  This paper studies closed 3-manifolds which are the attractors of a system of
finitely many affine contractions that tile $\mathbb{R}^3$. Such attractors are
called self-affine tiles. Effective characterization and recognition theorems
for these 3-manifolds as well as theoretical generalizations of these results
to higher dimensions are established. The methods developed build a bridge
linking geometric topology with iterated function systems and their attractors.
  A method to model self-affine tiles by simple iterative systems is developed
in order to study their topology. The model is functorial in the sense that
there is an easily computable map that induces isomorphisms between the natural
subdivisions of the attractor of the model and the self-affine tile. It has
many beneficial qualities including ease of computation allowing one to
determine topological properties of the attractor of the model such as
connectedness and whether it is a manifold. The induced map between the
attractor of the model and the self-affine tile is a quotient map and can be
checked in certain cases to be monotone or cell-like. Deep theorems from
geometric topology are applied to characterize and develop algorithms to
recognize when a self-affine tile is a topological or generalized manifold in
all dimensions. These new tools are used to check that several self-affine
tiles in the literature are 3-balls. An example of a wild 3-dimensional
self-affine tile is given whose boundary is a topological 2-sphere but which is
not itself a 3-ball. The paper describes how any 3-dimensional handlebody can
be given the structure of a self-affine 3-manifold. It is conjectured that
every self-affine tile which is a manifold is a handlebody.
