In our context Poincaré duality means that we can apply facts we know in general about discrete surfaces to the dual surface $M^*$ and obtain information about $M$ in this way.

A discrete surface $M$ is connected if and only if $E$ has no proper subsets that are invariant under both $s$ and $\rho$ (see an earlier post). Using this characterization of connectedness it is easy to see that for connected $M$ also $M^*$ is connected.

As an immediate application, we note that for a connected $M$ the kernel of $\partial_2$ is one-dimensional: it consists of the constant dual 0-forms and these can also be viewed as ordinary 0-forms on $M^*$. Now the image of a linear map between vector spaces is always the orthogonal complement of the kernel of its adjoint map. In our case this yields

$\textrm{Im }d_1=(\textrm{Ker }\partial_2)^\perp$.

Theorem: A 2-form $\sigma$ on a connected surface $M$ is exact if and only if

$\int_M \sigma = 0$.

In particular

$\textrm{dim(Im }d_1) = \#F-1$.