View source for Index formulas

Relations between analytic and topological invariants of operators of a certain class. More precisely, index formulas establish a relation between the analytic index of a linear operator

<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/i050/i050650/i0506501.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i0506502.png" /> are topological vector spaces), defined by the formula

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and measuring in this way the "difference" between the defective subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i0506504.png" /> (namely, the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i0506505.png" /> and its cokernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i0506506.png" />), and a topological index, namely some topological characteristic of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i0506507.png" /> and the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i0506508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i0506509.png" />. For a general elliptic differential operator on a closed manifold, the problem of finding index formulas was posed towards the end of the 1950's [[#References|[1]]] and solved in 1963 (see [[#References|[2]]]), although special forms of index formulas were known even earlier, for example, the [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]] and its multi-dimensional variants. Subsequently a number of generalizations of index formulas were obtained for objects of a more complex nature; in these cases, instead of the index, which is an integer, arbitrary complex numbers and more general objects (e.g. functions) may feature.

==Elementary index formulas.==


1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065010.png" /> be the differentiable boundary of a bounded region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065011.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065012.png" /> be an elliptic pseudo-differential operator mapping the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065013.png" /> of differentiable complex-valued vector functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065014.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065015.png" /> into itself. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065016.png" /> be the manifold of tangent vectors to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065017.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065018.png" />, oriented by means of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065019.png" />-form

<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/i050/i050650/i05065020.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065021.png" /> are local coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065023.png" /> are the corresponding coordinates in the tangent space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065024.png" /> be the oriented boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065025.png" /> formed by the unit tangent vectors. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065026.png" /> is elliptic, its symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065027.png" /> is a non-singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065028.png" />-matrix function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065029.png" />. It turns out that the following Dynin–Fedosov formula holds for the index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065030.png" /> [[#References|[7]]]:

<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/i050/i050650/i05065031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065032.png" /> is the exterior power of the matrix exterior form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065034.png" /> denotes the trace of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065035.png" />-matrix form. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065036.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065037.png" /> is a differential operator on an odd-dimensional manifold, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065038.png" /> (this is not true, in general, for a pseudo-differential operator).

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065039.png" /> be an elliptic differential operator of the form

<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/i050/i050650/i05065040.png" /></td> </tr></table>

(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065041.png" /> is a multi-index) in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065042.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065043.png" /> be boundary differential operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065044.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065045.png" /> of the form

<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/i050/i050650/i05065046.png" /></td> </tr></table>

The family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065047.png" /> defines an elliptic boundary value problem if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065048.png" /> is non-singular on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065049.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065050.png" /> are the coefficients of the polynomials

<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/i050/i050650/i05065051.png" /></td> </tr></table>

that are the remainders after division of the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065052.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065053.png" />) by the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065054.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065055.png" />), where

<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/i050/i050650/i05065056.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065057.png" /> is defined from the factorization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065058.png" />, where

<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/i050/i050650/i05065059.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065060.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065062.png" /> are, respectively, a unit tangent vector and the inward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065063.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065064.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065065.png" />) is a polynomial (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065066.png" />) without zeros in the upper (respectively, lower) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065067.png" />-half-plane. By the index of the above-described boundary value problem one means the index of the corresponding linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065068.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065069.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065070.png" /> taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065071.png" /> into the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065072.png" />. It turns out that the index of the elliptic boundary value problem is the same as that of the elliptic pseudo-differential operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065073.png" /> whose symbol is given by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065074.png" />. In particular, the index of the Dirichlet problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065075.png" /> is zero. There are general index formulas for boundary value problems [[#References|[16]]], , [[#References|[27]]].

==The Atiyah–Singer index formulas.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065077.png" /> be the spaces of infinitely-differentiable sections of the vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065079.png" /> over a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065080.png" />-dimensional differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065081.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065082.png" /> be a (pseudo-differential) elliptic operator acting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065083.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065084.png" />. The topological index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065085.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065086.png" /> is defined as follows. Because of the ellipticity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065087.png" /> the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065088.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065089.png" /> determines an isomorphism of the lifted vector bundles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065090.png" />:

<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/i050/i050650/i05065091.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065092.png" /> is the bundle of unit spheres of the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065093.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065094.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065095.png" /> be the bundle of unit balls in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065096.png" />; this is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065097.png" />-dimensional manifold with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065098.png" />. By glueing the copies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i05065099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650100.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650101.png" /> along their common boundary, one obtains a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650102.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650103.png" /> over which the vector bundle

