Indifferential geometry, theAtiyah–Singer index theorem, proved byMichael Atiyah andIsadore Singer (1963),^[1] states that for anelliptic differential operator on acompact manifold, theanalytical index (related to the dimension of the space of solutions) is equal to thetopological index (defined in terms of some topological data). It includes many other theorems, such as theChern–Gauss–Bonnet theorem andRiemann–Roch theorem, as special cases, and has applications totheoretical physics.^[2][3]

The index problem for elliptic differential operators was posed byIsrael Gel'fand.^[4] He noticed the homotopy invariance of the index, and asked for a formula for it by means oftopological invariants. Some of the motivating examples included theRiemann–Roch theorem and its generalization theHirzebruch–Riemann–Roch theorem, and theHirzebruch signature theorem.Friedrich Hirzebruch andArmand Borel had proved the integrality of the genus of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of theDirac operator (which was rediscovered by Atiyah and Singer in 1961).

The Atiyah–Singer theorem was announced in 1963.^[1] The proof sketched in this announcement was never published by them, though it appears in Palais's book.^[5] It appears also in the "Séminaire Cartan-Schwartz 1963/64"^[6] that was held in Paris simultaneously with the seminar led byRichard Palais atPrinceton University. The last talk in Paris was by Atiyah on manifolds with boundary. Their first published proof^[7] replaced thecobordism theory of the first proof withK-theory, and they used this to give proofs of various generalizations in another sequence of papers.^[8]

• 1965:Sergey P. Novikov published his results on the topological invariance of the rationalPontryagin classes on smooth manifolds.^[9]
• Robion Kirby andLaurent C. Siebenmann's results,^[10] combined withRené Thom's paper^[11] proved the existence of rational Pontryagin classes on topological manifolds. The rational Pontryagin classes are essential ingredients of the index theorem on smooth and topological manifolds.
• 1969: Michael Atiyah defines abstract elliptic operators on arbitrary metric spaces. Abstract elliptic operators became protagonists in Kasparov's theory and Connes's noncommutative differential geometry.^[12]
• 1971: Isadore Singer proposes a comprehensive program for future extensions of index theory.^[13]
• 1972: Gennadi G. Kasparov publishes his work on the realization of K-homology by abstract elliptic operators.^[14]
• 1973: Atiyah,Raoul Bott, andVijay Patodi gave a new proof of the index theorem^[15] using theheat equation, described in a paper by Melrose.^[16]
• 1977:Dennis Sullivan establishes his theorem on the existence and uniqueness of Lipschitz andquasiconformal structures on topological manifolds of dimension different from 4.^[17]
• 1983:Ezra Getzler^[18] motivated by ideas of Edward Witten^[19] andLuis Alvarez-Gaume, gave a short proof of the local index theorem for operators that are locallyDirac operators; this covers many of the useful cases.
• 1983: Nicolae Teleman proves that the analytical indices of signature operators with values in vector bundles are topological invariants.^[20]
• 1984: Teleman establishes the index theorem on topological manifolds.^[21]
• 1986:Alain Connes publishes his fundamental paper onnoncommutative geometry.^[22]
• 1989:Simon K. Donaldson and Sullivan study Yang–Mills theory on quasiconformal manifolds of dimension 4. They introduce the signature operatorS defined on differential forms of degree two.^[23]
• 1990: Connes and Henri Moscovici prove the local index formula in the context of non-commutative geometry.^[24]
• 1994: Connes, Sullivan, and Teleman prove the index theorem for signature operators on quasiconformal manifolds.^[25]

• X is acompact smoothmanifold (without boundary).
• E andF are smoothvector bundles overX.
• D is an elliptic differential operator fromE toF. So in local coordinates it acts as a differential operator, taking smooth sections ofE to smooth sections ofF.

IfD is a differential operator on a Euclidean space of ordern ink variables${\displaystyle x_1,\dots ,x_k}$, then itssymbol is the function of 2k variables${\displaystyle x_1,\dots ,x_k,y_1,\dots ,y_k}$, given by dropping all terms of order less thann and replacing${\displaystyle \mathrm{\partial }/\mathrm{\partial }{x}_i}$ by${\displaystyle y_i}$. So the symbol is homogeneous in the variablesy, of degreen. The symbol is well defined even though${\displaystyle \mathrm{\partial }/\mathrm{\partial }{x}_i}$ does not commute with${\displaystyle x_i}$ because we keep only the highest order terms and differential operators commute "up to lower-order terms". The operator is calledelliptic if the symbol is nonzero whenever at least oney is nonzero.

