  We introduce a notion of homogeneous topological order, which is obeyed by
most, if not all, known examples of topological order including fracton phases
on quantum spins (qudits). The notion is a condition on the ground state
subspace, rather than on the Hamiltonian, and demands that given a collection
of ball-like regions, any linear transformation on the ground space be realized
by an operator that avoids the ball-like regions. We derive a bound on the
ground state degeneracy $\mathcal D$ for systems with homogeneous topological
order on an arbitrary closed Riemannian manifold of dimension $d$, which reads
\[ \log \mathcal D \le c \mu (L/a)^{d-2}.\] Here, $L$ is the diameter of the
system, $a$ is the lattice spacing, and $c$ is a constant that only depends on
the isometry class of the manifold, and $\mu$ is a constant that only depends
on the density of degrees of freedom. If $d=2$, the constant $c$ is the
(demi)genus of the space manifold. This bound is saturated up to constants by
known examples.

Quantum Physics;Strongly Correlated Electrons