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The area of mathematics whose main object of study is the index of operators (cf. also [[Index of an operator|Index of an operator]]; [[Index formulas|Index formulas]]).
 
The area of mathematics whose main object of study is the index of operators (cf. also [[Index of an operator|Index of an operator]]; [[Index formulas|Index formulas]]).
  
The main question in index theory is to provide [[Index formulas|index formulas]] for classes of Fredholm operators (cf. also [[Fredholm-operator(2)|Fredholm operator]]), but this is not the only interesting question. First of all, to be able to provide index formulas, one has to specify what meaning of "index" is agreed upon, then one has to specify to what classes of operators these formulas will apply, and, finally, one has to explain how to use these formulas in applications.
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The main question in index theory is to provide [[Index formulas|index formulas]] for classes of Fredholm operators (cf. also [[Fredholm-operator(2)|Fredholm operator]]), but this is not the only interesting question. First of all, to be able to provide index formulas, one has to specify what meaning of "index" is agreed upon, then one has to specify to what classes of operators these formulas will apply, and, finally, one has to explain how to use these formulas in applications.
  
A consequence of this is that index theory also studies various generalizations of the concept of Fredholm index, including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i1300301.png" />-theoretical and cyclic homology indices, for example. Moreover, the study of the analytic properties necessary for the index to be defined are an important part of index theory. Here one includes the study of conditions for being Fredholm or non-Fredholm for classes of operators that nevertheless have finite-dimensional kernels. Soon after (1970s), other invariants of elliptic operators have been defined that are similar in nature to the analytic index. The study of these related invariants is also commonly considered to be part of index theory. The most prominent of these new, related invariants are the Ray–Singer analytic torsion and the [[Eta-invariant|eta-invariant]]. Fixed-point formulas are also usually considered part of index theory, see [[#References|[a9]]]. Finally, one of the most important goals of index theory is to study applications of the index theorems to geometry, physics, group representations, analysis, and other fields. There is a very long and fast growing list of papers dealing with these applications.
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A consequence of this is that index theory also studies various generalizations of the concept of Fredholm index, including $K$-theoretical and cyclic homology indices, for example. Moreover, the study of the analytic properties necessary for the index to be defined are an important part of index theory. Here one includes the study of conditions for being Fredholm or non-Fredholm for classes of operators that nevertheless have finite-dimensional kernels. Soon after (1970s), other invariants of elliptic operators have been defined that are similar in nature to the analytic index. The study of these related invariants is also commonly considered to be part of index theory. The most prominent of these new, related invariants are the Ray–Singer analytic torsion and the [[Eta-invariant|eta-invariant]]. Fixed-point formulas are also usually considered part of index theory, see [[#References|[a9]]]. Finally, one of the most important goals of index theory is to study applications of the index theorems to geometry, physics, group representations, analysis, and other fields. There is a very long and fast growing list of papers dealing with these applications.
  
Index theory has become a subject on its own only after M.F. Atiyah and I. Singer published their index theorems in the sequence of papers [[#References|[a4]]], [[#References|[a6]]], [[#References|[a7]]] (cf. also [[Index formulas|Index formulas]]). These theorems had become possible only due to progress in the related fields of [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i1300302.png" />-theory]] [[#References|[a10]]], [[#References|[a5]]] and pseudo-differential operators (cf. also [[Pseudo-differential operator|Pseudo-differential operator]]) [[#References|[a35]]], [[#References|[a37]]], [[#References|[a46]]]. Important particular cases of the Atiyah–Singer index theorems were known before. Among them, Hirzebruch's signature theorem (cf. also [[Signature|Signature]]) occupies a special place (see [[#References|[a33]]], especially for topics such as multiplicative genera and the Langlands formula for the dimension of spaces of automorphic forms). Hirzebruch's theorem was generalized by A. Grothendieck (see [[#References|[a22]]]), who introduced many of the ideas that proved to be fundamental for the proof of the index theorems. All these theorems turned out to be consequences of the Atiyah–Singer index theorems (see also [[Index formulas|Index formulas]] for some index formulas that preceded the Atiyah–Singer index formula).
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Index theory has become a subject on its own only after M.F. Atiyah and I. Singer published their index theorems in the sequence of papers [[#References|[a4]]], [[#References|[a6]]], [[#References|[a7]]] (cf. also [[Index formulas|Index formulas]]). These theorems had become possible only due to progress in the related fields of [[K-theory|$K$-theory]] [[#References|[a10]]], [[#References|[a5]]] and pseudo-differential operators (cf. also [[Pseudo-differential operator|Pseudo-differential operator]]) [[#References|[a35]]], [[#References|[a37]]], [[#References|[a46]]]. Important particular cases of the Atiyah–Singer index theorems were known before. Among them, Hirzebruch's signature theorem (cf. also [[Signature|Signature]]) occupies a special place (see [[#References|[a33]]], especially for topics such as multiplicative genera and the Langlands formula for the dimension of spaces of automorphic forms). Hirzebruch's theorem was generalized by A. Grothendieck (see [[#References|[a22]]]), who introduced many of the ideas that proved to be fundamental for the proof of the index theorems. All these theorems turned out to be consequences of the Atiyah–Singer index theorems (see also [[Index formulas|Index formulas]] for some index formulas that preceded the Atiyah–Singer index formula).
  
 
==Atiyah–Singer index formulas.==
 
==Atiyah–Singer index formulas.==
 
A common characteristic of the first three main index formulas of Atiyah–Singer and Atiyah–Segal is that they depend only on the principal symbol of the operator whose index they compute. (For a differential operator, the principal symbol is given by the terms involving only the highest-order differentials and is independent of the choice of a coordinate system; cf. also [[Principal part of a differential operator|Principal part of a differential operator]]; [[Symbol of an operator|Symbol of an operator]].) The main theorems mentioned above are:
 
A common characteristic of the first three main index formulas of Atiyah–Singer and Atiyah–Segal is that they depend only on the principal symbol of the operator whose index they compute. (For a differential operator, the principal symbol is given by the terms involving only the highest-order differentials and is independent of the choice of a coordinate system; cf. also [[Principal part of a differential operator|Principal part of a differential operator]]; [[Symbol of an operator|Symbol of an operator]].) The main theorems mentioned above are:
  
the index theorem for a single elliptic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i1300303.png" /> acting between sections of vector bundles on a smooth, compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i1300304.png" /> (Atiyah–Singer, [[#References|[a4]]]);
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the index theorem for a single elliptic operator $P$ acting between sections of vector bundles on a smooth, compact manifold $M$ (Atiyah–Singer, [[#References|[a4]]]);
  
the equivariant index theorem for a single elliptic operator equivariant with respect to a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i1300305.png" /> (Atiyah–Segal, [[#References|[a5]]]); and
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the equivariant index theorem for a single elliptic operator equivariant with respect to a compact group $G$ (Atiyah–Segal, [[#References|[a5]]]); and
  
the index theorem for families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i1300306.png" /> of elliptic operators acting on the fibres of a fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i1300307.png" /> (Atiyah–Singer, [[#References|[a7]]]). These results are briefly reviewed below.
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the index theorem for families $( P _ { b } ) _ { b \in B }$ of elliptic operators acting on the fibres of a fibre bundle $Y \rightarrow B$ (Atiyah–Singer, [[#References|[a7]]]). These results are briefly reviewed below.
  
 
===A single elliptic operator acting between sections of vector bundles.===
 
===A single elliptic operator acting between sections of vector bundles.===
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i1300308.png" /> is an elliptic differential, or, more generally, an elliptic [[Pseudo-differential operator|pseudo-differential operator]] acting between sections of two smooth vector bundles (cf. also [[Elliptic operator|Elliptic operator]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i1300309.png" /> defines a continuous operator between suitable Sobolev spaces with closed range and finite-dimensional kernel and cokernel, that is, a [[Fredholm-operator(2)|Fredholm operator]]. The first of the index theorems gives an explicit formula for the Fredholm, or analytic, index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003011.png" />:
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If $P$ is an elliptic differential, or, more generally, an elliptic [[Pseudo-differential operator|pseudo-differential operator]] acting between sections of two smooth vector bundles (cf. also [[Elliptic operator|Elliptic operator]]), then $P$ defines a continuous operator between suitable Sobolev spaces with closed range and finite-dimensional kernel and cokernel, that is, a [[Fredholm-operator(2)|Fredholm operator]]. The first of the index theorems gives an explicit formula for the Fredholm, or analytic, index $\operatorname{ind} ( P )$ of $P$:
  
<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/i/i130/i130030/i13003012.png" /></td> </tr></table>
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\begin{equation*} \operatorname{ind} ( P ) : = \operatorname { dim } ( \operatorname{ker} ( P ) ) - \operatorname { dim } ( \operatorname { coker } ( P ) ). \end{equation*}
  
Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003013.png" /> the [[Todd class|Todd class]] of the complexification of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003016.png" /> is an elliptic operator as above, its principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003017.png" /> defines a [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003018.png" />-theory]] class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003019.png" /> with compact supports on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003020.png" /> whose [[Chern character|Chern character]], denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003021.png" />, is in the even [[Cohomology|cohomology]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003022.png" /> with compact supports. The Atiyah–Singer index formula of [[#References|[a6]]] then states that
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Denote by $\mathcal{T} ( M )$ the [[Todd class|Todd class]] of the complexification of the tangent bundle $T M$ of $M$. If $P$ is an elliptic operator as above, its principal symbol $a = \sigma ( P )$ defines a [[K-theory|$K$-theory]] class $[ a ]$ with compact supports on $T ^ { * } M$ whose [[Chern character|Chern character]], denoted $\operatorname{Ch} ( [ a ] )$, is in the even [[Cohomology|cohomology]] of $T ^ { * } M$ with compact supports. The Atiyah–Singer index formula of [[#References|[a6]]] then states that
  
<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/i/i130/i130030/i13003023.png" /></td> </tr></table>
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\begin{equation*} \operatorname{ind} ( P ) = ( - 1 ) ^ { n } \operatorname{Ch} ( [ a ] ) {\cal T} ( M ) [ T ^ { * } M ], \end{equation*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003024.png" /> being the dimension of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003026.png" /> being the [[Fundamental class|fundamental class]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003027.png" />. (The factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003028.png" /> reflects the choice of the [[Orientation|orientation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003029.png" /> in the original articles. Other choices for this orientation will lead to different signs.) In other words, the index is obtained by evaluating the compactly supported cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003030.png" /> on the fundamental class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003031.png" />.
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$n$ being the dimension of the manifold $M$ and $[ T ^ { * } M ]$ being the [[Fundamental class|fundamental class]] of $T ^ { * } M$. (The factor $( - 1 ) ^ { n }$ reflects the choice of the [[Orientation|orientation]] of $T ^ { * } M$ in the original articles. Other choices for this orientation will lead to different signs.) In other words, the index is obtained by evaluating the compactly supported cohomology class $\operatorname{Ch} ( [ a ] ) \mathcal{T} ( M )$ on the fundamental class of $T ^ { * } M$.
  