Example: The Laplace operator ink variables has symbol${\displaystyle y_1^2+\cdots +y_k^2}$, and so is elliptic as this is nonzero whenever any of the${\displaystyle y_i}$'s are nonzero. The wave operator has symbol${\displaystyle -y_1^2+\cdots +y_k^2}$, which is not elliptic if${\displaystyle k\ge 2}$, as the symbol vanishes for some non-zero values of theys.

The symbol of a differential operator of ordern on a smooth manifoldX is defined in much the same way using local coordinate charts, and is a function on thecotangent bundle ofX, homogeneous of degreen on each cotangent space. (In general, differential operators transform in a rather complicated way under coordinate transforms (seejet bundle); however, the highest order terms transform like tensors so we get well defined homogeneous functions on the cotangent spaces that are independent of the choice of local charts.) More generally, the symbol of a differential operator between two vector bundlesE andF is a section of the pullback of the bundle Hom(E,F) to the cotangent space ofX. The differential operator is calledelliptic if the element of Hom(E_x,F_x) is invertible for all non-zero cotangent vectors at any pointx ofX.

A key property of elliptic operators is that they are almost invertible; this is closely related to the fact that their symbols are almost invertible. More precisely, an elliptic operatorD on a compact manifold has a (non-unique)parametrix (orpseudoinverse)D′ such thatDD′ -1andD′D -1 are both compact operators. An important consequence is that the kernel ofD is finite-dimensional, because all eigenspaces of compact operators, other than the kernel, are finite-dimensional. (The pseudoinverse of an elliptic differential operator is almost never a differential operator. However, it is an ellipticpseudodifferential operator.)

As the elliptic differential operatorD has a pseudoinverse, it is aFredholm operator. Any Fredholm operator has anindex, defined as the difference between the (finite) dimension of thekernel ofD (solutions ofDf = 0), and the (finite) dimension of thecokernel ofD (the constraints on the right-hand-side of an inhomogeneous equation likeDf =g, or equivalently the kernel of the adjoint operator). In other words,

Index(D) = dim Ker(D) − dim Coker(D) = dim Ker(D) − dim Ker(D*).

This is sometimes called theanalytical index ofD.

Example: Suppose that the manifold is the circle (thought of asR/Z), andD is the operator d/dx − λ for some complex constant λ. (This is the simplest example of an elliptic operator.) Then the kernel is the space of multiples of exp(λx) if λ is an integral multiple of 2πi and is 0 otherwise, and the kernel of the adjoint is a similar space with λ replaced by its complex conjugate. SoD has index 0. This example shows that the kernel and cokernel of elliptic operators can jump discontinuously as the elliptic operator varies, so there is no nice formula for their dimensions in terms of continuous topological data. However the jumps in the dimensions of the kernel and cokernel are the same, so the index, given by the difference of their dimensions, does indeed vary continuously, and can be given in terms of topological data by the index theorem.

Thetopological index of an elliptic differential operator${\displaystyle D}$ between smooth vector bundles${\displaystyle E}$ and${\displaystyle F}$ on an${\displaystyle n}$-dimensional compact manifold${\displaystyle X}$ is given by

${\displaystyle (-1{)}^n\mathrm{ch}(D)\mathrm{Td}(X)[X]=(-1{)}^n{\int }_X\mathrm{ch}(D)\mathrm{Td}(X)}$

in other words the value of the top dimensional component of the mixedcohomology class${\displaystyle \mathrm{ch}(D)\mathrm{Td}(X)}$ on thefundamental homology class of the manifold${\displaystyle X}$ up to a difference of sign.Here,

