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# Homotopy theory of topological stacks/orbifoldstacks/orbifolds

## Motivation $\newcommand{\T}{\mathscr{T}}$

I have many times found myself saying some variant of the following. Let $\T_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma_g$ its mapping class group. The quotient $\T_g/\Gamma_g$ is the moduli space of curves $M_g$. It is a deep fact that $\T_g$ is in fact diffeomorphic a complex ball, which implies that $M_g$ is a model for the classifying space $B\Gamma_g$ of the mapping class group. In particular, the cohomology of $M_g$ is just the group cohomology of $\Gamma_g$.

...well, almost. Since $\Gamma_g$ does not act freely, $M_g$ is in fact not a $B\Gamma_g$. However, all the stabilizers are finite groups, and this implies via a spectral sequence argument that the rational cohomology of $M_g$ coincides with the rational cohomology of $\Gamma_g$.

Most algebraic geometers seem to ignore these issues by working instead with stacks or orbifolds. Indeed, the stack quotient $[\T_g/\Gamma_g]$ is the moduli stack of curves $\mathcal M_g$, which is in any case the more fundamental object of study.

## Question

My question is whether the topological arguments in the first two paragraph can be carried out in a more highbrow way using orbifolds or topological stacks. I am vaguely aware that Noohi's work on topological stacks includes setting up a homotopy theory of topological stacks, but I know almost nothing about any of this. So the question should be interpreted as "does there exist a developed homotopy theory of topological stacks where the following question can be asked and answered".

Question 1. Are any two quotients $[E/G]$ and $[E'/G]$, where $E$ and $E'$ are contractible spaces with a not necessarily free group action by $G$, homotopy equivalent as topological stacks?

My second question is more speculative since I know even less about rational homotopy theory.

Question 2. Let $\mathcal X$ be a topological stack, with coarse moduli space $X$. Suppose that all isotropy groups of $\mathcal X$ are torsion. Is $\mathcal X \to X$ a rational homotopy equivalence?

1

# Homotopy theory of topological stacks/orbifold

## Motivation $\newcommand{\T}{\mathscr{T}}$

I have many times found myself saying some variant of the following. Let $\T_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma_g$ its mapping class group. The quotient $\T_g/\Gamma_g$ is the moduli space of curves $M_g$. It is a deep fact that $\T_g$ is in fact a complex ball, which implies that $M_g$ is a model for the classifying space $B\Gamma_g$ of the mapping class group. In particular, the cohomology of $M_g$ is just the group cohomology of $\Gamma_g$.

...well, almost. Since $\Gamma_g$ does not act freely, $M_g$ is in fact not a $B\Gamma_g$. However, all the stabilizers are finite groups, and this implies via a spectral sequence argument that the rational cohomology of $M_g$ coincides with the rational cohomology of $\Gamma_g$.

Most algebraic geometers seem to ignore these issues by working instead with stacks or orbifolds. Indeed, the stack quotient $[\T_g/\Gamma_g]$ is the moduli stack of curves $\mathcal M_g$, which is in any case the more fundamental object of study.

## Question

My question is whether the topological arguments in the first two paragraph can be carried out in a more highbrow way using orbifolds or topological stacks. I am vaguely aware that Noohi's work on topological stacks includes setting up a homotopy theory of topological stacks, but I know almost nothing about any of this. So the question should be interpreted as "does there exist a developed homotopy theory of topological stacks where the following question can be asked and answered".

Question 1. Are any two quotients $[E/G]$ and $[E'/G]$, where $E$ and $E'$ are contractible spaces with a not necessarily free group action by $G$, homotopy equivalent as topological stacks?

My second question is more speculative since I know even less about rational homotopy theory.

Question 2. Let $\mathcal X$ be a topological stack, with coarse moduli space $X$. Suppose that all isotropy groups of $\mathcal X$ are torsion. Is $\mathcal X \to X$ a rational homotopy equivalence?