  We discuss the topology of the parameter space of invertible phases with an
onsite symmetry $G$, i.e., quantum many-body ground states that have neither
fractionalization nor spontaneous breaking of the symmetry. The classification
of invertible phases is known to be obtained by counting the connected
components in the parameter space of the invertible phases. We consider its
generalization -- the deformation classes of the mappings from $n$-dimensional
spheres $S^n$ to this parameter space for arbitrary integer $n$. We argue a
direct one-to-one correspondence in the framework of lattice models between the
non-contractibility of $S^n$ and (i) the classification of invertible phases in
$d$ dimensions when $d\geq n$; or (ii) zero-dimensional invertible Hamiltonians
parametrized by $S^{n-d}$ when $d<n$, using an isotropic modulating Hamiltonian
approach. Explicitly, we construct the noncontractible spheres of
two-dimensional invertible phases, i.e., $n=2$ and $d=2$. We also propose a
large class of crystalline topological phases protected by a generalized
magnetic translations.