<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/i050/i050650/i050650104.png" /></td> </tr></table>

is constructed, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650106.png" /> is used to identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650108.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650109.png" />. This vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650110.png" /> carries all the topological information required for the definition of the topological index. Namely:

<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/i050/i050650/i050650111.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650112.png" /> is the cohomological [[Chern character|Chern character]] of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650113.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650114.png" /> is the cohomological [[Todd class|Todd class]] of the complexified cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650115.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650116.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650117.png" />. The right-hand side represents the value of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650118.png" />-dimensional component of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650119.png" /> on the [[Fundamental cycle|fundamental cycle]] of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650120.png" />. Thus, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650121.png" /> determines a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650122.png" /> that is trivial on the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650123.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650124.png" /> is the [[Grothendieck group|Grothendieck group]] generated by complex vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650125.png" />.

The Atiyah–Singer index theorem states:

<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/i050/i050650/i050650126.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Formula (2) admits a number of modifications. The rational cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650127.png" />, depending on the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650128.png" />, is introduced as follows. With the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650129.png" /> one can associate a difference element (cf. [[Difference-element-in-K-theory|Difference element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650130.png" />-theory]]), which can be regarded as the first [[Obstruction|obstruction]] to extending the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650131.png" /> to the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650132.png" />,

<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/i050/i050650/i050650133.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650134.png" /> is the tangent bundle, which (by means of the Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650135.png" />) can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650136.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650137.png" /> is the relative Grothendieck group of vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650138.png" />, and hence for the Chern character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650139.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650140.png" />. The formula for the topological index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650141.png" /> now takes the form:

<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/i050/i050650/i050650142.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650144.png" />.

The [[Thom isomorphism|Thom isomorphism]]

<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/i050/i050650/i050650145.png" /></td> </tr></table>

then enables one to write (4) in the form

<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/i050/i050650/i050650146.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

(As before, on the right-hand side of (4) and (5) are the values of the corresponding elements on the fundamental cycles, as in (2).)

The topological index is expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650147.png" />-theory as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650148.png" /> be a differentiable imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650149.png" /> in a Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650150.png" /> a tubular neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650151.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650152.png" />, which can be regarded as a real vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650153.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650154.png" /> is isomorphic (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650155.png" />) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650156.png" />, the complexification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650157.png" /> lifted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650158.png" /> by the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650159.png" />. Composition of the Thom isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650160.png" /> with the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650161.png" /> induced by the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650162.png" /> induces a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650163.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650164.png" /> be the Bott periodicity isomorphism. Then the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650165.png" /> does not depend on the imbedding and

<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/i050/i050650/i050650166.png" /></td> </tr></table>

==Examples.==


3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650167.png" /> be a closed oriented Riemannian manifold, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650168.png" /> be the bundle of complex exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650169.png" />-forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650170.png" /> and 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/i050/i050650/i050650171.png" /></td> </tr></table>

be the exterior differentiation operator and its adjoint, respectively. The operator

<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/i050/i050650/i050650172.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650173.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650174.png" />, is elliptic and the index formula (3) holds for it; furthermore the topological index is equal to the [[Euler characteristic|Euler characteristic]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650175.png" /> (the Hodge–de Rham theorem). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650176.png" /> the Gauss–Bonnet theorem follows.

4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650177.png" /> be the eigen <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650178.png" />-spaces of the involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650179.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650180.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650181.png" /> is the duality operator determined by the metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650182.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650183.png" />. The restriction of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650184.png" /> to an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650185.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650186.png" />, called the signature operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650187.png" />, is an elliptic operator for which the index formula (3) holds; furthermore, the analytic index is equal to the [[Signature|signature]] of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650188.png" />, while the topological index is equal to the [[L-genus|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650189.png" />-genus]] (Hirzebruch's theorem).

5) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650190.png" /> be a holomorphic vector bundle over the complex compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650191.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650192.png" /> be the bundle of differential forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650193.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650194.png" /> be the bundle of forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650195.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650196.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650197.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650198.png" />-module of smooth sections of this bundle. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650199.png" /> be the Cauchy–Riemann–Dolbeault operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650200.png" /> its adjoint, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650202.png" />. Then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650203.png" /> is an elliptic operator for which (3) holds; furthermore, the analytic index is equal to the Euler characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650204.png" /> with coefficients in the sheaf of germs of holomorphic sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650205.png" />, while the topological index is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650206.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650207.png" /> is the Chern character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650209.png" /> is the Todd class of the tangent bundle to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650210.png" /> (the Riemann–Roch–Hirzebruch theorem).

