JLO Cocycle

In noncommutative geometry, the JLO cocycle is a cocycle (and thus defines a cohomology class) in entire cyclic cohomology. It is a non-commutative version of the classic Chern character of the conventional differential geometry. In noncommutative geometry, the concept of a manifold is replaced by a noncommutative algebra of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra contains the information about the topology of that noncommutative space, very much as the deRham cohomology contains the information about the topology of a conventional manifold.

The JLO cocycle is associated with a metric structure of non-commutative differential geometry known as a -summable Fredholm module (also known as a -summable spectral triple).