• ${\displaystyle \mathrm{Td}(X)}$ is theTodd class of the complexified tangent bundle of${\displaystyle X}$.
• ${\displaystyle \mathrm{ch}(D)}$ is equal to${\displaystyle {\phi }^{-1}(\mathrm{ch}(d(p^{\ast }E,p^{\ast }F,\sigma (D))))}$, where
  • ${\displaystyle \phi :H^k(X;\mathbb{Q})\to H^{n+k}(B(X)/S(X);\mathbb{Q})}$ is theThom isomorphism for the sphere bundle${\displaystyle p:B(X)/S(X)\to X}$
  • ${\displaystyle \mathrm{ch}:K(X)\otimes \mathbb{Q}\to H^{\ast }(X;\mathbb{Q})}$ is theChern character
  • ${\displaystyle d(p^{\ast }E,p^{\ast }F,\sigma (D))}$ is the "difference element" in${\displaystyle K(B(X)/S(X))}$ associated to two vector bundles${\displaystyle p^{\ast }E}$ and${\displaystyle p^{\ast }F}$ on${\displaystyle B(X)}$ and an isomorphism${\displaystyle \sigma (D)}$ between them on the subspace${\displaystyle S(X)}$.
  • ${\displaystyle \sigma (D)}$ is the symbol of${\displaystyle D}$

In some situations, it is possible to simplify the above formula for computational purposes. In particular, if${\displaystyle X}$ is a${\displaystyle 2m}$-dimensional orientable (compact) manifold with non-zeroEuler class${\displaystyle e(TX)}$, then applying theThom isomorphism and dividing by the Euler class,^[26][27] the topological index may be expressed as

${\displaystyle (-1{)}^m{\int }_X\frac{\mathrm{ch}(E)-\mathrm{ch}(F)}{e(TX)}\mathrm{Td}(X)}$

where division makes sense by pulling${\displaystyle e(TX{)}^{-1}}$ back from the cohomology ring of theclassifying space${\displaystyle BSO}$.

One can also define the topological index using only K-theory (and this alternative definition is compatible in a certain sense with the Chern-character construction above). IfX is a compact submanifold of a manifoldY then there is a pushforward (or "shriek") map from K(TX) to K(TY). The topological index of an element of K(TX) is defined to be the image of this operation withY some Euclidean space, for which K(TY) can be naturally identified with the integersZ (as a consequence of Bott-periodicity). This map is independent of the embedding ofX in Euclidean space. Now a differential operator as above naturally defines an element of K(TX), and the image inZ under this map "is" the topological index.

As usual,D is an elliptic differential operator between vector bundlesE andF over a compact manifoldX.

Theindex problem is the following: compute the (analytical) index ofD using only the symbols andtopological data derived from the manifold and the vector bundle. The Atiyah–Singer index theorem solves this problem, and states:

The analytical index ofD is equal to its topological index.

In spite of its formidable definition, the topological index is usually straightforward to evaluate explicitly. So this makes it possible to evaluate the analytical index. (The cokernel and kernel of an elliptic operator are in general extremely hard to evaluate individually; the index theorem shows that we can usually at least evaluate theirdifference.) Many important invariants of a manifold (such as the signature) can be given as the index of suitable differential operators, so the index theorem allows us to evaluate these invariants in terms of topological data.

Although the analytical index is usually hard to evaluate directly, it is at least obviously an integer. The topological index is by definition a rational number, but it is usually not at all obvious from the definition that it is also integral. So the Atiyah–Singer index theorem implies some deep integrality properties, as it implies that the topological index is integral.

The index of an elliptic differential operator obviously vanishes if the operator is self adjoint. It also vanishes if the manifoldX has odd dimension, though there arepseudodifferential elliptic operators whose index does not vanish in odd dimensions.

TheGrothendieck–Riemann–Roch theorem was one of the main motivations behind the index theorem because the index theorem is the counterpart of this theorem in the setting of real manifolds. Now, if there's a map${\displaystyle f:X\to Y}$ of compact stably almost complex manifolds, then there is a commutative diagram^[28]

if${\displaystyle Y=\ast }$ is a point, then we recover the statement above. Here${\displaystyle K(X)}$ is theGrothendieck group of complex vector bundles. This commutative diagram is formally very similar to the GRR theorem because the cohomology groups on the right are replaced by theChow ring of a smooth variety, and the Grothendieck group on the left is given by the Grothendieck group of algebraic vector bundles.