==Elliptic complexes.==
In the more general situation which arises naturally, for example, in differential geometry, instead of a single operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650211.png" /> one considers a complex of (pseudo-differential) operators

<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/i050/i050650/i050650212.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650213.png" /> are differentiable vector bundles over the closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650214.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650215.png" />. By the symbol of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650216.png" /> one means the corresponding sequence of principal symbols

<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/i050/i050650/i050650217.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650218.png" /> is the lifting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650219.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650220.png" /> by the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650221.png" />. The complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650222.png" /> is called elliptic if its symbol is an acyclic complex, that is, if it is exact everywhere outside the zero section. By the analytic index of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650223.png" /> one means its Euler characteristic:

<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/i050/i050650/i050650224.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650225.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650226.png" />-th cohomology group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650227.png" />. Two important examples of elliptic complexes are the de Rham complex and its complex analogue, the Dolbeault complex. The problem of computing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650228.png" /> in terms of the class of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650229.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650230.png" /> can be reduced to computing the index for a single operator [[#References|[3]]].

If a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650231.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650232.png" /> (and commutes with the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650233.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650234.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650235.png" />-complex), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650236.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650237.png" />-module, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650238.png" /> is defined as an element of the ring of characters of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650239.png" />. This is a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650240.png" />. Here it turns out that the index theorem can be regarded as a generalization of the [[Lefschetz theorem|Lefschetz theorem]] on fixed points, since the topological index at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650241.png" /> can be expressed in terms of the index of the restriction of the symbol to the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650242.png" /> of fixed points of the mapping defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650243.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650244.png" /> be a topological cyclic group, that is, there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650245.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650246.png" /> whose powers are dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650247.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650248.png" /> be the normal bundle to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650249.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650250.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650251.png" /> be the class of the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650252.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650253.png" /> be its restriction and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650254.png" /> be the class generated by the standard complex of exterior powers of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650255.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650256.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650257.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650258.png" />). Then the Lefschetz number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650259.png" />, which is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650260.png" />, is given by the formula

<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/i050/i050650/i050650261.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650262.png" /> is the natural extension of the topological index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650263.png" />. The cohomological version of this formula is given by:

<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/i050/i050650/i050650264.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

Without the compactness condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650265.png" />, but under the hypothesis that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650266.png" /> is a zero-dimensional submanifold and that the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650267.png" /> is non-degenerate (that is, the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650268.png" /> is transversal to the diagonal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650269.png" />), there is an analogous formula, which can be expressed as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650270.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650271.png" /> leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650272.png" /> fixed while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650273.png" /> induces a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650274.png" /> on the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650275.png" />, and

<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/i050/i050650/i050650276.png" /></td> </tr></table>

Finally, it is possible to weaken the condition of ellipticity of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650277.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650278.png" /> by considering so-called transversally-elliptic complexes; in this case, the index turns out to be a generalized function on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650279.png" /> (see [[#References|[8]]]). In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650280.png" /> is finite, then transversal ellipticity is to equivalent to ellipticity, so that the previous formulas are applicable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650281.png" /> is a homogeneous space, then all the complexes of operators are transversally elliptic and in this case the index formula is in essence the same as the Frobenius reciprocity formula for the induced representations of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650282.png" />.

==Non-Fredholm operators.==
In this case it is also sometimes possible to give another definition of the analytic index and to obtain corresponding index formulas.

==Examples.==


6) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650283.png" /> be a uniformly-elliptic operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650284.png" /> with almost-periodic coefficients. The analytic index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650285.png" /> is introduced by means of the relative dimension in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650286.png" />-factor (see [[Von Neumann algebra|von Neumann algebra]]) and is a real number (see [[#References|[11]]]). There is a formula analogous to (1), but instead of the integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650287.png" /> the average value of the almost-periodic function is used.

