Higherindex

Higherindex is a term used primarily in mathematics to describe a family of invariants that generalize the analytic index of elliptic operators by incorporating group actions and noncommutative geometry. In its most developed form, the Mishchenko–Fomenko higher index promotes the index of an elliptic operator on a closed manifold to a K-theory class in the C*-algebra of the manifold’s fundamental group.

Construction typically begins with a closed manifold M and its universal cover \tilde M, which carries an action of the fundamental group G = π1(M). Using the Mishchenko line bundle associated to the reduced group C*-algebra C*_r(G), one forms a G-equivariant Dirac operator on \tilde M. The analytic index Ind(D) then takes values in K_*(C*_r(G)), providing a refined invariant that encodes information about the topology of M and the geometry of the covering. Pairing this higher index with cyclic cocycles yields numerical higher indices, which can be compared with classical index theory.

Applications of higher indices include obstructions to positive scalar curvature, rigidity phenomena in geometry, and connections between index theory and noncommutative geometry. They also relate to broader programmatic themes such as the Baum–Connes conjecture and coarse geometric methods, offering tools to study manifolds via operator-algebraic methods.

History and development center on the work of Mishchenko and Fomenko in the 1960s–70s, with significant later contributions from Connes, Moscovici, Roe, and others. While the term is most closely associated with these operator-algebraic invariants, the phrase “higher index” can appear in various mathematical contexts, always aiming to capture a richer, non-integer invariant extending the classical index.