  There is a well known link between (maximal) polar representations and
isotropy representations of symmetric spaces provided by Dadok. Moreover, the
theory by Tits and Burns-Spatzier provides a link between irreducible symmetric
spaces of non-compact type of rank at least three and irreducible topological
spherical buildings of rank at least three.
  We discover and exploit a rich structure of a (connected) chamber system of
finite (Coxeter) type M associated with any polar action of cohomogeneity at
least two on any simply connected closed positively curved manifold. Although
this chamber system is typically not a Tits geometry of type M, we prove that
in all cases but two that its universal Tits cover indeed is a building. We
construct a topology on this universal cover making it into a compact spherical
building in the sense of Burns and Spatzier. Using this structure we classify
up to equivariant diffeomorphism all polar actions on (simply connected)
positively curved manifolds of cohomogeneity at least two.
