Connes-Moscovici index theorem
Gamma index theorem, index theorem
A theorem [a3] which computes the pairing of a cyclic cocycle of the group algebra with the algebraic -theory index of an invariant (pseudo-) differential operator on a covering with Galois group (or group of deck transformations) (cf. also Cohomology).
The ingredients of this theorem are stated in more detail below. Let be a smooth compact manifold.
First, any -invariant, elliptic partial differential operator (cf. Elliptic partial differential equation) on has an algebraic -theory index . The definition of is obtained using the boundary mapping on applied to , the principal symbol of (cf. also Symbol of an operator). This gives
Secondly, it is known [a2] that any group-cohomology -cocycle of can be represented by an anti-symmetric function, and hence it defines a cyclic cocycle on the group algebra of the group . Moreover, the class of this cocycle in the periodic cyclic cohomology group , also denoted by , depends only on the class of in . Here, as customary, denotes the classifying space of , whose simplicial cohomology is known to be isomorphic to , the group cohomology of .
Finally, any element gives rise to a group morphism , see [a2]. In particular, any group cocycle gives rise to a mapping
using also the trace on .
The Connes–Moscovici index theorem now states [a3]): Let be the mapping classifying the covering , let be the Todd class of , and let be the Chern character of the element in defined by , as in the Atiyah–Singer index theorem (see [a1] and Index formulas). Then
is a pairing of a compactly supported cohomology class with the fundamental class of . Here, .
The Connes–Moscovici index theorem is sometimes called the higher index theorem for coverings and is the prototype of a higher index theorem.
|[a1]||M.F. Atiyah, I.M. Singer, "The index of elliptic operators I" Ann. of Math. , 93 (1971) pp. 484–530|
|[a2]||A. Connes, "Non-commutative differential geometry" Publ. Math. IHES , 62 (1985) pp. 41–144|
|[a3]||A. Connes, H. Moscovici, "Cyclic cohomology, the Novikov conjecture and hyperbolic groups" Topology , 29 (1990) pp. 345–388|
|[a4]||G. Lusztig, "Novikov's higher signature and families of elliptic operators" J. Diff. Geom. , 7 (1972) pp. 229–256|
Connes-Moscovici index theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connes-Moscovici_index_theorem&oldid=13937