 
===Equivariant index theorem.===
 
===Equivariant index theorem.===
The second of the index formulas refines the index when the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003032.png" /> above is invariant with respect to a compact [[Lie group|Lie group]], see [[#References|[a5]]], [[#References|[a6]]]. Recall that the representation ring of a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003033.png" /> is defined as the ring of formal linear combinations with integer coefficients of equivalence classes of irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003034.png" /> (cf. also [[Irreducible representation|Irreducible representation]]). For operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003035.png" /> equivariant with respect to a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003036.png" />, the kernel and cokernel are representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003037.png" />, so their difference can now be regarded as an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003038.png" />, called the equivariant index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003039.png" />. The Atiyah–Singer index formula in [[#References|[a6]]] gives the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003040.png" /> of the (character of the) index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003041.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003042.png" /> in terms of invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003043.png" />, the set of fixed points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003044.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003045.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003046.png" /> the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003047.png" /> to the cotangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003048.png" /> and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003049.png" /> the Todd class of the complexification of the cotangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003050.png" />. In addition to these ingredients, which are similar to the ingredients appearing in the formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003051.png" /> above, the formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003052.png" /> involves also a Lefschetz-type contribution, denoted below by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003053.png" />, obtained from the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003054.png" /> on the normal bundle to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003055.png" />:
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The second of the index formulas refines the index when the operator $P$ above is invariant with respect to a compact [[Lie group|Lie group]], see [[#References|[a5]]], [[#References|[a6]]]. Recall that the representation ring of a compact group $G$ is defined as the ring of formal linear combinations with integer coefficients of equivalence classes of irreducible representations of $G$ (cf. also [[Irreducible representation|Irreducible representation]]). For operators $P$ equivariant with respect to a compact group $G$, the kernel and cokernel are representations of $G$, so their difference can now be regarded as an element of $R ( G )$, called the equivariant index of $P$. The Atiyah–Singer index formula in [[#References|[a6]]] gives the value $\text{ind}_{ g } ( P )$ of the (character of the) index of $P$ at $g \in G$ in terms of invariants of $M ^ { g }$, the set of fixed points of $g$ in $M$. Denote by $a | _ { T ^{*} M ^{ g }}$ the restriction of $a$ to the cotangent bundle of $M ^ { g }$ and by $\mathcal{T} ( M ^ { g } )$ the Todd class of the complexification of the cotangent bundle of $M ^ { g }$. In addition to these ingredients, which are similar to the ingredients appearing in the formula for $\operatorname{ind} ( P )$ above, the formula for $\text{ind}_{ g } ( P )$ involves also a Lefschetz-type contribution, denoted below by $L ( N , g )$, obtained from the action of $g$ on the normal bundle to the set $M ^ { g }$:
  
<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/i/i130/i130030/i13003056.png" /></td> </tr></table>
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\begin{equation*} \operatorname { ind } _ { g } ( P ) = ( - 1 ) ^ { n } \operatorname { Ch } ( [ a | _ { T ^ { * } M ^ { g } } ] ) \mathcal{T} ( M ^ { g } ) L ( N , g ) [ T ^ { * } M ^ { g } ]. \end{equation*}
  
 
===Families of elliptic operators.===
 
===Families of elliptic operators.===
For families of elliptic operators acting on the fibres of a fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003057.png" /> (cf. also [[Fibration|Fibration]]), a first problem is to make sense of the index. The solution proposed by Atiyah and Singer in [[#References|[a7]]] is to define the index as an element of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003058.png" />-theory group, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003059.png" /> in this case (cf. also [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003060.png" />-theory]]). This fortunate choice has opened the way for many other developments in index theory. Actually, in the two index theorems mentioned above, the index can also be interpreted using a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003061.png" />-theory group, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003062.png" />-theory of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003063.png" /> of complex numbers in the first index theorem and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003064.png" />-theory group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003065.png" />, the norm closure of the convolution algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003066.png" />, in the equivariant index theorem. For the [[Chern character|Chern character]] of the family index of a family of elliptic operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003067.png" /> as above, there is a formula similar to the formula for the index of a single elliptic operator. The principal symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003068.png" /> of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003069.png" /> define, in this case, a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003070.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003071.png" />-theory with compact supports of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003072.png" />, the vertical cotangent bundle to the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003073.png" />, as in the case of a single elliptic operator. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003074.png" /> the Todd class of the complexification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003075.png" /> and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003076.png" /> the morphism induced by integration along the fibres, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003077.png" /> being the common dimension of the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003078.png" />. Then
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For families of elliptic operators acting on the fibres of a fibre bundle $\pi : Y \rightarrow B$ (cf. also [[Fibration|Fibration]]), a first problem is to make sense of the index. The solution proposed by Atiyah and Singer in [[#References|[a7]]] is to define the index as an element of a $K$-theory group, namely $K ^ { 0 } ( B )$ in this case (cf. also [[K-theory|$K$-theory]]). This fortunate choice has opened the way for many other developments in index theory. Actually, in the two index theorems mentioned above, the index can also be interpreted using a $K$-theory group, the $K$-theory of the algebra $\mathbf{C}$ of complex numbers in the first index theorem and the $K$-theory group of $C ^ { * } ( G )$, the norm closure of the convolution algebra of $G$, in the equivariant index theorem. For the [[Chern character|Chern character]] of the family index of a family of elliptic operators $( P _ { b } )$ as above, there is a formula similar to the formula for the index of a single elliptic operator. The principal symbols $a _ { b } = \sigma ( P _ { b } )$ of the operators $P _ { b }$ define, in this case, a class $[ a ]$ in the $K$-theory with compact supports of $T _ { \text{vert} } ^ { * } Y : = T ^ { * } Y / \pi ^ { * } ( T ^ { * } B )$, the vertical cotangent bundle to the fibres of $\pi : Y \rightarrow B$, as in the case of a single elliptic operator. Denote by $\mathcal{T} ( M | B )$ the Todd class of the complexification of $T _ { \text { vert } } ^ { * } Y$ and by $\pi_{ *} : H _ { c } ^ { * } ( T _ { \text { vert } } ^ { * } Y ) \rightarrow H ^ { * - 2 n} ( B )$ the morphism induced by integration along the fibres, with $n$ being the common dimension of the fibres of $\pi$. 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/i/i130/i130030/i13003079.png" /></td> </tr></table>
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\begin{equation*} \operatorname{Ch} ( \operatorname{ ind } ( P ) ) = ( - 1 ) ^ { n } \pi_{ *} ( \operatorname { ind } ( [ a ] ) {\cal T} ( M | B ) ). \end{equation*}
  
 
This completes the discussion of these three main theorems of Atiyah and Singer.
 
This completes the discussion of these three main theorems of Atiyah and Singer.
  
==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003080.png" />-theory in index theory.==
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==$K$-theory in index theory.==
The role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003081.png" />-theory in the proof and applications of the index theorems can hardly be overstated and certainly does not stop at providing an interpretation of the index as an element of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003082.png" />-theory group. A far-reaching consequence of the use of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003083.png" />-theory, which depends on Bott periodicity (or more precisely, the [[Thom isomorphism|Thom isomorphism]], cf. also [[Bott periodicity theorem|Bott periodicity theorem]]), is that all elliptic operators can be connected, by a homotopy of Fredholm operators, to certain operators of a very particular kind, the so-called generalized Dirac operators (see below). It is thus sufficient to prove the index theorems for generalized Dirac operators.
+
The role of $K$-theory in the proof and applications of the index theorems can hardly be overstated and certainly does not stop at providing an interpretation of the index as an element of a $K$-theory group. A far-reaching consequence of the use of $K$-theory, which depends on Bott periodicity (or more precisely, the [[Thom isomorphism|Thom isomorphism]], cf. also [[Bott periodicity theorem|Bott periodicity theorem]]), is that all elliptic operators can be connected, by a homotopy of Fredholm operators, to certain operators of a very particular kind, the so-called generalized Dirac operators (see below). It is thus sufficient to prove the index theorems for generalized Dirac operators.
  
 
Due to their differential-geometric properties, it is possible to give more concrete proofs of the Atiyah–Singer index theorem for generalized Dirac operators, using heat kernels, for example (cf. also [[Heat content asymptotics|Heat content asymptotics]]). The generalized Dirac operator with coefficients in the spin bundle is called simply the Dirac operator (sometimes called the Atiyah–Singer operator). See below for more about generalized Dirac operators.
 
Due to their differential-geometric properties, it is possible to give more concrete proofs of the Atiyah–Singer index theorem for generalized Dirac operators, using heat kernels, for example (cf. also [[Heat content asymptotics|Heat content asymptotics]]). The generalized Dirac operator with coefficients in the spin bundle is called simply the Dirac operator (sometimes called the Atiyah–Singer operator). See below for more about generalized Dirac operators.
Line 47: Line 55:
 
After the publication of the first papers by Atiyah and Singer, index theory has evolved into essentially three directions:
 
After the publication of the first papers by Atiyah and Singer, index theory has evolved into essentially three directions:
  
a direction which consists of applications and new proofs of the index theorems (especially "local" proofs using heat kernels);
+
a direction which consists of applications and new proofs of the index theorems (especially "local" proofs using heat kernels);
  
 
a direction which studies invariants other than the index; and
 
a direction which studies invariants other than the index; and
  
a direction which aims at more general index theorems. There is a very large number of applications of index theorems to topology and other areas of mathematics. A few examples follow. In [[#References|[a12]]], Atiyah and W. Schmid used Atiyah's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003084.png" />-index theorem for coverings [[#References|[a11]]] to construct discrete series representations. In [[#References|[a34]]], N.J. Hitchin used the families index theorem to prove that there exist metrics whose associated Dirac operators have non-trivial kernels (in suitable dimensions). An index theorem for foliations that is close in spirit to Atiyah's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003086.png" />-index theorem was obtained by A. Connes [[#References|[a27]]].
+
a direction which aims at more general index theorems. There is a very large number of applications of index theorems to topology and other areas of mathematics. A few examples follow. In [[#References|[a12]]], Atiyah and W. Schmid used Atiyah's $L^{2}$-index theorem for coverings [[#References|[a11]]] to construct discrete series representations. In [[#References|[a34]]], N.J. Hitchin used the families index theorem to prove that there exist metrics whose associated Dirac operators have non-trivial kernels (in suitable dimensions). An index theorem for foliations that is close in spirit to Atiyah's $L^{2}$-index theorem was obtained by A. Connes [[#References|[a27]]].
  
