Connes-Moscovici index theorem

Gamma index theorem, $\Gamma$ index theorem

A theorem [a3] which computes the pairing of a cyclic cocycle $\varphi$ of the group algebra $\mathbf{C} [ \Gamma ]$ with the algebraic $K$-theory index of an invariant (pseudo-) differential operator on a covering $\tilde { M } \rightarrow M$ with Galois group (or group of deck transformations) $\Gamma$ (cf. also Cohomology).

The ingredients of this theorem are stated in more detail below. Let $M$ be a smooth compact manifold.

First, any $\Gamma$-invariant, elliptic partial differential operator (cf. Elliptic partial differential equation) $D$ on $\tilde { M }$ has an algebraic $K$-theory index $\operatorname {ind} ( D )$. The definition of $\operatorname {ind} ( D )$ is obtained using the boundary mapping on $K _ { 1 }$ applied to $\sigma ( D )$, the principal symbol of $D$ (cf. also Symbol of an operator). This gives

\begin{equation*} \operatorname {ind}( D ) \in K _ { 0 } ^ { \text{alg} } ( \mathcal{C} _ { 1 } \bigotimes \mathbf{C} [ \Gamma ] ), \end{equation*}

where $C _ { 1 }$ is the algebra of trace-class operators on $L ^ { 2 } ( M )$ (cf. also Trace). More generally, one can assume that $D$ is an invariant pseudo-differential operator on $\tilde { M }$ (with nice support).

Secondly, it is known [a2] that any group-cohomology $q$-cocycle $\varphi : \Gamma ^ { q + 1 } \rightarrow \mathbf{C}$ of $\Gamma$ can be represented by an anti-symmetric function, and hence it defines a cyclic cocycle on the group algebra $\mathbf{C} [ \Gamma ]$ of the group $\Gamma$. Moreover, the class of this cocycle in the periodic cyclic cohomology group $\operatorname{HP} ^ { q } ( \mathbf{C} [ \Gamma ] )$, also denoted by $\varphi$, depends only on the class of $\varphi$ in $H ^ { q } ( B \Gamma , \mathbf{C} ) \simeq H ^ { q } ( \Gamma , \mathbf{C} )$. Here, as customary, $B \Gamma$ denotes the classifying space of $\Gamma$, whose simplicial cohomology is known to be isomorphic to $H ^ { q } ( \Gamma , \mathbf{C} )$, the group cohomology of $\Gamma$.

Finally, any element $\varphi \in \operatorname{HP} ^ { 0 } ( A )$ gives rise to a group morphism $\varphi_{ * } : K _ { 0 } ^ { \text{alg} } ( A ) \rightarrow \mathbf{C}$, see [a2]. In particular, any group cocycle $\varphi \in H ^ { 2 m } ( \Gamma , {\bf C} )$ gives rise to a mapping

\begin{equation*} \varphi_{ *} : K _ { 0 } ^ { \text{alg} } ( \mathcal{C} _ { 1 } \bigotimes \mathbf{C} [ \Gamma ] ) \rightarrow \mathbf{C}, \end{equation*}

using also the trace on $\mathcal{C} _ { 1 }$.

The Connes–Moscovici index theorem now states [a3]): Let $f : M \rightarrow B \Gamma$ be the mapping classifying the covering $\tilde { M }$, let $\mathcal{T} ( M )$ be the Todd class of $M$, and let $\operatorname {Ch} ( D ) \in H _ { c } ^ { * } ( T M )$ be the Chern character of the element in $K ( T M )$ defined by $\sigma ( D )$, as in the Atiyah–Singer index theorem (see [a1] and Index formulas). Then

\begin{equation*} \phi_{ *} ( \operatorname { ind } ( D ) ) = c _ { q } \left( \operatorname { Ch } ( D ) \mathcal{T} ( M ) f ^ { * } ( \phi ) \right) [ T M ] \end{equation*}

is a pairing of a compactly supported cohomology class with the fundamental class of $T M$. Here, $c _ { q } = ( - 1 ) ^ { q } q ! / ( 2 q ) !$.

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