  We use the heat kernel (on differential forms) on a compact Riemannian
manifold to assign a real number to a k-tuple of cycles on the manifold
satisfying certain conditions. If k is 2, this number is the ordinary
topological linking number, an integer but if k is at least 3 the number is
real and depends on the metric. We study how the number changes if the cycles
vary in their homology classes (preserving the conditions) - it changes by
specifically described linking integers. The main condition is that the
intersection of all k cycles is empty and the intersection of any k-1 bounds on
the intersection of any k-2 of these. The real number is expressed by iterated
integrals of the Poincare dual harmonic forms. If the manifold is complex
Kahler the number depends only on the complex structure and not on the metric :
in this case the forms need not be harmonic; also we are reduced to considering
just those cycles which are not complex.
