  We produce examples of complex algebraic surfaces with isolated singularities
such that these singularities are not metrically conic, i.e. the germs of the
surfaces near singular points are not bi-Lipschitz equivalent, with respect to
the inner metric, to cones. The technique used to prove the nonexistence of the
metric conic structure is related to a development of Metric Homology. The
class of the examples is rather large and it includes some surfaces of
Brieskorn.