The index of Dirac (or Atiyah–Singer) operators was used to formulate and then prove the Gromov–Lawson conjecture [[#References|[a32]]], which states that a compact, spin, simply connected manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003087.png" /> admits a metric of positive scalar curvature if and only if the index of the spin Dirac operator (in an appropriate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003088.png" />-theory group) is zero. This conjecture was proved by S. Stolz, [[#References|[a48]]]. Dirac operators have been used to give a concrete construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003090.png" />-homology [[#References|[a16]]].
+
The index of Dirac (or Atiyah–Singer) operators was used to formulate and then prove the Gromov–Lawson conjecture [[#References|[a32]]], which states that a compact, spin, simply connected manifold of dimension $\geq 5$ admits a metric of positive scalar curvature if and only if the index of the spin Dirac operator (in an appropriate $K$-theory group) is zero. This conjecture was proved by S. Stolz, [[#References|[a48]]]. Dirac operators have been used to give a concrete construction of $K$-homology [[#References|[a16]]].
  
Some of the applications of the index theorems require new proofs of these theorems, usually relying on the "heat-kernel method" . The main idea of this method is as follows. H. McKean and Singer [[#References|[a39]]] stated the problem of investigating the behaviour, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003092.png" />, of the (super-trace of the) heat kernel. More precisely, let
+
Some of the applications of the index theorems require new proofs of these theorems, usually relying on the "heat-kernel method" . The main idea of this method is as follows. H. McKean and Singer [[#References|[a39]]] stated the problem of investigating the behaviour, as $t \rightarrow 0$, $t &gt; 0$, of the (super-trace of the) heat kernel. More precisely, let
  
<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/i/i130/i130030/i13003093.png" /></td> </tr></table>
+
\begin{equation*} k _ { t } ( x , y ) = \operatorname { str } ( e ^ { - t D ^ { 2 } } ) = \operatorname { tr } ( e ^ { - t D _ { + } ^ { * } D _ { + } } ) - \operatorname { tr } ( e ^ { - t D _ { + } D _ { + } ^ { * } } ) \end{equation*}
  
be the well-known term appearing in the McKean–Singer index formula, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003094.png" /> is a self-adjoint geometric operator (cf. also [[Self-adjoint operator|Self-adjoint operator]]) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003095.png" /> mapping the subspace of even sections to the subspace of odd sections. They considered the case of the de Rham operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003096.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003097.png" /> is then the de Rham differential (cf. also [[De Rham cohomology|de Rham cohomology]]). It was known that the integral over the whole manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003098.png" /> gives the analytic index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i13003099.png" />, and they expressed the hope that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030100.png" /> will have a definite limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030101.png" />. This was proved for various particular cases by V. Patodi in [[#References|[a54]]] and then by P. Gilkey [[#References|[a29]]], [[#References|[a30]]] using invariant theory (see [[#References|[a31]]] for an exposition of this method). This method was finally refined in [[#References|[a1]]] to give a clear and elegant proof of the local index theorem for all Dirac operators.
+
be the well-known term appearing in the McKean–Singer index formula, where $D = D _ { + } + D _ { + } ^ { * }$ is a self-adjoint geometric operator (cf. also [[Self-adjoint operator|Self-adjoint operator]]) with $D _ { + }$ mapping the subspace of even sections to the subspace of odd sections. They considered the case of the de Rham operator $D _ { + } + D _ { + } ^ { * }$, where $D _ { + }$ is then the de Rham differential (cf. also [[De Rham cohomology|de Rham cohomology]]). It was known that the integral over the whole manifold of $k _ { t } ( x , x )$ gives the analytic index of $D _ { + }$, and they expressed the hope that $k _ { t } ( x , x )$ will have a definite limit as $t \rightarrow 0$. This was proved for various particular cases by V. Patodi in [[#References|[a54]]] and then by P. Gilkey [[#References|[a29]]], [[#References|[a30]]] using invariant theory (see [[#References|[a31]]] for an exposition of this method). This method was finally refined in [[#References|[a1]]] to give a clear and elegant proof of the local index theorem for all Dirac operators.
  
 
Inspired by a talk of Atiyah, J.-M. Bismut investigated connections between [[Probability theory|probability theory]] and index theory. He was able to use the stochastic calculus (cf. also [[Malliavin calculus|Malliavin calculus]]) to give a new proof of the local index theorem [[#References|[a17]]]. His methods then generalized to give proofs of the local index theorem for families of Dirac operators [[#References|[a18]]] using Quillen's theory of super-connections [[#References|[a43]]], and of the Atiyah–Bott fixed-point formulas [[#References|[a19]]]. An application of his results is the determination of the Quillen metric on the determinant bundle [[#References|[a21]]].
 
Inspired by a talk of Atiyah, J.-M. Bismut investigated connections between [[Probability theory|probability theory]] and index theory. He was able to use the stochastic calculus (cf. also [[Malliavin calculus|Malliavin calculus]]) to give a new proof of the local index theorem [[#References|[a17]]]. His methods then generalized to give proofs of the local index theorem for families of Dirac operators [[#References|[a18]]] using Quillen's theory of super-connections [[#References|[a43]]], and of the Atiyah–Bott fixed-point formulas [[#References|[a19]]]. An application of his results is the determination of the Quillen metric on the determinant bundle [[#References|[a21]]].
  
The local index theorems have many connections to physics, where Dirac operators play a prominent role. Actually, several physicists have come up with arguments for a proof of the local index theorem based on supersymmetry and functional integration, see [[#References|[a8]]] and [[#References|[a53]]], for example. Building on these arguments, E. Getzler has obtained a short and elegant proof of the local index theorem [[#References|[a14]]], [[#References|[a31]]], which also uses supersymmetry. Moreover, ideas inspired from physics have lead E. Witten to conjecture that certain twisted Dirac operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030102.png" />-manifolds have an index that is a trivial representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030103.png" />, see [[#References|[a52]]]. This was proved by C.H. Taubes [[#References|[a49]]] (see also [[#References|[a23]]] and [[#References|[a51]]]). For the Dirac operator, this had been proved before by Atiyah and F. Hirzebruch [[#References|[a2]]].
+
The local index theorems have many connections to physics, where Dirac operators play a prominent role. Actually, several physicists have come up with arguments for a proof of the local index theorem based on supersymmetry and functional integration, see [[#References|[a8]]] and [[#References|[a53]]], for example. Building on these arguments, E. Getzler has obtained a short and elegant proof of the local index theorem [[#References|[a14]]], [[#References|[a31]]], which also uses supersymmetry. Moreover, ideas inspired from physics have lead E. Witten to conjecture that certain twisted Dirac operators on $S ^ { 1 }$-manifolds have an index that is a trivial representation of $S ^ { 1 }$, see [[#References|[a52]]]. This was proved by C.H. Taubes [[#References|[a49]]] (see also [[#References|[a23]]] and [[#References|[a51]]]). For the Dirac operator, this had been proved before by Atiyah and F. Hirzebruch [[#References|[a2]]].
  
 
==Other invariants.==
 
==Other invariants.==
 
Heat-kernel methods have proved very useful in dealing with non-compact and singular spaces. A common feature of these spaces is that the index formulas for the natural operators on them depend on more than just the principal symbol, which leads to the appearance of non-local invariants in these index formulas. In general, there exists no good understanding, at this time (2000), of what these non-local invariants are, except in particular cases. The most prominent of these particular cases is the Atiyah–Patodi–Singer index theorem for manifolds with boundary. Other results in these directions were obtained in [[#References|[a25]]], [[#References|[a40]]], [[#References|[a41]]], [[#References|[a47]]]. In all these cases, eta-invariants of certain boundary operators must be included in the formula for the index. Moreover, one has to either work on complete manifolds or to include boundary conditions to make the given problems Fredholm. The Atiyah–Patodi–Singer index theorem [[#References|[a3]]], e.g., requires such boundary conditions; see below.
 
Heat-kernel methods have proved very useful in dealing with non-compact and singular spaces. A common feature of these spaces is that the index formulas for the natural operators on them depend on more than just the principal symbol, which leads to the appearance of non-local invariants in these index formulas. In general, there exists no good understanding, at this time (2000), of what these non-local invariants are, except in particular cases. The most prominent of these particular cases is the Atiyah–Patodi–Singer index theorem for manifolds with boundary. Other results in these directions were obtained in [[#References|[a25]]], [[#References|[a40]]], [[#References|[a41]]], [[#References|[a47]]]. In all these cases, eta-invariants of certain boundary operators must be included in the formula for the index. Moreover, one has to either work on complete manifolds or to include boundary conditions to make the given problems Fredholm. The Atiyah–Patodi–Singer index theorem [[#References|[a3]]], e.g., requires such boundary conditions; see below.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030104.png" /> be a compact [[Manifold|manifold]] with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030105.png" /> and [[Metric|metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030106.png" /> which is a product metric in a suitable cylindrical neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030107.png" />. Fix a Clifford module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030108.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030109.png" /> (cf. also [[Clifford algebra|Clifford algebra]]) and an admissible [[Connection|connection]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030110.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030111.png" /> the generalized Dirac operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030112.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030113.png" /> is the Clifford multiplication and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030114.png" /> is a local orthonormal basis (cf. also [[Orthogonal basis|Orthogonal basis]]). Also, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030115.png" /> be the corresponding generalized Dirac operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030116.png" />, which is (essentially) self-adjoint because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030117.png" /> is compact without boundary. Then the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030118.png" /> will form a discrete subset of the real numbers; denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030119.png" /> the spectral projection corresponding to the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030120.png" /> that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030121.png" />. Decompose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030122.png" /> using the natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030123.png" />-grading on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030124.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030125.png" />, the chiral Dirac operator, acts from sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030126.png" /> to sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030127.png" />, and has an infinite-dimensional kernel. Because of that, Atiyah, Patodi and Singer have introduced a non-local boundary condition of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030128.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030129.png" /> a smooth section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030130.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030131.png" />, which is a compact perturbation of the Calderón projection boundary condition. The effect of this boundary condition is that the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030132.png" /> to the subspace of sections satisfying this boundary condition is Fredholm. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030133.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030135.png" /> with spinor bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030136.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030137.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030138.png" /> denote the dimension of the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030139.png" />. The index of the resulting operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030140.png" /> with the above boundary conditions is then
+
Let $M$ be a compact [[Manifold|manifold]] with boundary $\partial M$ and [[Metric|metric]] $g$ which is a product metric in a suitable cylindrical neighbourhood of $\partial M$. Fix a Clifford module $W$ on $M$ (cf. also [[Clifford algebra|Clifford algebra]]) and an admissible [[Connection|connection]] $\nabla$. Denote by $D : = \sum c ( e _ { i } ) \nabla _ { e_i }$ the generalized Dirac operator on $W$, where $c : T ^ { * } M \cong T M \rightarrow \operatorname { End } ( W )$ is the Clifford multiplication and $e _ { i }$ is a local orthonormal basis (cf. also [[Orthogonal basis|Orthogonal basis]]). Also, let $D _ { 0 }$ be the corresponding generalized Dirac operator on $\partial M$, which is (essentially) self-adjoint because $\partial M$ is compact without boundary. Then the eigenvalues of $D _ { 0 }$ will form a discrete subset of the real numbers; denote by $P _ { + }$ the spectral projection corresponding to the eigenvalues of $D _ { 0 }$ that are $\geq 0$. Decompose $D = D _ { + } + D _ { + } ^ { * }$ using the natural ${\bf Z} / 2 {\bf Z}$-grading on $W$. The operator $D _ { + }$, the chiral Dirac operator, acts from sections of $W _ { + }$ to sections of $W_-$, and has an infinite-dimensional kernel. Because of that, Atiyah, Patodi and Singer have introduced a non-local boundary condition of the form $P _ { + } f = 0$, for $f$ a smooth section of $W _ { + }$ over $\partial M$, which is a compact perturbation of the Calderón projection boundary condition. The effect of this boundary condition is that the restriction of $D$ to the subspace of sections satisfying this boundary condition is Fredholm. Assume that $M$ is $\operatorname {spin}^ { c }$ with spinor bundle $S$, such that $W = S \otimes E$, and let $h$ denote the dimension of the kernel of $D _ { 0 }$. The index of the resulting operator $D _ { + }$ with the above boundary conditions is 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/i/i130/i130030/i130030141.png" /></td> </tr></table>
+
\begin{equation*} \operatorname{ind}_{\alpha} ( D _ { + } ) = \int _ { M } \hat { A } ( M ) \operatorname{Ch} ( E ) - \frac { \eta ( D _ { 0 } ) + h } { 2 }. \end{equation*}
  
