  Deformable objects present a formidable challenge for robotic manipulation
due to the lack of canonical low-dimensional representations and the difficulty
of capturing, predicting, and controlling such objects. We construct compact
topological representations to capture the state of highly deformable objects
that are topologically nontrivial. We develop an approach that tracks the
evolution of this topological state through time. Under several mild
assumptions, we prove that the topology of the scene and its evolution can be
recovered from point clouds representing the scene. Our further contribution is
a method to learn predictive models that take a sequence of past point cloud
observations as input and predict a sequence of topological states, conditioned
on target/future control actions. Our experiments with highly deformable
objects in simulation show that the proposed multistep predictive models yield
more precise results than those obtained from computational topology libraries.
These models can leverage patterns inferred across various objects and offer
fast multistep predictions suitable for real-time applications.
