[Paper Review] Group actions on stacks and applications to equivariant string topology for stacks

Rafael J. L. Morcillo, Behrang Noohi|arXiv (Cornell University)|Jun 25, 2012

Homotopy and Cohomology in Algebraic Topology29 references8 citations

TL;DR

This paper establishes a graded Lie algebra structure on the $S^1$-equivariant homology of the free loop stack $\mathrm{L}{\mathfrak{X}}$ of an oriented differentiable stack ${\mathfrak{X}}$, generalizing the Chas-Sullivan string bracket to stacks. It constructs this via a transfer map and quotient stack techniques, proving that $H^{S^1}_{*+\dim{\mathfrak{X}}-2}(\mathrm{L}{\mathfrak{X}})$ is a graded Lie algebra, with an explicit Goldman-type bracket for 2-dimensional orbifolds.

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 $H^{SO(2)}_{*+dim(X) -2}(L(X))$ 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.

Motivation & Objective

• To extend Chas-Sullivan's string topology construction to differentiable stacks, particularly by defining an $S^1$-equivariant string bracket on the free loop stack $\mathrm{L}{\mathfrak{X}}$.
• To develop a general theory of group actions and quotients on topological stacks, independent of the main application, for broader applicability in equivariant topology.
• To provide a geometric realization of the string bracket for 2-dimensional orbifolds, generalizing the classical Goldman bracket to non-manifold stacks.
• To define equivariant (co)homology for stacks using the homotopy type of quotient stacks, enabling the use of transfer maps and Gysin sequences.

Proposed method

• Constructs the quotient stack $[G\backslash{\mathfrak{X}}]$ for a group $G$ acting on a stack ${\mathfrak{X}}$, showing it is a topological stack and geometric if ${\mathfrak{X}}$ is geometric.
• Uses the homotopy type of the quotient stack $[S^1\backslash\mathrm{L}{\mathfrak{X}}]$ to define equivariant homology for stacks via classifying spaces of mapping stacks.
• Introduces a transfer map $T: H^{S^1}_*(\mathrm{L}{\mathfrak{X}})[2-d] \to H_*(\mathrm{L}{\mathfrak{X}})[1-d]$ using the quotient map $q: \mathrm{L}{\mathfrak{X}} \to [S^1\backslash\mathrm{L}{\mathfrak{X}}]$.
• Defines the string bracket via $\{x,y\} := (-1)^{|x|} q(T(x) \star T(y))$, where $\star$ is the Pontryagin product, and proves it endows the shifted equivariant homology with a graded Lie algebra structure.
• Applies the framework to 2-dimensional orbifolds by showing that the fundamental group of the quotient stack $[S^1\backslash\mathrm{L}{\mathfrak{X}}]$ is isomorphic to that of the interior of the underlying surface, enabling a Goldman-type description.
• Uses van Kampen's theorem and presentation of fundamental groups to identify the Lie algebra structure on $H_0^{S^1}(\mathrm{L}{\mathfrak{X}})$ with the free module on conjugacy classes of the fundamental group of the surface.

Experimental results

Research questions

• RQ1 Can the Chas-Sullivan string bracket construction be generalized from manifolds to differentiable stacks, particularly via equivariant homology?
• RQ2 How can equivariant (co)homology for stacks be defined using group actions and quotient stacks, especially when standard groupoid models fail to be functorial?
• RQ3 What is the explicit geometric description of the string bracket for 2-dimensional orbifolds, and how does it relate to the classical Goldman bracket?
• RQ4 Does the transfer map from equivariant to non-equivariant homology preserve the Lie algebra structure in the stack setting?
• RQ5 To what extent does the homotopy type of the quotient stack $[S^1\backslash\mathrm{L}{\mathfrak{X}}]$ determine the equivariant homology and its algebraic structure?

Key findings

• The $S^1$-equivariant homology $H^{S^1}_{*+\dim{\mathfrak{X}}-2}(\mathrm{L}{\mathfrak{X}})$ is a graded Lie algebra under the bracket $\{x,y\} := (-1)^{|x|} q(T(x) \star T(y))$.
• The transfer map $T: H^{S^1}_*(\mathrm{L}{\mathfrak{X}})[2-d] \to H_*(\mathrm{L}{\mathfrak{X}})[1-d]$ is a Lie algebra homomorphism, preserving the algebraic structure.
• For a 2-dimensional orbifold ${\mathfrak{X}}_{g,h}$, the Lie algebra $H_0^{S^1}(\mathrm{L}{\mathfrak{X}}_{g,h})$ is isomorphic to $\mathbb{Z}[\operatorname{Conj}(F(\gamma_1,\dots,\gamma_{2g}, a_1,\dots,a_{h-1}))]$, with bracket computable via geometric intersections of loops.
• The inclusion $(\Sigma_{g,h,0})^0 \hookrightarrow {\mathfrak{X}}_{g,h}$ induces an isomorphism on $\pi_1$, and thus on the Goldman Lie algebra after taking free loops.
• The homology $H_3({\mathfrak{X}}_{g,h})$ is isomorphic to $\mathbb{Z}^{h-1} \oplus \mathbb{Z}$, with no nontrivial extensions, confirming the associated graded structure is fully determined.
• The string bracket of generators $\gamma_i$ and $\gamma_j$ in $H_0^{S^1}(\mathrm{L}{\mathfrak{X}}_{g,h})$ is nontrivial in general, reflecting non-abelian intersection behavior.