This formula was generalized by Bismut and J. Cheeger in [[#References|[a20]]] to families of manifolds with boundary, the result being expressed using the "eta form" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030142.png" />. More precisely, using the notation above, they proved that
+
This formula was generalized by Bismut and J. Cheeger in [[#References|[a20]]] to families of manifolds with boundary, the result being expressed using the "eta form" $\hat{\eta}$. More precisely, using the notation above, they proved that
  
<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/i/i130/i130030/i130030143.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030143.png"/></td> </tr></table>
  
 
provided that all Dirac operator associated to the boundaries of the fibres are invertible.
 
provided that all Dirac operator associated to the boundaries of the fibres are invertible.
  
Presently (2000), cyclic homology (cf. also [[Cyclic cohomology|Cyclic cohomology]]) is probably the only general tool to deal with index problems in which the index belongs to an abstract, possibly unknown, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030144.png" />-theory group, or to deal with index theorems involving non-local invariants. See [[#References|[a26]]], [[#References|[a36]]], [[#References|[a38]]], or [[#References|[a50]]] for the basic results on cyclic homology. The relation between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030145.png" />-theory of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030146.png" /> and the cyclic homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030147.png" /> is via Chern characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030148.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030149.png" />, and is due to Connes and M. Karoubi.
+
Presently (2000), cyclic homology (cf. also [[Cyclic cohomology|Cyclic cohomology]]) is probably the only general tool to deal with index problems in which the index belongs to an abstract, possibly unknown, $K$-theory group, or to deal with index theorems involving non-local invariants. See [[#References|[a26]]], [[#References|[a36]]], [[#References|[a38]]], or [[#References|[a50]]] for the basic results on cyclic homology. The relation between the $K$-theory of the algebra $A$ and the cyclic homology of $A$ is via Chern characters $\operatorname{Ch} : K _ { 0 } ( A ) \rightarrow  \operatorname{HC} _ { 2 n } ( A )$, $n \geq 0$, and is due to Connes and M. Karoubi.
  
 
==Generalized index theorems.==
 
==Generalized index theorems.==
In [[#References|[a24]]], Connes and H. Moscovici have generalized Atiyah's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030150.png" />-index theorem, which allowed them to obtain a proof of the Novikov conjecture (cf. also [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030151.png" />-algebra]]) for certain classes of groups. The index theorem, also called the higher index theorem for coverings, is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030152.png" /> be a [[Covering|covering]] of a compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030153.png" /> with group of deck transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030154.png" /> (cf. also [[Monodromy transformation|Monodromy transformation]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030155.png" /> is an elliptic differential operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030156.png" /> invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030157.png" /> (such as the signature operator), then it has an index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030158.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030159.png" />-group of the closure of the group algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030160.png" /> acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030161.png" />. This index was defined by A.T. Fomenko and A. Mishchenko in [[#References|[a28]]]. This index can be refined to an index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030162.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030163.png" /> is the algebra of infinite matrices with complex entries and with rapid decrease. Using cyclic cohomology and the Chern character in cyclic homology, every cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030164.png" /> gives rise to a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030165.png" />, and the problem is to determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030166.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030167.png" />, then this number is exactly the von Neumann index appearing in Atiyah's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030168.png" />-index formula.
+
In [[#References|[a24]]], Connes and H. Moscovici have generalized Atiyah's $L^{2}$-index theorem, which allowed them to obtain a proof of the Novikov conjecture (cf. also [[C*-algebra|$C ^ { * }$-algebra]]) for certain classes of groups. The index theorem, also called the higher index theorem for coverings, is as follows. Let $\tilde { M } \rightarrow M$ be a [[Covering|covering]] of a compact manifold $M$ with group of deck transformations $\Gamma$ (cf. also [[Monodromy transformation|Monodromy transformation]]). If $D$ is an elliptic differential operator on $M$ invariant with respect to $\Gamma$ (such as the signature operator), then it has an index $\operatorname{ind} ( D ) \in K _ { 0 } ( C _ { r } ^ { * } ( \Gamma ) )$, the $K _ { 0 }$-group of the closure of the group algebra of $\Gamma$ acting on $l ^ { 2 } ( \Gamma )$. This index was defined by A.T. Fomenko and A. Mishchenko in [[#References|[a28]]]. This index can be refined to an index in $K_0({\cal R}\otimes {\bf C}[\Gamma])$, where $\mathcal{R}$ is the algebra of infinite matrices with complex entries and with rapid decrease. Using cyclic cohomology and the Chern character in cyclic homology, every cohomology class $\phi \in H ^ { * } ( \Gamma ) = H ^ { * } ( B \Gamma )$ gives rise to a morphism $\phi * : K _ { 0 } ( {\cal R} \otimes {\bf C} [ \Gamma ] ) \rightarrow \bf C$, and the problem is to determine $\phi_{*} ( \text { ind } ( D ) )$. If $\phi = 1 \in H ^ { 0 } ( \Gamma )$, then this number is exactly the von Neumann index appearing in Atiyah's $L^{2}$-index formula.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030169.png" /> be the mapping that classifies the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030170.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030171.png" /> be the Todd class of the complexification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030172.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030173.png" /> is an elliptic invariant differential operator, its principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030174.png" /> defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030175.png" />-theory class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030176.png" /> with compact supports on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030177.png" />, whose Chern character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030178.png" /> is in the even cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030179.png" /> with compact supports, as in the case of the Atiyah–Singer index theorem for a single elliptic operator. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030180.png" />; then the Connes–Moscovici higher index theorem for coverings [[#References|[a24]]] states that
+
Let $f : M \rightarrow B \Gamma$ be the mapping that classifies the covering $\tilde { M } \rightarrow M$ and let $\mathcal{T} ( M )$ be the Todd class of the complexification of $T _ { \text { vert } } ^ { * } Y$. If $D$ is an elliptic invariant differential operator, its principal symbol $a = \sigma ( P )$ defines a $K$-theory class $[ a ]$ with compact supports on $T ^ { * } M$, whose Chern character $\operatorname{Ch} ( [ a ] )$ is in the even cohomology of $T ^ { * } M$ with compact supports, as in the case of the Atiyah–Singer index theorem for a single elliptic operator. Suppose $\phi \in H ^ { 2 m } ( \Gamma )$; then the Connes–Moscovici higher index theorem for coverings [[#References|[a24]]] states that
  
<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/i/i130/i130030/i130030181.png" /></td> </tr></table>
+
\begin{equation*} \phi _ { * } ( \text { ind } ( D ) ) = ( - 1 ) ^ { n } \left( 2 \pi i ) ^ { - m } ( \operatorname {Ch} ( [ a ] ) \mathcal{T} ( M ) f ^ { * } \phi \right) [ T ^ { * } M ]. \end{equation*}
  
 
The Chern character in cyclic cohomology turns out to be a natural mapping, and this can be interpreted as a general index theorem in cyclic cohomology [[#References|[a42]]]. It is hoped that this general index theorem will help explain the ubiquity of the Todd class in index theorems.
 
