  The algebra of observables of planar electrons subject to a constant
background magnetic field B is given by A_theta(R^2) x A_theta(R^2) the product
of two mutually commuting Moyal algebras. It describes the free Hamiltonian and
the guiding centre coordinates. We argue that A_theta(R^2) itself furnishes a
representation space for the actions of these two Moyal algebras, and suggest
physical arguments for this choice of the representation space. We give the
proper setup to couple the matter fields based on A_theta(R^2) to
electromagnetic fields which are described by the abelian commutative gauge
group G_c(U(1)), i.e. gauge fields based on A_0(R^2). This enables us to give a
manifestly gauge covariant formulation of integer quantum Hall effect (IQHE).
Thus, we can view IQHE as an elementary example of interacting quantum
topologies, where matter and gauge fields based on algebras A_theta^prime with
different theta^prime appear. Two-particle wave functions in this approach are
based on A_theta(R^2) x A_theta(R^2). We find that the full symmetry group in
IQHE, which is the semi-direct product SO(2) \ltimes G_c(U(1)) acts on this
tensor product using the twisted coproduct Delta_theta. Consequently, as we
show, many particle sectors of each Landau level have twisted statistics. As an
example, we find the twisted two particle Laughlin wave functions.