Due to (Teleman 1983), (Teleman 1984):

For any abstract elliptic operator (Atiyah 1970) on a closed, oriented, topological manifold, the analytical index equals the topological index.

The proof of this result goes through specific considerations, including the extension of Hodge theory on combinatorial and Lipschitz manifolds (Teleman 1980), (Teleman 1983), the extension of Atiyah–Singer's signature operator to Lipschitz manifolds (Teleman 1983), Kasparov's K-homology (Kasparov 1972) and topological cobordism (Kirby & Siebenmann 1977).

This result shows that the index theorem is not merely a differentiability statement, but rather a topological statement.

Due to (Donaldson & Sullivan 1989), (Connes, Sullivan & Teleman 1994):

For any quasiconformal manifold there exists a local construction of the Hirzebruch–Thom characteristic classes.

This theory is based on a signature operatorS, defined on middle degree differential forms on even-dimensional quasiconformal manifolds (compare (Donaldson & Sullivan 1989)).

Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds (see page 678 of (Connes, Sullivan & Teleman 1994)). The work (Connes, Sullivan & Teleman 1994) "provides local constructions for characteristic classes based on higher dimensional relatives of the measurable Riemann mapping in dimension two and the Yang–Mills theory in dimension four."

These results constitute significant advances along the lines of Singer's programProspects in Mathematics (Singer 1971). At the same time, they provide, also, an effective construction of the rational Pontrjagin classes on topological manifolds. The paper (Teleman 1985) provides a link between Thom's original construction of the rational Pontrjagin classes (Thom 1956) and index theory.

It is important to mention that the index formula is a topological statement. The obstruction theories due to Milnor, Kervaire, Kirby, Siebenmann, Sullivan, Donaldson show that only a minority of topological manifolds possess differentiable structures and these are not necessarily unique. Sullivan's result on Lipschitz and quasiconformal structures (Sullivan 1979) shows that any topological manifold in dimension different from 4 possesses such a structure which is unique (up to isotopy close to identity).

The quasiconformal structures (Connes, Sullivan & Teleman 1994) and more generally theL^p-structures,p >n(n+1)/2, introduced by M. Hilsum (Hilsum 1999), are the weakest analytical structures on topological manifolds of dimensionn for which the index theorem is known to hold.

• The Atiyah–Singer theorem applies to ellipticpseudodifferential operators in much the same way as for elliptic differential operators. In fact, for technical reasons most of the early proofs worked with pseudodifferential rather than differential operators: their extra flexibility made some steps of the proofs easier.
• Instead of working with an elliptic operator between two vector bundles, it is sometimes more convenient to work with anelliptic complex$${\displaystyle 0\to E_0\to E_1\to E_2\to \cdots \to E_m\to 0}$$ of vector bundles. The difference is that the symbols now form an exact sequence (off the zero section). In the case when there are just two non-zero bundles in the complex this implies that the symbol is an isomorphism off the zero section, so an elliptic complex with 2 terms is essentially the same as an elliptic operator between two vector bundles. Conversely the index theorem for an elliptic complex can easily be reduced to the case of an elliptic operator: the two vector bundles are given by the sums of the even or odd terms of the complex, and the elliptic operator is the sum of the operators of the elliptic complex and their adjoints, restricted to the sum of the even bundles.
• If themanifold is allowed to have boundary, then some restrictions must be put on the domain of the elliptic operator in order to ensure a finite index. These conditions can be local (like demanding that the sections in the domain vanish at the boundary) or more complicated global conditions (like requiring that the sections in the domain solve some differential equation). The local case was worked out by Atiyah and Bott, but they showed that many interesting operators (e.g., thesignature operator) do not admit local boundary conditions. To handle these operators,Atiyah,Patodi andSinger introduced global boundary conditions equivalent to attaching a cylinder to the manifold along the boundary and then restricting the domain to those sections that are square integrable along the cylinder. This point of view is adopted in the proof ofMelrose (1993) of theAtiyah–Patodi–Singer index theorem.
• Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some spaceY. In this case the index is an element of the K-theory ofY, rather than an integer. If the operators in the family are real, then the index lies in the real K-theory ofY. This gives a little extra information, as the map from the real K-theory ofY to the complex K-theory is not always injective.
• If there is agroup action of a groupG on the compact manifoldX, commuting with the elliptic operator, then one replaces ordinary K-theory withequivariant K-theory. Moreover, one gets generalizations of theLefschetz fixed-point theorem, with terms coming from fixed-point submanifolds of the groupG. See also:equivariant index theorem.
• Atiyah (1976) showed how to extend the index theorem to some non-compact manifolds, acted on by a discrete group with compact quotient. The kernel of the elliptic operator is in general infinite dimensional in this case, but it is possible to get a finite index using the dimension of a module over avon Neumann algebra; this index is in general real rather than integer valued. This version is called theL² index theorem, and was used byAtiyah & Schmid (1977) to rederive properties of thediscrete series representations ofsemisimple Lie groups.
• TheCallias index theorem is an index theorem for a Dirac operator on a noncompact odd-dimensional space. The Atiyah–Singer index is only defined on compact spaces, and vanishes when their dimension is odd. In 1978Constantine Callias, at the suggestion of his Ph.D. advisorRoman Jackiw, used theaxial anomaly to derive this index theorem on spaces equipped with aHermitian matrix called theHiggs field.^[29] The index of the Dirac operator is a topological invariant which measures the winding of the Higgs field on a sphere at infinity. IfU is the unit matrix in the direction of the Higgs field, then the index is proportional to the integral ofU(dU)ⁿ⁻¹ over the (n−1)-sphere at infinity. Ifn is even, it is always zero.
  • The topological interpretation of this invariant and its relation to theHörmander index proposed byBoris Fedosov, as generalized byLars Hörmander, was published byRaoul Bott andRobert Thomas Seeley.^[30]