The Chern character in cyclic cohomology turns out to be a natural mapping, and this can be interpreted as a general index theorem in cyclic cohomology [[#References|[a42]]]. It is hoped that this general index theorem will help explain the ubiquity of the Todd class in index theorems.
Line 92: Line 100:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Atiyah,   R. Bott,   V. Patodi,   "On the heat equation and the index theorem" ''Invent. Math.'' , '''19''' (1973) pp. 279–330 (Erata ibid. 28 (1975), 277-280)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Atiyah,   F. Hirzebruch,   "Spin manifolds and group actions" , ''Essays in Topology and Related subjects'' , Springer (1994) pp. 18–28</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Atiyah,   V. Patodi,   I. Singer,   "Spectral asymmetry and Riemannian geometry I" ''Math. Proc. Cambridge Philos. Soc.'' , '''77''' (1975) pp. 43–69</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Atiyah,   I. Singer,   "The index of elliptic operators I" ''Ann. of Math.'' , '''87''' (1968) pp. 484–530</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Atiyah,   G. Segal,   "The index of elliptic operators II" ''Ann. of Math.'' , '''87''' (1968) pp. 531–545</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Atiyah,   I. Singer,   "The index of elliptic operators III" ''Ann. of Math.'' , '''93''' (1968) pp. 546–604</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> M. Atiyah,   I. Singer,   "The index of elliptic operators IV" ''Ann. of Math.'' , '''93''' (1971) pp. 119–138</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> L. Alvarez-Gaumé,   "Supersymmetry and the Atiyah–Singer index theorem" ''Comm. Math. Phys.'' , '''90''' (1983) pp. 161–173</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Atiyah,   R. Bott,   "A Lefschetz fixed-point formula for elliptic complexes II: Applications." ''Ann. of Math.'' , '''88''' (1968) pp. 451–491</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Atiyah,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030182.png" />-theory" , Benjamin (1967)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> M. Atiyah,   "Elliptic operators, discrete subgroups, and von Neumann algebras" ''Astérisque'' , '''32/33''' (1969) pp. 43–72</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> M. Atiyah,   W. Schmid,   "A geometric construction of the discrete series" ''Invent. Math.'' , '''42''' (1977) pp. 1–62</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> B. Booss–Bavnbek,   D. Bleecker,   "Topology and analysis. The Atiyah–Singer index formula and gauge-theoretic physics" , ''Universitext'' , Springer (1985)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> N. Berline,   E. Getzler,   M. Vèrgne,   "Heat kernels and Dirac operator" , ''Grundl. Math. Wissenschaft.'' , '''298''' , Springer (1996)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> B. Booss–Bavnbek,   K. Wojciechowski,   "Elliptic boundary problems for Dirac operators" , ''Math. Th. Appl.'' , Birkhäuser (1993)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> P. Baum,   R. Douglas,   "Index theory, bordism, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030183.png" />-homology" , ''Operator Algebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030184.png" />-Theory (San Francisco, Calif., 1981)'' , ''Contemp. Math.'' , '''10''' , Amer. Math. Soc. (1982) pp. 1–31</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> J.-M. Bismut,   "The Atiyah–Singer theorems: a probabilistic approach" ''J. Funct. Anal.'' , '''57''' (1984) pp. 56–99</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> J.-M. Bismut,   "The index theorem for families of Dirac operators: two heat equation proofs" ''Invent. Math.'' , '''83''' (1986) pp. 91–151</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> J.-M. Bismut,   "The Atiyah–Singer theorems: a probabilistic approach. II. The Lefschetz fixed point formulas" ''J. Funct. Anal.'' , '''57''' : 3 (1984) pp. 329–348</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> J.-M. Bismut,   J. Cheeger,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030185.png" />-invariants and their adiabatic limits"  ''J. Amer. Math. Soc.'' , '''2''' (1989) pp. 33–70</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> J.-M. Bismut,   D. Freed,   "The analysis of elliptic families: Metrics and connections on determinant bundles" ''Comm. Math. Phys.'' , '''106''' (1986) pp. 103–163</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> A. Borel,   J.-P. Serre,   "Le téorème de Riemann–Roch (d'apreès Grothendieck)" ''Bull. Soc. Math. France'' , '''86''' (1958) pp. 97–136</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> R. Bott,   C. Taubes,   "On the rigidity theorems of Witten" ''J. Amer. Math. Soc.'' , '''2''' : 1 (1989) pp. 137–186</TD></TR><TR><TD valign="top">[a24]</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">[a25]</TD> <TD valign="top"> J. Cheeger,   "On the Hodge theory of Riemannian pseudomanifolds" , ''Geometry of the Laplace operator (Univ. Hawaii, 1979)'' , ''Proc. Symp. Pure Math.'' , '''XXXVI''' , Amer. Math. Soc. (1980) pp. 91–146</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> A. Connes,   "Non-commutative differential geometry" ''Publ. Math. IHES'' , '''62''' (1985) pp. 41–144</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> A. Connes,   "Sur la théorie noncommutative de l'intégration" , ''Algèbres d'Opérateurs'' , ''Lecture Notes in Mathematics'' , '''725''' , Springer (1982) pp. 19–143</TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top"> A. Miščenko,   A. Fomenko,   "The index of elliptic operators over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030186.png" />-algebras"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''43''' (1979) pp. 831–859</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> P. Gilkey,   "Curvature and the eigenvalues of the Laplacian for elliptic complexes" ''Adv. Math.'' , '''10''' (1973) pp. 344–382</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> P. Gilkey,   "Curvature and the eigenvalues of the Dolbeault complex for Kaehler manifolds" ''Adv. Math.'' , '''11''' (1973) pp. 311–325</TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top"> P. Gilkey,   "Invariance theory, the heat equation, and the Atiyah–Singer index theorem" , CRC (1994)</TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top"> M. Gromov,   H. Lawson Jr.,   "The classification of simply connected manifolds of positive scalar curvature" ''Ann. of Math.'' , '''111''' (1980) pp. 423–434</TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top"> F. Hirzebruch,   "Topological methods in algebraic geometry" , ''Grundl. Math. Wissenschaft.'' , '''131''' , Springer (1966) (Edition: Third)</TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top"> N. Hitchin,   "Harmonic spinors" ''Adv. Math.'' , '''14''' (1974) pp. 1–55</TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top"> L. Hörmander,   "Pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''18''' (1965) pp. 501–517</TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top"> M. Karoubi,   "Homology cyclique et K-theorie" ''Astérisque'' , '''149''' (1987) pp. 1–147</TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top"> J. Kohn,   L. Nirenberg,   "An algebra of pseudodifferential operators" ''Commun. Pure Appl. Math.'' , '''18''' (1965) pp. 269–305</TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top"> J.-L. Loday,   D. Quillen,   "Cyclic homology and the Lie homology of matrices" ''Comment. Math. Helv.'' , '''59''' (1984) pp. 565–591</TD></TR><TR><TD valign="top">[a39]</TD> <TD valign="top"> H. McKean Jr.,   I. Singer,   "Curvature and the eigenvalues of the Laplacian" ''J. Diff. Geom.'' , '''1''' (1967) pp. 43–69</TD></TR><TR><TD valign="top">[a40]</TD> <TD valign="top"> R. Melrose,   "The Atiyah–Patodi–Singer index theorem" , Peters (1993)</TD></TR><TR><TD valign="top">[a41]</TD> <TD valign="top"> W. Müller,   "Manifolds with cusps of rank one, spectral theory and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030187.png" />-index theorem" , ''Lecture Notes in Mathematics'' , '''1244''' , Springer (1987)</TD></TR><TR><TD valign="top">[a42]</TD> <TD valign="top"> V. Nistor,   "Higher index theorems and the boundary map in cyclic cohomology" ''Documenta Math.'' (1997) pp. 263–296 ((electronic))</TD></TR><TR><TD valign="top">[a43]</TD> <TD valign="top"> D. Quillen,   "Superconnections and the Chern character" ''Topology'' , '''24''' (1985) pp. 89–95</TD></TR><TR><TD valign="top">[a44]</TD> <TD valign="top"> D. Ray,   I. Singer,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030188.png" />-torsion and the laplacian on Riemannian manifolds"  ''Adv. Math.'' , '''7''' (1971) pp. 145–210</TD></TR><TR><TD valign="top">[a45]</TD> <TD valign="top"> J. Roe,   "Elliptic operators, topology and asymptotic methods" , ''Pitman Res. Notes in Math. Ser.'' , '''179''' , Longman (1988)</TD></TR><TR><TD valign="top">[a46]</TD> <TD valign="top"> R.T. Seeley,   "Refinement of the functional calculus of Calderòn and Zygmund" ''Indag. Math.'' , '''27''' (1965) pp. 521–531 ''Nederl. Akad. Wetensch. Proc. Ser. A'' , '''68''' (1965)</TD></TR><TR><TD valign="top">[a47]</TD> <TD valign="top"> Mark Stern,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030189.png" />-index theorems on locally symmetric spaces" ''Invent. Math.'' , '''96''' (1989) pp. 231–282</TD></TR><TR><TD valign="top">[a48]</TD> <TD valign="top"> S. Stolz,   "Simply connected manifolds of positive scalar curvature" ''Ann. of Math.'' , '''136''' : 2 (1992) pp. 511–540</TD></TR><TR><TD valign="top">[a49]</TD> <TD valign="top"> C. Taubes,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130030/i130030190.png" /> actions and elliptic genera" ''Comm. Math. Phys.'' , '''122''' (1989) pp. 455–526</TD></TR><TR><TD valign="top">[a50]</TD> <TD valign="top"> B.L. Tsygan,   "Homology of matrix Lie algebras over rings and Hochschild homology" ''Uspekhi Mat. Nauk.'' , '''38''' (1983) pp. 217–218</TD></TR><TR><TD valign="top">[a51]</TD> <TD valign="top"> E. Witten,   "Supersymmetry and Morse theory" ''J. Diff. Geom.'' , '''17''' (1982) pp. 661–692</TD></TR><TR><TD valign="top">[a52]</TD> <TD valign="top"> E. Witten,   "Elliptic genera and quantum field theory" ''Comm. Math. Phys.'' , '''109''' (1987) pp. 525–536</TD></TR><TR><TD valign="top">[a53]</TD> <TD valign="top"> E. Witten,   "Constraints on supersymmetry breaking" ''Nucl. Phys. B'' , '''202''' (1982) pp. 253–316</TD></TR><TR><TD valign="top">[a54]</TD> <TD valign="top"> V. Patodi,   "Curvature and the eigenforms of the Laplace operator" ''J. Diff. Geom.'' , '''5''' (1971) pp. 233–249</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> M. Atiyah, R. Bott, V. Patodi, "On the heat equation and the index theorem" ''Invent. Math.'' , '''19''' (1973) pp. 279–330 (Erata ibid. 28 (1975), 277-280) {{MR|0650828}} {{ZBL|0364.58016}} {{ZBL|0257.58008}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> M. Atiyah, F. Hirzebruch, "Spin manifolds and group actions" , ''Essays in Topology and Related subjects'' , Springer (1994) pp. 18–28 {{MR|0278334}} {{ZBL|0193.52401}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> M. Atiyah, V. Patodi, I. Singer, "Spectral