (3)

• Principal actions of stacky Lie groupoids
• Topological T-duality for Stacks using a Gysin Sequence
• Derived deformation theory of algebraic structures