Suppose that${\displaystyle M}$ is a compact oriented manifold of dimension${\displaystyle n=2r}$. If we take${\displaystyle {\mathrm{\Lambda }}^{\text{even}}}$ to be the sum of the even exterior powers of the cotangent bundle, and${\displaystyle {\mathrm{\Lambda }}^{\text{odd}}}$ to be the sum of the odd powers, define${\displaystyle D=d+d^{\ast }}$, considered as a map from${\displaystyle {\mathrm{\Lambda }}^{\text{even}}}$ to${\displaystyle {\mathrm{\Lambda }}^{\text{odd}}}$. Then the analytical index of${\displaystyle D}$ is theEuler characteristic${\displaystyle \chi (M)}$ of theHodge cohomology of${\displaystyle M}$, and the topological index is the integral of theEuler class over the manifold. The index formula for this operator yields theChern–Gauss–Bonnet theorem.

The concrete computation goes as follows: according to one variation of thesplitting principle, if${\displaystyle E}$ is a real vector bundle of dimension${\displaystyle n=2r}$, in order to prove assertions involving characteristic classes, we may suppose that there are complex line bundles${\displaystyle l_1,\dots ,l_r}$ such that${\displaystyle E\otimes \mathbb{C}=l_1\oplus \overline{l_1}\oplus \cdots l_r\oplus \overline{l_r}}$. Therefore, we can consider the Chern roots${\displaystyle x_i(E\otimes \mathbb{C})=c_1(l_i)}$,${\displaystyle x_{r+i}(E\otimes \mathbb{C})=c_1\left(\overline{l_i}\right)=-x_i(E\otimes \mathbb{C})}$,${\displaystyle i=1,\dots ,r}$.

Using Chern roots as above and the standard properties of the Euler class, we have that$e(TM)=\prod _i^r{x}_i(TM\otimes \mathbb{C})$. As for the Chern character and the Todd class,^[31]

Applying the index theorem,

${\displaystyle \chi (M)=(-1{)}^r{\int }_M\frac{\prod _i^n\left(1-e^{-x_i}\right)}{\prod _i^r{x}_i}\prod _i^n\frac{x_i}{1-e^{-x_i}}(TM\otimes \mathbb{C})=(-1{)}^r{\int }_M(-1{)}^r\prod _i^r{x}_i(TM\otimes \mathbb{C})={\int }_{M}e(TM)}$

which is the "topological" version of the Chern-Gauss-Bonnet theorem (the geometric one being obtained by applying theChern-Weil homomorphism).