asymmetry and Riemannian geometry I" ''Math. Proc. Cambridge Philos. Soc.'' , '''77''' (1975) pp. 43–69 {{MR|0397797}} {{MR|0397798}} {{MR|0397799}} {{ZBL|0297.58008}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> M. Atiyah, I. Singer, "The index of elliptic operators I" ''Ann. of Math.'' , '''87''' (1968) pp. 484–530 {{MR|0236950}} {{MR|0232402}} {{ZBL|0164.24001}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> M. Atiyah, G. Segal, "The index of elliptic operators II" ''Ann. of Math.'' , '''87''' (1968) pp. 531–545 {{MR|0236953}} {{MR|0236951}} {{ZBL|0164.24201}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> M. Atiyah, I. Singer, "The index of elliptic operators III" ''Ann. of Math.'' , '''93''' (1968) pp. 546–604 {{MR|0236952}} {{ZBL|0164.24301}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> M. Atiyah, I. Singer, "The index of elliptic operators IV" ''Ann. of Math.'' , '''93''' (1971) pp. 119–138 {{MR|0279833}} {{ZBL|0212.28603}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> L. Alvarez-Gaumé, "Supersymmetry and the Atiyah–Singer index theorem" ''Comm. Math. Phys.'' , '''90''' (1983) pp. 161–173 {{MR|}} {{ZBL|0528.58034}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> M. Atiyah, R. Bott, "A Lefschetz fixed-point formula for elliptic complexes II: Applications." ''Ann. of Math.'' , '''88''' (1968) pp. 451–491 {{MR|0232406}} {{ZBL|0167.21703}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> M. Atiyah, "$K$-theory" , Benjamin (1967) {{MR|0224084}} {{MR|0224083}} {{ZBL|0159.53401}} {{ZBL|0159.53302}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> M. Atiyah, "Elliptic operators, discrete subgroups, and von Neumann algebras" ''Astérisque'' , '''32/33''' (1969) pp. 43–72</td></tr><tr><td valign="top">[a12]</td> <td valign="top"> M. Atiyah, W. Schmid, "A geometric construction of the discrete series" ''Invent. Math.'' , '''42''' (1977) pp. 1–62 {{MR|0463358}} {{ZBL|0373.22001}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> B. Booss–Bavnbek, D. Bleecker, "Topology and analysis. The Atiyah–Singer index formula and gauge-theoretic physics" , ''Universitext'' , Springer (1985) {{MR|0771117}} {{ZBL|0551.58031}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> N. Berline, E. Getzler, M. Vèrgne, "Heat kernels and Dirac operator" , ''Grundl. Math. Wissenschaft.'' , '''298''' , Springer (1996) {{MR|2273508}} {{MR|1215720}} {{ZBL|}} </td></tr><tr><td valign="top">[a15]</td> <td valign="top"> B. Booss–Bavnbek, K. Wojciechowski, "Elliptic boundary problems for Dirac operators" , ''Math. Th. Appl.'' , Birkhäuser (1993) {{MR|1233386}} {{ZBL|0797.58004}} </td></tr><tr><td valign="top">[a16]</td> <td valign="top"> P. Baum, R. Douglas, "Index theory, bordism, and $K$-homology" , ''Operator Algebras and $K$-Theory (San Francisco, Calif., 1981)'' , ''Contemp. Math.'' , '''10''' , Amer. Math. Soc. (1982) pp. 1–31 {{MR|0658506}} {{ZBL|}} </td></tr><tr><td valign="top">[a17]</td> <td valign="top"> J.-M. Bismut, "The Atiyah–Singer theorems: a probabilistic approach" ''J. Funct. Anal.'' , '''57''' (1984) pp. 56–99 {{MR|0756173}} {{MR|0744920}} {{ZBL|0556.58027}} {{ZBL|0538.58033}} </td></tr><tr><td valign="top">[a18]</td> <td valign="top"> J.-M. Bismut, "The index theorem for families of Dirac operators: two heat equation proofs" ''Invent. Math.'' , '''83''' (1986) pp. 91–151 {{MR|}} {{ZBL|0592.58047}} </td></tr><tr><td valign="top">[a19]</td> <td valign="top"> J.-M. Bismut, "The Atiyah–Singer theorems: a probabilistic approach. II. The Lefschetz fixed point formulas" ''J. Funct. Anal.'' , '''57''' : 3 (1984) pp. 329–348 {{MR|0744920}} {{MR|0756173}} {{ZBL|0556.58027}} </td></tr><tr><td valign="top">[a20]</td> <td valign="top"> J.-M. Bismut, J. Cheeger, "$ \eta $-invariants and their adiabatic limits" ''J. Amer. Math. Soc.'' , '''2''' (1989) pp. 33–70 {{MR|0966608}} {{ZBL|0671.58037}} </td></tr><tr><td valign="top">[a21]</td> <td valign="top"> J.-M. Bismut, D. Freed, "The analysis of elliptic families: Metrics and connections on determinant bundles" ''Comm. Math. Phys.'' , '''106''' (1986) pp. 103–163 {{MR|853982}} {{ZBL|}} </td></tr><tr><td valign="top">[a22]</td> <td valign="top"> A. Borel, J.-P. Serre, "Le téorème de Riemann–Roch (d'apreès Grothendieck)" ''Bull. Soc. Math. France'' , '''86''' (1958) pp. 97–136</td></tr><tr><td valign="top">[a23]</td> <td valign="top"> R. Bott, C. Taubes, "On the rigidity theorems of Witten" ''J. Amer. Math. Soc.'' , '''2''' : 1 (1989) pp. 137–186 {{MR|0954493}} {{ZBL|0667.57009}} </td></tr><tr><td valign="top">[a24]</td> <td valign="top"> A. Connes, H. Moscovici, "Cyclic cohomology, the Novikov conjecture and hyperbolic groups" ''Topology'' , '''29''' (1990) pp. 345–388 {{MR|1066176}} {{ZBL|0759.58047}} </td></tr><tr><td valign="top">[a25]</td> <td valign="top"> J. Cheeger, "On the Hodge theory of Riemannian pseudomanifolds" , ''Geometry of the Laplace operator (Univ. Hawaii, 1979)'' , ''Proc. Symp. Pure Math.'' , '''XXXVI''' , Amer. Math. Soc. (1980) pp. 91–146 {{MR|0573430}} {{ZBL|0461.58002}} </td></tr><tr><td valign="top">[a26]</td> <td valign="top"> A. Connes, "Non-commutative differential geometry" ''Publ. Math. IHES'' , '''62''' (1985) pp. 41–144 {{MR|823176}} {{ZBL|0592.46056}} {{ZBL|0564.58002}} </td></tr><tr><td valign="top">[a27]</td> <td valign="top"> A. Connes, "Sur la théorie noncommutative de l'intégration" , ''Algèbres d'Opérateurs'' , ''Lecture Notes in Mathematics'' , '''725''' , Springer (1982) pp. 19–143</td></tr><tr><td valign="top">[a28]</td> <td valign="top"> A. Miščenko, A. Fomenko, "The index of elliptic operators over $C ^ { * }$-algebras" ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''43''' (1979) pp. 831–859 {{MR|548506}} {{ZBL|}} </td></tr><tr><td valign="top">[a29]</td> <td valign="top"> P. Gilkey, "Curvature and the eigenvalues of the Laplacian for elliptic complexes" ''Adv. Math.'' , '''10''' (1973) pp. 344–382 {{MR|0324731}} {{ZBL|0259.58010}} </td></tr><tr><td valign="top">[a30]</td> <td valign="top"> P. Gilkey, "Curvature and the eigenvalues of the Dolbeault complex for Kaehler manifolds" ''Adv. Math.'' , '''11''' (1973) pp. 311–325 {{MR|0334290}} {{ZBL|0285.53044}} </td></tr><tr><td valign="top">[a31]</td> <td valign="top"> P. Gilkey, "Invariance theory, the heat equation, and the Atiyah–Singer index theorem" , CRC (1994) {{MR|1396308}} {{MR|0783634}} {{ZBL|0856.58001}} {{ZBL|0565.58035}} </td></tr><tr><td valign="top">[a32]</td> <td valign="top"> M. Gromov, H. Lawson Jr., "The classification of simply connected manifolds of positive scalar curvature" ''Ann. of Math.'' , '''111''' (1980) pp. 423–434 {{MR|0577131}} {{ZBL|0463.53025}} </td></tr><tr><td valign="top">[a33]</td> <td valign="top"> F. Hirzebruch, "Topological methods in algebraic geometry" , ''Grundl. Math. Wissenschaft.'' , '''131''' , Springer (1966) (Edition: Third) {{MR|0202713}} {{ZBL|0138.42001}} </td></tr><tr><td valign="top">[a34]</td> <td valign="top"> N. Hitchin, "Harmonic spinors" ''Adv. Math.'' , '''14''' (1974) pp. 1–55 {{MR|0358873}} {{ZBL|0284.58016}} </td></tr><tr><td valign="top">[a35]</td> <td valign="top"> L. Hörmander, "Pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''18''' (1965) pp. 501–517 {{MR|0180740}} {{ZBL|0125.33401}} </td></tr><tr><td valign="top">[a36]</td> <td valign="top"> M. Karoubi, "Homology cyclique et K-theorie" ''Astérisque'' , '''149''' (1987) pp. 1–147 {{MR|}} {{ZBL|0601.18007}} </td></tr><tr><td valign="top">[a37]</td> <td valign="top"> J. Kohn, L. Nirenberg, "An algebra of pseudodifferential operators" ''Commun. Pure Appl. Math.'' , '''18''' (1965) pp. 269–305 {{MR|176362}} {{ZBL|}} </td></tr><tr><td valign="top">[a38]</td> <td valign="top"> J.-L. Loday, D. Quillen, "Cyclic homology and the Lie homology of matrices" ''Comment. Math. Helv.'' , '''59''' (1984) pp. 565–591 {{MR|780077}} {{ZBL|0565.17006}} </td></tr><tr><td valign="top">[a39]</td> <td valign="top"> H. McKean Jr., I. Singer, "Curvature and the eigenvalues of the Laplacian" ''J. Diff. Geom.'' , '''1''' (1967) pp. 43–69</td></tr><tr><td valign="top">[a40]</td> <td valign="top"> R. Melrose, "The Atiyah–Patodi–Singer index theorem" , Peters (1993) {{MR|1348401}} {{ZBL|0796.58050}} </td></tr><tr><td valign="top">[a41]</td> <td valign="top"> W. Müller, "Manifolds with cusps of rank one, spectral theory and an $L^{2}$-index theorem" , ''Lecture Notes in Mathematics'' , '''1244''' , Springer (1987)</td></tr><tr><td valign="top">[a42]</td> <td valign="top"> V. Nistor, "Higher index theorems and the boundary map in cyclic cohomology" ''Documenta Math.'' (1997) pp. 263–296 ((electronic)) {{MR|1480038}} {{ZBL|0893.19002}} </td></tr><tr><td valign="top">[a43]</td> <td valign="top"> D. Quillen, "Superconnections and the Chern character" ''Topology'' , '''24''' (1985) pp. 89–95 {{MR|0790678}} {{ZBL|0569.58030}} </td></tr><tr><td valign="top">[a44]</td> <td valign="top"> D. Ray, I. Singer, "$R$-torsion and the laplacian on Riemannian manifolds" ''Adv. Math.'' , '''7''' (1971) pp. 145–210 {{MR|0295381}} {{ZBL|0239.58014}} </td></tr><tr><td valign="top">[a45]</td> <td valign="top"> J. Roe, "Elliptic operators, topology and asymptotic methods" , ''Pitman Res. Notes in Math. Ser.'' , '''179''' , Longman (1988) {{MR|0960889}} {{ZBL|0654.58031}} </td></tr><tr><td valign="top">[a46]</td> <td valign="top"> R.T. Seeley, "Refinement of the functional calculus of Calderòn and Zygmund" ''Indag. Math.'' , '''27''' (1965) pp. 521–531 ''Nederl. Akad. Wetensch. Proc. Ser. A'' , '''68''' (1965) {{MR|0226450}} {{ZBL|0141.13302}} </td></tr><tr><td valign="top">[a47]</td> <td valign="top"> Mark Stern, "$L^{2}$-index theorems on locally symmetric spaces" ''Invent. Math.'' , '''96''' (1989) pp. 231–282 {{MR|0989698}} {{ZBL|0694.58039}} </td></tr><tr><td valign="top">[a48]</td> <td valign="top"> S. Stolz, "Simply connected manifolds of positive scalar curvature" ''Ann. of Math.'' , '''136''' : 2 (1992) pp. 511–540 {{MR|1189863}} {{ZBL|0784.53029}} </td></tr><tr><td valign="top">[a49]</td> <td valign="top"> C. Taubes, "$S ^ { 1 }$ actions and elliptic genera" ''Comm. Math. Phys.'' , '''122''' (1989) pp. 455–526 {{MR|0998662}} {{ZBL|0683.58043}} </td></tr><tr><td valign="top">[a50]</td> <td valign="top"> B.L. Tsygan, "Homology of matrix Lie algebras over rings and Hochschild homology" ''Uspekhi Mat. Nauk.'' , '''38''' (1983) pp. 217–218 {{MR|0695483}} {{ZBL|0526.17006}} </td></tr><tr><td valign="top">[a51]</td> <td valign="top"> E. Witten, "Supersymmetry and Morse theory" ''J. Diff. Geom.'' , '''17''' (1982) pp. 661–692 {{MR|0683171}} {{ZBL|0499.53056}} </td></tr><tr><td valign="top">[a52]</td> <td valign="top"> E. Witten, "Elliptic genera and quantum field theory" ''Comm. Math. Phys.'' , '''109''' (1987) pp. 525–536 {{MR|0885560}} {{ZBL|0625.57008}} </td></tr><tr><td valign="top">[a53]</td> <td valign="top"> E. Witten, "Constraints on supersymmetry breaking" ''Nucl. Phys. B'' , '''202''' (1982) pp. 253–316 {{MR|0668987}} {{ZBL|}} </td></tr><tr><td valign="top">[a54]</td> <td valign="top"> V. Patodi, "Curvature and the eigenforms of the Laplace operator" ''J. Diff. Geom.'' , '''5''' (1971) pp. 233–249 {{MR|0292114}} {{ZBL|0211.53901}} </td></tr></table>

