  A model aimed at understanding quantum gravity in terms of Birkhoff's
approach is discussed. The geometry of this model is constructed by using a
winding map of Minkowski space into a $\mathbb{R}^{3} \times S^{1}$-cylinder.
The basic field of this model is a field of unit vectors defined through the
velocity field of a flow wrapping the cylinder. The degeneration of some parts
of the flow into circles (topological features) results in inhomogeneities and
gives rise to a scalar field, analogous to the gravitational field. The
geometry and dynamics of this field are briefly discussed. We treat the
intersections between the topological features and the observer's 3-space as
matter particles and argue that these entities are likely to possess some
quantum properties.
