Atiyah–Singer index theorem

6557:, 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. 4899: 4221: 4894:{\displaystyle {\begin{aligned}\operatorname {ch} {\mathord {\left(\Lambda ^{\text{even}}-\Lambda ^{\text{odd}}\right)}}&=1-\operatorname {ch} (T^{*}M\otimes \mathbb {C} )+\operatorname {ch} {\mathord {\left(\Lambda ^{2}T^{*}M\otimes \mathbb {C} \right)}}-\ldots +(-1)^{n}\operatorname {ch} {\mathord {\left(\Lambda ^{n}T^{*}M\otimes \mathbb {C} \right)}}\\&=1-\sum _{i}^{n}e^{-x_{i}}(TM\otimes \mathbb {C} )+\sum _{i<j}e^{-x_{i}}e^{-x_{j}}(TM\otimes \mathbb {C} )+\ldots +(-1)^{n}e^{-x_{1}}\dotsm e^{-x_{n}}(TM\otimes \mathbb {C} )\\&=\prod _{i}^{n}\left(1-e^{-x_{i}}\right)(TM\otimes \mathbb {C} )\\\operatorname {Td} (TM\otimes \mathbb {C} )&=\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}(TM\otimes \mathbb {C} )\end{aligned}}} 6146: 5741: 2878: 6141:{\displaystyle {\begin{aligned}\operatorname {ch} \left(\sum _{j}^{n}(-1)^{j}V\otimes \Lambda ^{j}{\overline {T^{*}X}}\right)&=\operatorname {ch} (V)\prod _{j}^{n}\left(1-e^{x_{j}}\right)(TX)\\\operatorname {Td} (TX\otimes \mathbb {C} )=\operatorname {Td} (TX)\operatorname {Td} \left({\overline {TX}}\right)&=\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}\prod _{j}^{n}{\frac {-x_{j}}{1-e^{x_{j}}}}(TX)\end{aligned}}} 5251: 3214:

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.

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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

1103:

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

2585:

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

6501:

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

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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

2593:

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

6517:

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

6467:

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 Â genus for spin

2873:{\displaystyle {\begin{array}{ccc}&&&\\&K(X)&{\xrightarrow{{\text{Td}}(X)\cdot {\text{ch}}}}&H(X;\mathbb {Q} )&\\&f_{*}{\Bigg \downarrow }&&{\Bigg \downarrow }f_{*}\\&K(Y)&{\xrightarrow{}}&H(Y;\mathbb {Q} )&\\&&&\\\end{array}}} 3071:

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.

3303:

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 a

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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.

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The Atiyah–Singer theorem was announced in 1963. The proof sketched in this announcement was never published by them, though it appears in Palais's book. It appears also in the "Séminaire Cartan-Schwartz 1963/64" that was held in Paris simultaneously with the seminar led by

1354: 4090: 5246:{\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)} 913:); 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 bundles 3211: 2410: 3892: 3238:

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 of

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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 elliptic

3120:

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

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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.

5658: 1875: 6237: 1739: 1625: 2590:.) 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. 6682:

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 of

4226: 5356: 5374: 4213: 3132: 1407: 227:'s paper 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. 5663:

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

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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.

3970: 555: 5746: 3049:) "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." 8070:

Séminaire Henri Cartan. Théoreme d'Atiyah-Singer sur l'indice d'un opérateur différentiel elliptique. 16 annee: 1963/64 dirigee par Henri Cartan et Laurent Schwartz. Fasc. 1; Fasc. 2. (French)

6494:

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 the

3334:

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 1978

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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 operator

2298: 854: 1950: 768: 5733: 5368:

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

3800: 670: 597: 5519: 3795: 2617:

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

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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). If

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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., the

2005: 6337: 2455: 2183: 2123: 3553: 9133: 3659: 2647: 880: 6498:

of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the Fourier transforms of multiplication by more general functions.

2065: 2035: 6840:{\displaystyle \operatorname {index} (D)=\dim \operatorname {Ker} (D^{*})=\operatorname {Tr} \left(e^{-tD^{*}D}\right)-\operatorname {Tr} \left(e^{-tDD^{*}}\right)} 2286: 1104:

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.