Latest revision as of 17:45, 1 July 2020

The area of mathematics whose main object of study is the index of operators (cf. also Index of an operator; Index formulas).

The main question in index theory is to provide index formulas for classes of Fredholm operators (cf. also Fredholm operator), but this is not the only interesting question. First of all, to be able to provide index formulas, one has to specify what meaning of "index" is agreed upon, then one has to specify to what classes of operators these formulas will apply, and, finally, one has to explain how to use these formulas in applications.

A consequence of this is that index theory also studies various generalizations of the concept of Fredholm index, including $K$-theoretical and cyclic homology indices, for example. Moreover, the study of the analytic properties necessary for the index to be defined are an important part of index theory. Here one includes the study of conditions for being Fredholm or non-Fredholm for classes of operators that nevertheless have finite-dimensional kernels. Soon after (1970s), other invariants of elliptic operators have been defined that are similar in nature to the analytic index. The study of these related invariants is also commonly considered to be part of index theory. The most prominent of these new, related invariants are the Ray–Singer analytic torsion and the eta-invariant. Fixed-point formulas are also usually considered part of index theory, see [a9]. Finally, one of the most important goals of index theory is to study applications of the index theorems to geometry, physics, group representations, analysis, and other fields. There is a very long and fast growing list of papers dealing with these applications.

Index theory has become a subject on its own only after M.F. Atiyah and I. Singer published their index theorems in the sequence of papers [a4], [a6], [a7] (cf. also Index formulas). These theorems had become possible only due to progress in the related fields of $K$-theory [a10], [a5] and pseudo-differential operators (cf. also Pseudo-differential operator) [a35], [a37], [a46]. Important particular cases of the Atiyah–Singer index theorems were known before. Among them, Hirzebruch's signature theorem (cf. also Signature) occupies a special place (see [a33], especially for topics such as multiplicative genera and the Langlands formula for the dimension of spaces of automorphic forms). Hirzebruch's theorem was generalized by A. Grothendieck (see [a22]), who introduced many of the ideas that proved to be fundamental for the proof of the index theorems. All these theorems turned out to be consequences of the Atiyah–Singer index theorems (see also Index formulas for some index formulas that preceded the Atiyah–Singer index formula).

Atiyah–Singer index formulas.

A common characteristic of the first three main index formulas of Atiyah–Singer and Atiyah–Segal is that they depend only on the principal symbol of the operator whose index they compute. (For a differential operator, the principal symbol is given by the terms involving only the highest-order differentials and is independent of the choice of a coordinate system; cf. also Principal part of a differential operator; Symbol of an operator.) The main theorems mentioned above are:

the index theorem for a single elliptic operator $P$ acting between sections of vector bundles on a smooth, compact manifold $M$ (Atiyah–Singer, [a4]);

the equivariant index theorem for a single elliptic operator equivariant with respect to a compact group $G$ (Atiyah–Segal, [a5]); and

the index theorem for families $( P _ { b } ) _ { b \in B }$ of elliptic operators acting on the fibres of a fibre bundle $Y \rightarrow B$ (Atiyah–Singer, [a7]). These results are briefly reviewed below.

A single elliptic operator acting between sections of vector bundles.

If $P$ is an elliptic differential, or, more generally, an elliptic pseudo-differential operator acting between sections of two smooth vector bundles (cf. also Elliptic operator), then $P$ defines a continuous operator between suitable Sobolev spaces with closed range and finite-dimensional kernel and cokernel, that is, a Fredholm operator. The first of the index theorems gives an explicit formula for the Fredholm, or analytic, index $\operatorname{ind} ( P )$ of $P$:

\begin{equation*} \operatorname{ind} ( P ) : = \operatorname { dim } ( \operatorname{ker} ( P ) ) - \operatorname { dim } ( \operatorname { coker } ( P ) ). \end{equation*}

Denote by $\mathcal{T} ( M )$ the Todd class of the complexification of the tangent bundle $T M$ of $M$. If $P$ is an elliptic operator as above, its principal symbol $a = \sigma ( P )$ defines a $K$-theory class $[ a ]$ with compact supports on $T ^ { * } M$ whose Chern character, denoted $\operatorname{Ch} ( [ a ] )$, is in the even cohomology of $T ^ { * } M$ with compact supports. The Atiyah–Singer index formula of [a6] then states that

\begin{equation*} \operatorname{ind} ( P ) = ( - 1 ) ^ { n } \operatorname{Ch} ( [ a ] ) {\cal T} ( M ) [ T ^ { * } M ], \end{equation*}

$n$ being the dimension of the manifold $M$ and $[ T ^ { * } M ]$ being the fundamental class of $T ^ { * } M$. (The factor $( - 1 ) ^ { n }$ reflects the choice of the orientation of $T ^ { * } M$ in the original articles. Other choices for this orientation will lead to different signs.) In other words, the index is obtained by evaluating the compactly supported cohomology class $\operatorname{Ch} ( [ a ] ) \mathcal{T} ( M )$ on the fundamental class of $T ^ { * } M$.

Equivariant index theorem.

The second of the index formulas refines the index when the operator $P$ above is invariant with respect to a compact Lie group, see [a5], [a6]. Recall that the representation ring of a compact group $G$ is defined as the ring of formal linear combinations with integer coefficients of equivalence classes of irreducible representations of $G$ (cf. also Irreducible representation). For operators $P$ equivariant with respect to a compact group $G$, the kernel and cokernel are representations of $G$, so their difference can now be regarded as an element of $R ( G )$, called the equivariant index of $P$. The Atiyah–Singer index formula in [a6] gives the value $\text{ind}_{ g } ( P )$ of the (character of the) index of $P$ at $g \in G$ in terms of invariants of $M ^ { g }$, the set of fixed points of $g$ in $M$. Denote by $a | _ { T ^{*} M ^{ g }}$ the restriction of $a$ to the cotangent bundle of $M ^ { g }$ and by $\mathcal{T} ( M ^ { g } )$ the Todd class of the complexification of the cotangent bundle of $M ^ { g }$. In addition to these ingredients, which are similar to the ingredients appearing in the formula for $\operatorname{ind} ( P )$ above, the formula for $\text{ind}_{ g } ( P )$ involves also a Lefschetz-type contribution, denoted below by $L ( N , g )$, obtained from the action of $g$ on the normal bundle to the set $M ^ { g }$:

\begin{equation*} \operatorname { ind } _ { g } ( P ) = ( - 1 ) ^ { n } \operatorname { Ch } ( [ a | _ { T ^ { * } M ^ { g } } ] ) \mathcal{T} ( M ^ { g } ) L ( N , g ) [ T ^ { * } M ^ { g } ]. \end{equation*}

Families of elliptic operators.

For families of elliptic operators acting on the fibres of a fibre bundle $\pi : Y \rightarrow B$ (cf. also Fibration), a first problem is to make sense of the index. The solution proposed by Atiyah and Singer in [a7] is to define the index as an element of a $K$-theory group, namely $K ^ { 0 } ( B )$ in this case (cf. also $K$-theory). This fortunate choice has opened the way for many other developments in index theory. Actually, in the two index theorems mentioned above, the index can also be interpreted using a $K$-theory group, the $K$-theory of the algebra $\mathbf{C}$ of complex numbers in the first index theorem and the $K$-theory group of $C ^ { * } ( G )$, the norm closure of the convolution algebra of $G$, in the equivariant index theorem. For the Chern character of the family index of a family of elliptic operators $( P _ { b } )$ as above, there is a formula similar to the formula for the index of a single elliptic operator. The principal symbols $a _ { b } = \sigma ( P _ { b } )$ of the operators $P _ { b }$ define, in this case, a class $[ a ]$ in the $K$-theory with compact supports of $T _ { \text{vert} } ^ { * } Y : = T ^ { * } Y / \pi ^ { * } ( T ^ { * } B )$, the vertical cotangent bundle to the fibres of $\pi : Y \rightarrow B$, as in the case of a single elliptic operator. Denote by $\mathcal{T} ( M | B )$ the Todd class of the complexification of $T _ { \text { vert } } ^ { * } Y$ and by $\pi_{ *} : H _ { c } ^ { * } ( T _ { \text { vert } } ^ { * } Y ) \rightarrow H ^ { * - 2 n} ( B )$ the morphism induced by integration along the fibres, with $n$ being the common dimension of the fibres of $\pi$. Then

\begin{equation*} \operatorname{Ch} ( \operatorname{ ind } ( P ) ) = ( - 1 ) ^ { n } \pi_{ *} ( \operatorname { ind } ( [ a ] ) {\cal T} ( M | B ) ). \end{equation*}

This completes the discussion of these three main theorems of Atiyah and Singer.

$K$-theory in index theory.

The role of $K$-theory in the proof and applications of the index theorems can hardly be overstated and certainly does not stop at providing an interpretation of the index as an element of a $K$-theory group. A far-reaching consequence of the use of $K$-theory, which depends on Bott periodicity (or more precisely, the Thom isomorphism, cf. also Bott periodicity theorem), is that all elliptic operators can be connected, by a homotopy of Fredholm operators, to certain operators of a very particular kind, the so-called generalized Dirac operators (see below). It is thus sufficient to prove the index theorems for generalized Dirac operators.

Due to their differential-geometric properties, it is possible to give more concrete proofs of the Atiyah–Singer index theorem for generalized Dirac operators, using heat kernels, for example (cf. also Heat content asymptotics). The generalized Dirac operator with coefficients in the spin bundle is called simply the Dirac operator (sometimes called the Atiyah–Singer operator). See below for more about generalized Dirac operators.

Applications of index theorems.

