High Quality Content by WIKIPEDIA articles! 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 mathcal{A} of "functions" on the putative noncommutative space.Noncommutative geometry, or NCG, is a branch of mathematics concerned with the possible spatial interpretations of algebraic structures for which the commutative law fails, that is, for which xy does not always equal yx. Although one could technically construct geometries by simply removing this condition (commutativity), the results are typically trivial or uninteresting. The most common usage of the term, therefore, refers to what is properly called differential noncommutative geometry, a subject which was developed extensively by French mathematician Alain Connes. The challenge of NCG theory is to get around the lack of commutative multiplication, which is a requirement of previous geometric theories of algebraic structures.

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