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''Gamma index theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c1201902.png" /> index theorem''
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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]]).
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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]].
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''Gamma index theorem, $\Gamma$ index theorem''
  
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
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A theorem [[#References|[a3]]] which computes the pairing of a cyclic cocycle $\varphi$ of the [[Group algebra|group algebra]] $\mathbf{C} [ \Gamma ]$ with the algebraic [[K-theory|$K$-theory]] index of an invariant (pseudo-) differential operator on a covering $\tilde { M } \rightarrow M$ with [[Galois group|Galois group]] (or group of deck transformations) $\Gamma$ (cf. also [[Cohomology|Cohomology]]).
  
<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>
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The ingredients of this theorem are stated in more detail below. Let $M$ be a smooth compact [[Manifold|manifold]].
  
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).
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First, any $\Gamma$-invariant, elliptic partial differential operator (cf. [[Elliptic partial differential equation|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|Symbol of an operator]]). This gives
  
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" />.
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\begin{equation*} \operatorname {ind}( D ) \in K _ { 0 } ^ { \text{alg} } ( \mathcal{C} _ { 1 } \bigotimes \mathbf{C} [ \Gamma ] ), \end{equation*}
  
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
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where $C _ { 1 }$ is the algebra of trace-class operators on $L ^ { 2 } ( M )$ (cf. also [[Trace|Trace]]). More generally, one can assume that $D$ is an invariant [[Pseudo-differential operator|pseudo-differential operator]] on $\tilde { M }$ (with nice support).
  
<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>
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Secondly, it is known [[#References|[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|classifying space]] of $\Gamma$, whose simplicial cohomology is known to be isomorphic to $H ^ { q } ( \Gamma , \mathbf{C} )$, the group cohomology of $\Gamma$.
  
using also the trace on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120190/c12019041.png" />.
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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 [[#References|[a2]]]. In particular, any group cocycle $\varphi \in H ^ { 2 m } ( \Gamma , {\bf C} )$ gives rise to a mapping
  
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
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\begin{equation*} \varphi_{ *} : K _ { 0 } ^ { \text{alg} } ( \mathcal{C} _ { 1 } \bigotimes \mathbf{C} [ \Gamma ] ) \rightarrow \mathbf{C}, \end{equation*}
  
<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>
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using also the trace on $\mathcal{C} _ { 1 }$.
  
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" />.
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The Connes–Moscovici index theorem now states [[#References|[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|Chern character]] of the element in $K ( T M )$ defined by $\sigma ( D )$, as in the Atiyah–Singer index theorem (see [[#References|[a1]]] and [[Index formulas|Index formulas]]). Then
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\begin{equation*} \phi_{ *} ( \operatorname { ind } ( D ) ) = c _ { q } \left( \operatorname { Ch } ( D ) \mathcal{T} ( M ) f ^ { * } ( \phi ) \right) [ T M ] \end{equation*}
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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.
 
The Connes–Moscovici index theorem is sometimes called the higher index theorem for coverings and is the prototype of a higher index theorem.
  
 
====References====
 
====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>
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<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>

Latest revision as of 17:02, 1 July 2020

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
How to Cite This Entry:
Connes-Moscovici index theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connes-Moscovici_index_theorem&oldid=22303
This article was adapted from an original article by V. Nistor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article