  We describe polar homology groups for complex manifolds. The polar k-chains
are subvarieties of complex dimension k with meromorphic forms on them, while
the boundary operator is defined by taking the polar divisor and the Poincare
residue on it. The polar homology groups may be regarded as holomorphic
analogues of the homology groups in topology. We also describe the polar
homology groups for quasi-projective one-dimensional varieties (affine curves).
These groups obey the Mayer--Vietoris property. A complex counterpart of the
Gauss linking number of two curves in a three-fold and various gauge-theoretic
aspects of the above correspondence are also discussed.