After the publication of the first papers by Atiyah and Singer, index theory has evolved into essentially three directions:

a direction which consists of applications and new proofs of the index theorems (especially "local" proofs using heat kernels);

a direction which studies invariants other than the index; and

a direction which aims at more general index theorems. There is a very large number of applications of index theorems to topology and other areas of mathematics. A few examples follow. In [a12], Atiyah and W. Schmid used Atiyah's $L^{2}$-index theorem for coverings [a11] to construct discrete series representations. In [a34], N.J. Hitchin used the families index theorem to prove that there exist metrics whose associated Dirac operators have non-trivial kernels (in suitable dimensions). An index theorem for foliations that is close in spirit to Atiyah's $L^{2}$-index theorem was obtained by A. Connes [a27].

The index of Dirac (or Atiyah–Singer) operators was used to formulate and then prove the Gromov–Lawson conjecture [a32], which states that a compact, spin, simply connected manifold of dimension $\geq 5$ admits a metric of positive scalar curvature if and only if the index of the spin Dirac operator (in an appropriate $K$-theory group) is zero. This conjecture was proved by S. Stolz, [a48]. Dirac operators have been used to give a concrete construction of $K$-homology [a16].

Some of the applications of the index theorems require new proofs of these theorems, usually relying on the "heat-kernel method" . The main idea of this method is as follows. H. McKean and Singer [a39] stated the problem of investigating the behaviour, as $t \rightarrow 0$, $t > 0$, of the (super-trace of the) heat kernel. More precisely, let

\begin{equation*} k _ { t } ( x , y ) = \operatorname { str } ( e ^ { - t D ^ { 2 } } ) = \operatorname { tr } ( e ^ { - t D _ { + } ^ { * } D _ { + } } ) - \operatorname { tr } ( e ^ { - t D _ { + } D _ { + } ^ { * } } ) \end{equation*}

be the well-known term appearing in the McKean–Singer index formula, where $D = D _ { + } + D _ { + } ^ { * }$ is a self-adjoint geometric operator (cf. also Self-adjoint operator) with $D _ { + }$ mapping the subspace of even sections to the subspace of odd sections. They considered the case of the de Rham operator $D _ { + } + D _ { + } ^ { * }$, where $D _ { + }$ is then the de Rham differential (cf. also de Rham cohomology). It was known that the integral over the whole manifold of $k _ { t } ( x , x )$ gives the analytic index of $D _ { + }$, and they expressed the hope that $k _ { t } ( x , x )$ will have a definite limit as $t \rightarrow 0$. This was proved for various particular cases by V. Patodi in [a54] and then by P. Gilkey [a29], [a30] using invariant theory (see [a31] for an exposition of this method). This method was finally refined in [a1] to give a clear and elegant proof of the local index theorem for all Dirac operators.

Inspired by a talk of Atiyah, J.-M. Bismut investigated connections between probability theory and index theory. He was able to use the stochastic calculus (cf. also Malliavin calculus) to give a new proof of the local index theorem [a17]. His methods then generalized to give proofs of the local index theorem for families of Dirac operators [a18] using Quillen's theory of super-connections [a43], and of the Atiyah–Bott fixed-point formulas [a19]. An application of his results is the determination of the Quillen metric on the determinant bundle [a21].

The local index theorems have many connections to physics, where Dirac operators play a prominent role. Actually, several physicists have come up with arguments for a proof of the local index theorem based on supersymmetry and functional integration, see [a8] and [a53], for example. Building on these arguments, E. Getzler has obtained a short and elegant proof of the local index theorem [a14], [a31], which also uses supersymmetry. Moreover, ideas inspired from physics have lead E. Witten to conjecture that certain twisted Dirac operators on $S ^ { 1 }$-manifolds have an index that is a trivial representation of $S ^ { 1 }$, see [a52]. This was proved by C.H. Taubes [a49] (see also [a23] and [a51]). For the Dirac operator, this had been proved before by Atiyah and F. Hirzebruch [a2].

Other invariants.

Heat-kernel methods have proved very useful in dealing with non-compact and singular spaces. A common feature of these spaces is that the index formulas for the natural operators on them depend on more than just the principal symbol, which leads to the appearance of non-local invariants in these index formulas. In general, there exists no good understanding, at this time (2000), of what these non-local invariants are, except in particular cases. The most prominent of these particular cases is the Atiyah–Patodi–Singer index theorem for manifolds with boundary. Other results in these directions were obtained in [a25], [a40], [a41], [a47]. In all these cases, eta-invariants of certain boundary operators must be included in the formula for the index. Moreover, one has to either work on complete manifolds or to include boundary conditions to make the given problems Fredholm. The Atiyah–Patodi–Singer index theorem [a3], e.g., requires such boundary conditions; see below.

Let $M$ be a compact manifold with boundary $\partial M$ and metric $g$ which is a product metric in a suitable cylindrical neighbourhood of $\partial M$. Fix a Clifford module $W$ on $M$ (cf. also Clifford algebra) and an admissible connection $\nabla$. Denote by $D : = \sum c ( e _ { i } ) \nabla _ { e_i }$ the generalized Dirac operator on $W$, where $c : T ^ { * } M \cong T M \rightarrow \operatorname { End } ( W )$ is the Clifford multiplication and $e _ { i }$ is a local orthonormal basis (cf. also Orthogonal basis). Also, let $D _ { 0 }$ be the corresponding generalized Dirac operator on $\partial M$, which is (essentially) self-adjoint because $\partial M$ is compact without boundary. Then the eigenvalues of $D _ { 0 }$ will form a discrete subset of the real numbers; denote by $P _ { + }$ the spectral projection corresponding to the eigenvalues of $D _ { 0 }$ that are $\geq 0$. Decompose $D = D _ { + } + D _ { + } ^ { * }$ using the natural ${\bf Z} / 2 {\bf Z}$-grading on $W$. The operator $D _ { + }$, the chiral Dirac operator, acts from sections of $W _ { + }$ to sections of $W_-$, and has an infinite-dimensional kernel. Because of that, Atiyah, Patodi and Singer have introduced a non-local boundary condition of the form $P _ { + } f = 0$, for $f$ a smooth section of $W _ { + }$ over $\partial M$, which is a compact perturbation of the Calderón projection boundary condition. The effect of this boundary condition is that the restriction of $D$ to the subspace of sections satisfying this boundary condition is Fredholm. Assume that $M$ is $\operatorname {spin}^ { c }$ with spinor bundle $S$, such that $W = S \otimes E$, and let $h$ denote the dimension of the kernel of $D _ { 0 }$. The index of the resulting operator $D _ { + }$ with the above boundary conditions is then

\begin{equation*} \operatorname{ind}_{\alpha} ( D _ { + } ) = \int _ { M } \hat { A } ( M ) \operatorname{Ch} ( E ) - \frac { \eta ( D _ { 0 } ) + h } { 2 }. \end{equation*}

This formula was generalized by Bismut and J. Cheeger in [a20] to families of manifolds with boundary, the result being expressed using the "eta form" $\hat{\eta}$. More precisely, using the notation above, they proved that

provided that all Dirac operator associated to the boundaries of the fibres are invertible.

Presently (2000), cyclic homology (cf. also Cyclic cohomology) is probably the only general tool to deal with index problems in which the index belongs to an abstract, possibly unknown, $K$-theory group, or to deal with index theorems involving non-local invariants. See [a26], [a36], [a38], or [a50] for the basic results on cyclic homology. The relation between the $K$-theory of the algebra $A$ and the cyclic homology of $A$ is via Chern characters $\operatorname{Ch} : K _ { 0 } ( A ) \rightarrow \operatorname{HC} _ { 2 n } ( A )$, $n \geq 0$, and is due to Connes and M. Karoubi.

Generalized index theorems.

In [a24], Connes and H. Moscovici have generalized Atiyah's $L^{2}$-index theorem, which allowed them to obtain a proof of the Novikov conjecture (cf. also $C ^ { * }$-algebra) for certain classes of groups. The index theorem, also called the higher index theorem for coverings, is as follows. Let $\tilde { M } \rightarrow M$ be a covering of a compact manifold $M$ with group of deck transformations $\Gamma$ (cf. also Monodromy transformation). If $D$ is an elliptic differential operator on $M$ invariant with respect to $\Gamma$ (such as the signature operator), then it has an index $\operatorname{ind} ( D ) \in K _ { 0 } ( C _ { r } ^ { * } ( \Gamma ) )$, the $K _ { 0 }$-group of the closure of the group algebra of $\Gamma$ acting on $l ^ { 2 } ( \Gamma )$. This index was defined by A.T. Fomenko and A. Mishchenko in [a28]. This index can be refined to an index in $K_0({\cal R}\otimes {\bf C}[\Gamma])$, where $\mathcal{R}$ is the algebra of infinite matrices with complex entries and with rapid decrease. Using cyclic cohomology and the Chern character in cyclic homology, every cohomology class $\phi \in H ^ { * } ( \Gamma ) = H ^ { * } ( B \Gamma )$ gives rise to a morphism $\phi * : K _ { 0 } ( {\cal R} \otimes {\bf C} [ \Gamma ] ) \rightarrow \bf C$, and the problem is to determine $\phi_{*} ( \text { ind } ( D ) )$. If $\phi = 1 \in H ^ { 0 } ( \Gamma )$, then this number is exactly the von Neumann index appearing in Atiyah's $L^{2}$-index formula.

Let $f : M \rightarrow B \Gamma$ be the mapping that classifies the covering $\tilde { M } \rightarrow M$ and let $\mathcal{T} ( M )$ be the Todd class of the complexification of $T _ { \text { vert } } ^ { * } Y$. If $D$ is an elliptic invariant differential operator, its principal symbol $a = \sigma ( P )$ defines a $K$-theory class $[ a ]$ with compact supports on $T ^ { * } M$, whose Chern character $\operatorname{Ch} ( [ a ] )$ is in the even cohomology of $T ^ { * } M$ with compact supports, as in the case of the Atiyah–Singer index theorem for a single elliptic operator. Suppose $\phi \in H ^ { 2 m } ( \Gamma )$; then the Connes–Moscovici higher index theorem for coverings [a24] states that

\begin{equation*} \phi _ { * } ( \text { ind } ( D ) ) = ( - 1 ) ^ { n } \left( 2 \pi i ) ^ { - m } ( \operatorname {Ch} ( [ a ] ) \mathcal{T} ( M ) f ^ { * } \phi \right) [ T ^ { * } M ]. \end{equation*}

The Chern character in cyclic cohomology turns out to be a natural mapping, and this can be interpreted as a general index theorem in cyclic cohomology [a42]. It is hoped that this general index theorem will help explain the ubiquity of the Todd class in index theorems.

For more information on index theory, see, e.g., [a14], [a15], [a45]. To get a balanced point of view, see also [a13] for an account of the original approach to the Atiyah–Singer index theorems, which also gives all the necessary background a student needs.

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How to Cite This Entry:
Index theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Index_theory&oldid=12180
This article was adapted from an original article by Victor Nistor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article