7) Suppose that a discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650288.png" /> acts freely on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650289.png" /> and that the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650290.png" /> is compact; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650291.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650292.png" /> be vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650293.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650294.png" /> act on them in accordance with its action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650295.png" />. The analytic index of an elliptic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650296.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650297.png" /> commuting with the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650298.png" /> is defined by the formula

<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/i050/i050650/i050650299.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650300.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650301.png" /> are the orthogonal projections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650303.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650305.png" /> is any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650306.png" />-invariant smooth density on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650308.png" /> is defined, for any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650309.png" /> commuting with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650310.png" /> and having smooth kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650311.png" />, by the formula

<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/i050/i050650/i050650312.png" /></td> </tr></table>

(here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650313.png" /> is any fundamental domain of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650314.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650315.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650316.png" /> is the trace of the matrix). It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650317.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650318.png" /> is the operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650319.png" /> whose symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650320.png" /> induces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650321.png" /> under the lifting to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650322.png" /> by the canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650323.png" /> [[#References|[12]]]. Thus, the index formula for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650324.png" /> can be obtained from the index formula for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650325.png" /> on the compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650326.png" />. This result enables one to reveal the non-triviality of spaces in which representations of discrete series are realized [[#References|[13]]].

A formula of the same type can be obtained for invariant elliptic operators on homogeneous spaces of Lie groups, even without <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650327.png" /> being discrete, with a natural generalization of the analytic index [[#References|[20]]].

Another generalization of this situation can be obtained if one considers invariant operators on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650328.png" /> with an action of a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650329.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650330.png" /> is compact [[#References|[24]]].

8) If the coefficients of a uniformly-elliptic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650331.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650332.png" /> form a homogeneous measurable random field, then it is possible to introduce the analytic index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650333.png" />, which is a random variable (in the ergodic case, a real number) defined by formula (7) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650334.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650335.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650336.png" /> is constructed from the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650337.png" /> of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650338.png" /> by averaging over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650339.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650340.png" />. This example is a generalization of Example 6) and an analogous index formula holds for it .

9) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650341.png" /> be a compact manifold with a foliation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650342.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650343.png" /> a longitudinal elliptic differential operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650344.png" />, i.e. a differential operator containing only differentiations along the leaves and elliptic on every leaf. Suppose that there is a transverse measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650345.png" />. Then a real-valued analytic index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650346.png" /> can be defined and a formula of Atiyah–Singer type can be proved. Considering measured foliations, in this context one comes to a formula which generalizes that of Example 8), [[#References|[18]]], [[#References|[19]]].

==Index formulas with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650347.png" />-groups.==


10) If a family of elliptic operators is given, parametrized by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650348.png" /> of a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650349.png" />, then its analytic index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650350.png" /> has been defined (see [[#References|[15]]]). The topological index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650351.png" /> is constructed by analogy with formula (6) (all the constructions are carried out "fibrewise" over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650352.png" />) and the index theorem holds.

11) A more general theorem is obtained if one considers elliptic operators over a compact manifold acting in sections of vector bundles with fibres which are finitely-generated projective modules over a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650353.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650354.png" />. The analytic index here takes values in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650355.png" />. If one takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650356.png" /> with a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650357.png" />, then one obtains the formula of Example 10). Also the equivalent situation (with a compact Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650358.png" />) can be considered in this context [[#References|[26]]], [[#References|[29]]].

The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650359.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650360.png" />-factor is of particular interest [[#References|[28]]], implying the formula of Example 7).

12) There is a number of generalizations of the Atiyah–Singer formulas with the analytic index taking values in homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650361.png" />-groups or bivariant Kasparov <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650362.png" />-groups. Taking the Chern character and applying some kind of intersection index usually allows one to pass to the usual number-valued index formulas [[#References|[23]]], [[#References|[25]]]. Also, the longitudinal index theorem of Example 9) can be generalized in this manner [[#References|[21]]].

13) Consider two generalized Dirac operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650363.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650364.png" /> which coincide near infinity (in particular, they are defined on Riemannian manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650365.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650366.png" /> which coincide near infinity, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650367.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650368.png" /> are isometric for some compact subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650369.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650370.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650371.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650372.png" /> be positive near infinity and let there be the natural splittings

<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/i050/i050650/i050650373.png" /></td> </tr></table>

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650374.png" /> can be expressed by a formula of Atiyah–Singer type having important geometrical applications [[#References|[22]]].

