TITLE: When are (finite) simplicial complexes (smooth) manifolds?
QUESTION [18 upvotes]: Hi,
is there an algorithm that determines if a given simplicial complex is 
a.) a manifold
b.) a smooth manifold
c.) homotopy equivalent to a manifold
d.) a real algebraic variety
?

REPLY [4 votes]: (b) When does a PL manifold have a smooth structure?
I believe that this is computable. Smoothing theory, closely related to the Hirsch-Smale immersion theory (an h-principle), identifies the set of smooth structures on a PL manifold with the set vector bundles that are reasonable candidates for the tangent bundle in the sense that they are reductions of structure group of the PL tangent bundle. The theory of principal bundles reduces to homotopy theory. The set of homotopy classes of maps from a finite complex to a finite complex with abelian fundamental group and the induced maps between such sets are computable, but it is rarely implemented.