View source for Connes-Moscovici index theorem

''Gamma index theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c1201902.png" /> index theorem''

A theorem [[#References|[a3]]] which computes the pairing of a cyclic cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c1201903.png" /> of the [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c1201904.png" /> with the algebraic [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c1201905.png" />-theory]] index of an invariant (pseudo-) differential operator on a covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c1201906.png" /> with [[Galois group|Galois group]] (or group of deck transformations) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c1201907.png" /> (cf. also [[Cohomology|Cohomology]]).

The ingredients of this theorem are stated in more detail below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c1201908.png" /> be a smooth compact [[Manifold|manifold]].

First, any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c1201909.png" />-invariant, elliptic partial differential operator (cf. [[Elliptic partial differential equation|Elliptic partial differential equation]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019011.png" /> has an algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019013.png" />-theory index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019014.png" />. The definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019015.png" /> is obtained using the boundary mapping on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019016.png" /> applied to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019017.png" />, the principal symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019018.png" /> (cf. also [[Symbol of an operator|Symbol of an operator]]). This gives

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019020.png" /> is the algebra of trace-class operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019021.png" /> (cf. also [[Trace|Trace]]). More generally, one can assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019022.png" /> is an invariant [[Pseudo-differential operator|pseudo-differential operator]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019023.png" /> (with nice support).

Secondly, it is known [[#References|[a2]]] that any group-cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019024.png" />-cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019026.png" /> can be represented by an anti-symmetric function, and hence it defines a cyclic cocycle on the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019027.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019028.png" />. Moreover, the class of this cocycle in the periodic cyclic cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019029.png" />, also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019030.png" />, depends only on the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019032.png" />. Here, as customary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019033.png" /> denotes the [[Classifying space|classifying space]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019034.png" />, whose simplicial cohomology is known to be isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019035.png" />, the group cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019036.png" />.

Finally, any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019037.png" /> gives rise to a group morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019038.png" />, see [[#References|[a2]]]. In particular, any group cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019039.png" /> gives rise to a mapping

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019040.png" /></td> </tr></table>

using also the trace on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019041.png" />.

The Connes–Moscovici index theorem now states [[#References|[a3]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019042.png" /> be the mapping classifying the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019043.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019044.png" /> be the Todd class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019045.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019046.png" /> be the [[Chern character|Chern character]] of the element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019047.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019048.png" />, as in the Atiyah–Singer index theorem (see [[#References|[a1]]] and [[Index formulas|Index formulas]]). Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019049.png" /></td> </tr></table>

is a pairing of a compactly supported cohomology class with the fundamental class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019050.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019051.png" />.

The Connes–Moscovici index theorem is sometimes called the higher index theorem for coverings and is the prototype of a higher index theorem.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators I"  ''Ann. of Math.'' , '''93'''  (1971)  pp. 484–530</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Connes,   "Non-commutative differential geometry"  ''Publ. Math. IHES'' , '''62'''  (1985)  pp. 41–144</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Connes,   H. Moscovici,   "Cyclic cohomology, the Novikov conjecture and hyperbolic groups"  ''Topology'' , '''29'''  (1990)  pp. 345–388</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Lusztig,   "Novikov's higher signature and families of elliptic operators"  ''J. Diff. Geom.'' , '''7'''  (1972)  pp. 229–256</TD></TR></table>