==New analytic tools.==
The Atiyah–Bott formula

<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/i050/i050650/i050650375.png" /></td> </tr></table>

provides a local expression of the index if one uses the asymptotic expansion of the traces on the right-hand side as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650376.png" />. But this expression contains lower-order terms of the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650377.png" />, so it seems difficult to see how the corresponding integrals cancel. It occurred that cancellation is obtained by using some symmetry and supersymmetry arguments. Also a probabilistic approach is effective to work with the traces of heat kernels. Families of elliptic operators can be considered in this way too [[#References|[30]]]–[[#References|[42]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.M. Gel'fand, "On elliptic differential equations" ''Russian Math. Surveys'' , '''15''' : 3 (1960) pp. 113–123 ''Uspekhi Mat. Nauk'' , '''15''' : 3 (1960) pp. 121–132</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators on compact manifolds" ''Bull. Amer. Math. Soc.'' , '''69''' (1963) pp. 422–433 {{MR|0157392}} {{ZBL|0118.31203}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.F. Atiyah, I.M. 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">[4]</TD> <TD valign="top"> M.F. Atiyah, G.B. 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">[5]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators III" ''Ann. of Math.'' , '''87''' (1968) pp. 546–604 {{MR|0236952}} {{ZBL|0164.24301}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.F. Atiyah, I.M. 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">[7]</TD> <TD valign="top"> B.V. Fedosov, "Analytic formulas for the index of elliptic operators" ''Trans. Moscow Math. Soc.'' , '''30''' (1974) pp. 279–330 ''Trudy Moskov. Mat. Obshch.'' , '''30''' (1974) pp. 159–241 {{MR|0420731}} {{ZBL|0349.58006}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M.F. Atiyah, "Elliptic operators and compact groups" , ''Lect. notes in math.'' , '''401''' , Springer (1974) {{MR|0482866}} {{ZBL|0297.58009}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> R.S. Palais, "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965) {{MR|0198494}} {{ZBL|0137.17002}} </TD></TR><TR><TD valign="top">[10a]</TD> <TD valign="top"> M.F. Atiyah, R. Bott, V.K. Patodi, "On the heat equation and the index theorem" ''Invent. Math.'' , '''19''' (1973) pp. 279–330 {{MR|0650828}} {{ZBL|0364.58016}} {{ZBL|0257.58008}} </TD></TR><TR><TD valign="top">[10b]</TD> <TD valign="top"> M.F. Atiyah, R. Bott, V.K. Patodi, "Errata to "On the heat equation and the index theorem" " ''Invent. Math.'' , '''28''' (1975) pp. 277–280 {{MR|0650829}} {{ZBL|0301.58018}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> L. Coburn, R. Moyer, I.M. Singer, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050650/i050650378.png" />-algebras of almost-periodic pseudo-differential operators" ''Acta Math.'' , '''130''' : 3–4 (1973) pp. 279–307 {{MR|0415407}} {{ZBL|0263.47042}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> M.F. 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====Comments====
Several new proofs of the Atiyah–Singer index theorem have been given in recent years.

In his paper [[#References|[a5]]], E. Witten suggested that supersymmetric quantum theory might provide the framework for a simple proof of the index theorem. Such a proof was realized by L. Alvarez-Gaumé [[#References|[30]]] and subsequently by Friedan and Windey [[#References|[a4]]]. These theoretical physicists relied on formal manipulations inside path integrals (including fermionic path integrals). So their proofs were certainly not rigorous. E. Getzler [[#References|[38]]] found a rigorous version of their arguments which relied on [[Pseudo-differential operator|pseudo-differential operator]] theory and the theory of supermanifolds. More recently, Getzler [[#References|[39]]] found a proof whose geometric and algebraic parts are elementary and transparent. Independent of this work J.-M. Bismut [[#References|[34]]] found a related proof using probabilistic methods.

For further material cf. also [[#References|[a2]]], Chapt. XIX, [[#References|[a1]]], Chapt. 12 and [[#References|[a3]]], Chapt. 9.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, "Schrödinger operators" , Springer (1987) {{MR|0883643}} {{ZBL|0619.47005}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.V. Hörmander, "The analysis of linear partial differential operators" , '''III. Pseudo-differential operators''' , Springer (1985) {{MR|1540773}} {{MR|0781537}} {{MR|0781536}} {{ZBL|0612.35001}} {{ZBL|0601.35001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Kaku, "Introduction to superstrings" , Springer (1988) {{MR|0954610}} {{ZBL|0655.58001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Friedan, P. Windey, "Supersymmetric derivation of the Atiyah–Singer index theorem and the chiral anomaly" ''Nucl. Phys.'' , '''B253''' (1984) pp. 395–416 {{MR|888706}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</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></table>