Difference between revisions of "Connes-Moscovici index theorem"
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Revision as of 18:51, 24 March 2012
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
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where is the algebra of trace-class operators on
(cf. also Trace). More generally, one can assume that
is an invariant pseudo-differential operator on
(with nice support).
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.
References
[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