TakeX to be acomplex manifold of (complex) dimensionn with a holomorphic vector bundleV. We let the vector bundlesE andF be the sums of the bundles of differential forms with coefficients inV of type (0,i) withi even or odd, and we let the differential operatorD be the sum

${\displaystyle \overline{\mathrm{\partial }}+{\overline{\mathrm{\partial }}}^{\ast }}$

restricted toE.

This derivation of the Hirzebruch–Riemann–Roch theorem is more natural if we use the index theorem for elliptic complexes rather than elliptic operators. We can take the complex to be

${\displaystyle 0\to V\to V\otimes {\mathrm{\Lambda }}^{0,1}{T}^{\ast }(X)\to V\otimes {\mathrm{\Lambda }}^{0,2}{T}^{\ast }(X)\to \cdots }$

with the differential given by${\displaystyle \overline{\mathrm{\partial }}}$. Then thei'th cohomology group is just the coherent cohomology group Hⁱ(X,V), so the analytical index of this complex is theholomorphic Euler characteristic ofV:

${\displaystyle \mathrm{index}(D)=\sum _p(-1{)}^p\dim H^p(X,V)=\chi (X,V)}$

Since we are dealing with complex bundles, the computation of the topological index is simpler. Using Chern roots and doing similar computations as in the previous example, the Euler class is given by$e(TX)=\prod _i^n{x}_i(TX)$ and

Applying the index theorem, we obtain theHirzebruch-Riemann-Roch theorem:

${\displaystyle \chi (X,V)={\int }_X\mathrm{ch}(V)\mathrm{Td}(TX)}$

In fact we get a generalization of it to all complex manifolds: Hirzebruch's proof only worked forprojective complex manifoldsX.

TheHirzebruch signature theorem states that the signature of a compact oriented manifoldX of dimension 4k is given by theL genus of the manifold. This follows from the Atiyah–Singer index theorem applied to the followingsignature operator.

The bundlesE andF are given by the +1 and −1 eigenspaces of the operator on the bundle of differential forms ofX, that acts onk-forms as${\displaystyle i^{k(k-1)}}$ times theHodge star operator. The operatorD is theHodge Laplacian

${\displaystyle D\equiv \mathrm{\Delta }:={\left(\mathbf{d}+{\mathbf{d}}^{\mathbf{\ast }}\right)}^2}$

restricted toE, whered is the Cartanexterior derivative andd* is its adjoint.

The analytic index ofD is the signature of the manifoldX, and its topological index is the L genus ofX, so these are equal.

The genus is a rational number defined for any manifold, but is in general not an integer. Borel and Hirzebruch showed that it is integral for spin manifolds, and an even integer if in addition the dimension is 4 mod 8. This can be deduced from the index theorem, which implies that the  genus for spin manifolds is the index of a Dirac operator. The extra factor of 2 in dimensions 4 mod 8 comes from the fact that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so the index is even.

In dimension 4 this result impliesRochlin's theorem that the signature of a 4-dimensional spin manifold is divisible by 16: this follows because in dimension 4 the  genus is minus one eighth of the signature.

Pseudodifferential operators can be explained easily in the case of constant coefficient operators on Euclidean space. In this case, constant coefficient differential operators are just theFourier transforms of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the Fourier transforms of multiplication by more general functions.

Many proofs of the index theorem use pseudodifferential operators rather than differential operators. The reason for this is that for many purposes there are not enough differential operators. For example, a pseudoinverse of an elliptic differential operator of positive order is not a differential operator, but is a pseudodifferential operator. Also, there is a direct correspondence between data representing elements of K(B(X),S(X)) (clutching functions) and symbols of elliptic pseudodifferential operators.

Pseudodifferential operators have an order, which can be any real number or even −∞, and have symbols (which are no longer polynomials on the cotangent space), and elliptic differential operators are those whose symbols are invertible for sufficiently large cotangent vectors. Most versions of the index theorem can be extended from elliptic differential operators to elliptic pseudodifferential operators.

The initial proof was based on that of theHirzebruch–Riemann–Roch theorem (1954), and involvedcobordism theory andpseudodifferential operators.