795: 697: 624: 3747: 3460: 2936: 2152: 2094: 2907: 2484: 6356: 2251: 3718: 3683: 3627: 3431: 2228: 2203: 1490: 1431: 1214: 1194: 1174: 1154: 1134: 1091:

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(λ

2942:

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 the

6475:

that the signature of a 4-dimensional spin manifold is divisible by 16: this follows because in dimension 4 the Â genus is minus one eighth of the signature.

5547: 1807: 6854:. 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 positive 9138: 2614: 75: 8666:

This describes the original proof of the theorem (Atiyah and Singer never published their original proof themselves, but only improved versions of it.)

6161: 1632: 1529: 6527: 156: 3076:) shows that any topological manifold in dimension different from 4 possesses such a structure which is unique (up to isotopy close to identity). 2981:

The proof of this result goes through specific considerations, including the extension of Hodge theory on combinatorial and Lipschitz manifolds (

3041:

Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds (see page 678 of (

8730: 8697: 8660: 8611: 8390: 8143: 8029: 7952: 7923: 7248: 7592:

This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family.

3060:). At the same time, they provide, also, an effective construction of the rational Pontrjagin classes on topological manifolds. The paper ( 5314: 3690: 124: 71: 7630:. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information. 3354:. The index of the Dirac operator is a topological invariant which measures the winding of the Higgs field on a sphere at infinity. If 5484:{\displaystyle 0\rightarrow V\rightarrow V\otimes \Lambda ^{0,1}T^{*}(X)\rightarrow V\otimes \Lambda ^{0,2}T^{*}(X)\rightarrow \dotsm } 8981: 8591: 8439: 8307: 7977: 7349: 7279: 2597:

The index of an elliptic differential operator obviously vanishes if the operator is self adjoint. It also vanishes if the manifold

909:

on each cotangent space. (In general, differential operators transform in a rather complicated way under coordinate transforms (see

472: 4138: 699:

because we keep only the highest order terms and differential operators commute "up to lower-order terms". The operator is called

299:

Nicolae Teleman proves that the analytical indices of signature operators with values in vector bundles are topological invariants.

9021: 6502:

operator, but is a pseudodifferential operator. Also, there is a direct correspondence between data representing elements of K(B(

1365: 1349:{\displaystyle (-1)^{n}\operatorname {ch} (D)\operatorname {Td} (X)=(-1)^{n}\int _{X}\operatorname {ch} (D)\operatorname {Td} (X)} 5534: 9128: 6152: 4085:{\displaystyle x_{r+i}(E\otimes \mathbb {C} )=c_{1}{\mathord {\left({\overline {l_{i}}}\right)}}=-x_{i}(E\otimes \mathbb {C} )} 3897: 8570:, Annals of Mathematics Studies in Mathematics, vol. 88, Princeton: Princeton University Press and Tokio University Press 8621: 7973: 3320: 3282: 206: 9115:- A partial transcript of informal post–dinner conversation during a symposium held in Roskilde, Denmark, in September 1998. 2946:

of a smooth variety, and the Grothendieck group on the left is given by the Grothendieck group of algebraic vector bundles.

8207:; Teleman, N. (1994), "Quasiconformal mappings, operators on Hilbert space and local formulae for characteristic classes", 482: 9066: 8310:. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data. 3286: 3004:

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

3124:

Instead of working with an elliptic operator between two vector bundles, it is sometimes more convenient to work with an

6535: 6259: 3244: 327:

and Sullivan study Yang–Mills theory on quasiconformal manifolds of dimension 4. They introduce the signature operator

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which is the "topological" version of the Chern-Gauss-Bonnet theorem (the geometric one being obtained by applying the

2571:

data derived from the manifold and the vector bundle. The Atiyah–Singer index theorem solves this problem, and states:

9061: 8400:

Hamilton, M. J. D. (2020). "The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking".

8035:

This gives an elementary proof of the index theorem for the Dirac operator, using the heat equation and supersymmetry.

