A Comparison of Leibniz and Cyclic Homologies

Abstract

We relate Leibniz homology to cyclic homology by studying a map from a long exact sequence in the Leibniz theory to the Connes' periodicity (ISB) exact sequence in the cyclic theory. We then show that the Godbillon–Vey invariant, as detected by the Leibniz homology of formal vector fields, maps to the Godbillon–Vey invariant as detected by the cyclic homology of the universal enveloping algebra of these vector fields. Additionally the Leibniz theory maps surjectively to string topology where the latter is expressed as cyclic homology.

Key Words:

• Godbillon–Vey invariant
• Leibniz homology
• String topology

2000 Mathematics Subject Classification:

• 19D55
• 17B66
• 17A32