  In this paper we combine a survey of the most important topological
properties of kinematic maps that appear in robotics, with the exposition of
some basic results regarding the topological complexity of a map. In
particular, we discuss mechanical devices that consist of rigid parts connected
by joints and show how the geometry of the joints determines the forward
kinematic map that relates the configuration of joints with the pose of the
end-effector of the device. We explain how to compute the dimension of the
joint space and describe topological obstructions for a kinematic map to be a
fibration or to admit a continuous section. In the second part of the paper we
define the complexity of a continuous map and show how the concept can be
viewed as a measure of the difficulty to find a robust manipulation plan for a
given mechanical device. We also derive some basic estimates for the complexity
and relate it to the degree of instability of a manipulation plan.