6489: 3117: 994: 3206:{\displaystyle 0\rightarrow E_{0}\rightarrow E_{1}\rightarrow E_{2}\rightarrow \dotsm \rightarrow E_{m}\rightarrow 0} 2405:{\displaystyle (-1)^{m}\int _{X}{\frac {\operatorname {ch} (E)-\operatorname {ch} (F)}{e(TX)}}\operatorname {Td} (X)} 6627:

of some elliptic operator on the point. This reduces the index theorem to the case of a point, where it is trivial.

3749:, in order to prove assertions involving characteristic classes, we may suppose that there are complex line bundles 8633: 3270: 3250:

Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space

1410: 1099:

and is 0 otherwise, and the kernel of the adjoint is a similar space with λ replaced by its complex conjugate. So

7294: 3887:{\displaystyle E\otimes \mathbb {C} =l_{1}\oplus {\overline {l_{1}}}\oplus \dotsm l_{r}\oplus {\overline {l_{r}}}} 800: 9112: 7969: 3294: 1885: 717: 8061:

Bismut proves the theorem for elliptic complexes using probabilistic methods, rather than heat equation methods.

3025:

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

152: 128: 9077: 5666: 3258:, rather than an integer. If the operators in the family are real, then the index lies in the real K-theory of 2210:

In some situations, it is possible to simplify the above formula for computational purposes. In particular, if

6518:

index theorem can be extended from elliptic differential operators to elliptic pseudodifferential operators.

637: 564: 175:

of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the

5497: 3752: 8579: 7698:

Atiyah, M. F.; Bott, R. (1968), "A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications",

7267: 5257: 3558: 3465: 1497: 1439: 315: 191:. The last talk in Paris was by Atiyah on manifolds with boundary. Their first published proof replaced the 3585: 3492: 3331: 8675: 7226: 3358:

is the unit matrix in the direction of the Higgs field, then the index is proportional to the integral of

343:

Connes, Sullivan, and Teleman prove the index theorem for signature operators on quasiconformal manifolds.

273: 8318: 3335: 8563: 8534: 7700: 7605: 7567: 7529: 428: 220: 148: 92: 39: 4095: 245:

Gennadi G. Kasparov publishes his work on the realization of K-homology by abstract elliptic operators.

1748: 8996: 8799: 8511: 8330: 8286: 8065: 7880: 7837: 7792: 7749: 6472: 3630: 3394: 3378: 3324: 3034:, defined on middle degree differential forms on even-dimensional quasiconformal manifolds (compare ( 188: 164: 83: 9056: 8680: 7450:

This reformulates the result as a sort of Lefschetz fixed-point theorem, using equivariant K-theory.

7231: 6862:

tends to 0, giving a proof of the Atiyah–Singer index theorem. The asymptotic expansions for small

337:

Connes and Henri Moscovici prove the local index formula in the context of non-commutative geometry.

7660:. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex. 6433: 6242:

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

3697: 3382: 3305: 132: 1955: 8913: 8847: 8815: 8771: 8491: 8483: 8465: 8416: 8401: 8346: 8108: 8039: 8001: 7861: 7773: 7737: 7717: 7688: 7622: 7584: 7546: 7508: 7442: 7328: 7317: 6640: 6275: 3231: 3223: 2976:) on a closed, oriented, topological manifold, the analytical index equals the topological index. 2939: 286: 6300: 2418: 2159: 2099: 144: 3519: 9073: 9040: 8977: 8726: 8693: 8656: 8607: 8587: 8435: 8386: 8303: 8139: 8025: 7948: 7919: 7345: 7275: 7244: 6495: 6289:

are given by the +1 and −1 eigenspaces of the operator on the bundle of differential forms of

3635: 3386: 2620: 2458: 1010: 199:, and they used this to give proofs of various generalizations in another sequence of papers. 147:. He noticed the homotopy invariance of the index, and asked for a formula for it by means of 108: 859: 8948: 8905: 8885: 8875: 8855: 8839: 8807: 8779: 8763: 8736: 8685: 8575: 8546: 8519: 8475: 8366: 8338: 8294: 8263: 8253: 8226: 8216: 8189: 8179: 8149: 8116: 8100: 8073: 8051: 8009: 7993: 7888: 7845: 7800: 7757: 7709: 7680: 7650: 7614: 7576: 7538: 7500: 7470: 7434: 7388: 7263: 7236: 6531: 6274:

of the manifold. This follows from the Atiyah–Singer index theorem applied to the following

5273: 3662: 3347: 3126: 2289: 2040: 2010: 1742: 1360: 1018: 898: 210: 112: 8962: 7962: 7933: 7902: 7857: 7814: 7769: 7725:

These give the proofs and some applications of the results announced in the previous paper.