The idea of this first proof is roughly as follows. Consider the ring generated by pairs (X,V) whereV is a smooth vector bundle on the compact smooth oriented manifoldX, with relations that the sum and product of the ring on these generators are given by disjoint union and product of manifolds (with the obvious operations on the vector bundles), and any boundary of a manifold with vector bundle is 0. This is similar to the cobordism ring of oriented manifolds, except that the manifolds also have a vector bundle. The topological and analytical indices are both reinterpreted as functions from this ring to the integers. Then one checks that these two functions are in fact both ring homomorphisms. In order to prove they are the same, it is then only necessary to check they are the same on a set of generators of this ring. Thom's cobordism theory gives a set of generators; for example, complex vector spaces with the trivial bundle together with certain bundles over even dimensional spheres. So the index theorem can be proved by checking it on these particularly simple cases.

Atiyah and Singer's first published proof usedK-theory rather than cobordism. Ifi is any inclusion of compact manifolds fromX toY, they defined a 'pushforward' operationi_! on elliptic operators ofX to elliptic operators ofY that preserves the index. By takingY to be some sphere thatX embeds in, this reduces the index theorem to the case of spheres. IfY is a sphere andX is some point embedded inY, then any elliptic operator onY is the image underi_! of some elliptic operator on the point. This reduces the index theorem to the case of a point, where it is trivial.

Atiyah, Bott, and Patodi (1973) gave a new proof of the index theorem using theheat equation, see e.g.Berline, Getzler & Vergne (1992). The proof is also published in (Melrose 1993) and (Gilkey 1994).

IfD is a differential operator with adjointD*, thenD*D andDD* are self adjoint operators whose non-zero eigenvalues have the same multiplicities. However their zero eigenspaces may have different multiplicities, as these multiplicities are the dimensions of the kernels ofD andD*. Therefore, the index ofD is given by

${\displaystyle \mathrm{index}(D)=\dim \mathrm{Ker}(D)-\dim \mathrm{Ker}(D^{\ast })=\dim \mathrm{Ker}(D^{\ast }D)-\dim \mathrm{Ker}(D{D}^{\ast })=\mathrm{Tr}\left(e^{-t{D}^{\ast }D}\right)-\mathrm{Tr}\left(e^{-tD{D}^{\ast }}\right)}$

for any positivet. The right hand side is given by the trace of the difference of the kernels of two heat operators. These have an asymptotic expansion for small positivet, which can be used to evaluate the limit ast tends to 0, giving a proof of the Atiyah–Singer index theorem. The asymptotic expansions for smallt appear very complicated, but invariant theory shows that there are huge cancellations between the terms, which makes it possible to find the leading terms explicitly. These cancellations were later explained using supersymmetry.

• (-1)F – Term in quantum field theoryPages displaying short descriptions of redirect targets
• Witten index – Modified partition function

The papers by Atiyah are reprinted in volumes 3 and 4 of his collected works, (Atiyah 1988a,1988b)

• Mazzeo, Rafe."The Atiyah–Singer Index Theorem: What it is and why you should care"(PDF). Archived from the original on June 24, 2006. RetrievedJanuary 3, 2006.{{cite web}}: CS1 maint: bot: original URL status unknown (link) Pdf presentation.
• Voitsekhovskii, M.I.; Shubin, M.A. (2001) [1994],"Index formulas",Encyclopedia of Mathematics,EMS Press
• Wassermann, Antony."Lecture notes on the Atiyah–Singer Index Theorem". Archived fromthe original on March 29, 2017.

• Raussen, Martin; Skau, Christian (2005),"Interview with Michael Atiyah and Isadore Singer"(PDF),Notices of AMS, pp. 223–231
• R. R. Seeley and other (1999)Recollections from the early days of index theory and pseudo-differential operators - A partial transcript of informal post–dinner conversation during a symposium held in Roskilde, Denmark, in September 1998.

• Differential operators
• Elliptic partial differential equations
• Theorems in differential geometry

• Articles with short description
• Short description is different from Wikidata
• Pages displaying short descriptions of redirect targets via Module:Annotated link
• CS1: long volume value
• CS1 maint: bot: original URL status unknown