---
created: 2022-04-05T23:42
updated: 2024-04-16T20:35
aliases:
  - symmetric space
  - locally symmetric space
  - locally symmetric
tags:
  - homotopy/stable-homotopy
  - dissertation
---

- #why-care about the cohomology of locally symmetric spaces?
	- These provide integral structures on spaces of [automorphic forms](automorphic%20form.md), which are needed in order to use [p-adic](p-adic.md) methods.

# symmetric spaces

Consider the locally symmetric spaces $\Gamma \backslash X$ and $\Gamma^{\prime} \backslash X$. Since $\Gamma^{\prime}$ is normal in $\Gamma$, the finite group $\Gamma / \Gamma^{\prime}$ acts on $\Gamma^{\prime} \backslash X$ with quotient $\Gamma \backslash X$, i.e., the natural morphism $\pi_{\Gamma^{\prime} \mid \Gamma}: \Gamma^{\prime} \backslash X \longrightarrow \Gamma \backslash X$ is a [[Galois cover]]. Assuming $\Gamma$ is torsion free it is étale, while if $\Gamma$ has torsion this cover will be branched. 

![](2024-04-13-12.png)

Important examples:
- See [hyperbolic](hyperbolic%20geometry.md)
- See [modular curve](modular%20curve.md)
- See [Siegel upper half space](Hermitian%20symmetric%20domains.md)
## In AG/complex geometry

In complex geometry: **locally symmetric spaces** are $\dcoset\Gamma G K$ with $G$ a semisimple Lie group, $K$ a maximal compact, $\Lambda$ a lattice in $G$.


## In homotopy theory

See Barnes and Roitzheim 7.1.
A **symmetric space** $F$ is a $\Top_*$ [enriched functor](enriched%20functor) $F: \Sigma \to \Top_*$,  where $\Sigma$ is the category whose objects are $\NN$ and 
$$\Sigma(a, b)=\left\{\begin{array}{ll}\left(\Sigma_{a}\right)_{+} & \text {if } a=b \\ * & \text { if } a \neq b .\end{array}\right.$$

Equivalently, a sequence of pointed spaces $X_n$ with associative unital maps
$$
(\Sigma_n)_+ \smashprod X_n \to X_{n}
,$$
i.e. an action of $\Sigma_n \actson X_n$.