Crystalline equivalence principle for topological phases

Let $G$ be a spatial symmetry group acting on physical space $X$, usually $X=\mathbb{R}^d$. Let $X//G$ be the homotopy quotient, defined up to homotopy equivalence by

$$X//G=(X\times EG)/G,$$

where $EG$ is contractible with a free action of $G$, and $G$ acts diagonally on $X\times EG$. Let $\Theta_d$ be the space used to classify $d$-dimensional topological phases.

Crystalline equivalence conjecture. The classification of SPT or SET topological phases in $d$ dimensions with spatial symmetry $G$ acting on $X$ is given by homotopy classes of maps

$$f:X//G\to\Theta_d.$$

This extends the internal-symmetry classification to spatial symmetries and underlies the crystalline equivalence principle. For fermionic systems and bosonic systems with orientation-reversing symmetries, the paper says the statement must be modified so that the map becomes a section of a fiber bundle; the supplied text does not resolve the conjecture.