Heat Kernels and Cycles

Abstract

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.