Several notions of computability can be defined for compact
subsets of Euclidean spaces. Some of them turn out to be equivalent for
certain sets - we then say that the set has ``computable type''. Miller
(2002) proved that n-dimensional spheres have computable type, and
Iljazovic (2013) proved that closed manifolds have computable type. We
study the case of finite simplicial complexes and obtain several
topological characterizations of the complexes having computable type.
We also relate the notion of computable type to the descriptive
complexity of topological invariants, and investigate the expressiveness
of low complexity invariants. This work was done in collaboration with
Djamel Eddine Amir.