# Invertible phases of matter with spatial symmetry

## TL;DR

This paper introduces a comprehensive formula for classifying invertible topological phases of matter with spatial symmetry, applicable to various symmetry types and space configurations.

## Contribution

It provides a general classification formula for invertible topological phases considering arbitrary symmetry groups and spatial configurations.

## Key findings

- Derived a universal formula for invertible topological phases with symmetry.
- Applicable to Euclidean space and crystallographic groups.
- Clarified the concept of topological crystalline phases.

## Abstract

We propose a general formula for the group of invertible topological phases on a space $Y$, possibly equipped with the action of a group $G$. Our formula applies to arbitrary symmetry types. When $Y$ is Euclidean space and $G$ a crystallographic group, the term `topological crystalline phases' is sometimes used for these phases of matter.