  Polaritonic lattice configurations in dimensions $D=2$ are used as simulators
of topological phases, based on symmetry class A Hamiltonians. Numerical and
topological studies are performed in order to characterise the bulk topology of
insulating phases, which is predicted to be connected to non-trivial edge mode
states on the boundary. By using spectral flattened Hamiltonians on specific
lattice geometries with time reversal symmetry breaking, e.g. Kagome lattice,
we obtain maps from the Brillouin zone into Grassmannian spaces, which are
further investigated by the topological method of space fibrations. Numerical
evidence reveals a connection between the sum of valence band Chern numbers and
the index of the projection operator onto the valence band states. Along these
lines, we discover an index formula which resembles other index theorems and
the classical result of Atiyah-Singer, but without any Dirac operator and from
a different perspective. Through a combination of different tools, in
particular homotopy and homology-cohomology duality, we provide a comprehensive
mathematical framework, which fully addresses the source and structure of
topological phases in coupled polaritonic array systems. Based on these
results, it becomes possible to infer further designs and models of
two-dimensional single sheet Chern insulators, implemented as polariton
simulators.