7411: 7366:

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

6414:{\displaystyle D\equiv \Delta \mathrel {:=} \left(\mathbf {d} +\mathbf {d^{*}} \right)^{2}} 3064:) provides a link between Thom's original construction of the rational Pontrjagin classes ( 2525:

in Euclidean space. Now a differential operator as above naturally defines an element of K(

2259: 773: 675: 602: 123:(defined in terms of some topological data). It includes many other theorems, such as the 8958: 8889: 8859: 8783: 8747: 8740: 8718: 8267: 8241: 8237: 8230: 8204: 8193: 8153: 8120: 8077: 8013: 7958: 7929: 7898: 7853: 7825: 7810: 7765: 7407: 7392: 6541:

The idea of this first proof is roughly as follows. Consider the ring generated by pairs (

6348: 3723: 3436: 2912: 2128: 2070: 1878: 324: 269: 2886: 2660: 2463: 9097: 8803: 8515: 8396:

Free online textbook that proves the Atiyah–Singer theorem with a heat equation approach

8334: 8290: 7884: 7841: 7796: 7753: 7554:

This paper shows how to convert from the K-theory version to a version using cohomology.

4135:

Using Chern roots as above and the standard properties of the Euler class, we have that

2233: 897:

is defined in much the same way using local coordinate charts, and is a function on the

396:. So in local coordinates it acts as a differential operator, taking smooth sections of 9000: 8988: 8969: 8723:

J.C. Candrell, "Geometric Topology", Proc. Georgia Topology Conf. Athens, Georgia, 1977

8706: 8646: 8274: 8132: 7947:, Oxford Science Publications, New York: The Clarendon Press, Oxford University Press, 7940: 7918:, Oxford Science Publications, New York: The Clarendon Press, Oxford University Press, 7909: 7821: 7729: 7664: 7634: 7600: 7596: 7562: 7558: 7524: 7520: 7488: 7484: 7458: 7454: 7418: 7399: 7380: 3703: 3668: 3612: 3416: 3343: 3308:; this index is in general real rather than integer valued. This version is called the 3235: 3227: 2213: 2188: 1475: 1416: 1199: 1179: 1159: 1139: 1119: 290: 184: 176: 104: 100: 53: 49: 8502:

Kasparov, G.G. (1972), "Topological invariance of elliptic operators, I: K-homology",

8042:(1984), "The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem", 239:

Isadore Singer proposes a comprehensive program for future extensions of index theory.

9122: 8936: 8917: 8851: 8819: 8495: 8371: 8221: 8184: 8112: 8056: 7981: 7865: 7828:(1977), "A geometric construction of the discrete series for semisimple Lie groups", 7777: 6648: 5653:{\displaystyle \operatorname {index} (D)=\sum _{p}(-1)^{p}\dim H^{p}(X,V)=\chi (X,V)} 1870:{\displaystyle \operatorname {ch} :K(X)\otimes \mathbb {Q} \to H^{*}(X;\mathbb {Q} )} 375: 357: 260: 17: 8827: 8751: 8650: 8551: 8523: 8350: 8319:"Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem" 8298: 8088: 8005: 7655: 7475: 2521:(as a consequence of Bott-periodicity). This map is independent of the embedding of 9113:

Recollections from the early days of index theory and pseudo-differential operators

8775: 8559: 8530: 8314: 8200: 8160: 8127: 8084: 7422: 6885: 3339: 311: 282: 256: 216: 168: 8924: 3052:

These results constitute significant advances along the lines of Singer's program

224: 8415:

Kayani, U. (2020). "Dynamical supersymmetry enhancement of black hole horizons".

7913: 6232:{\displaystyle \chi (X,V)=\int _{X}\operatorname {ch} (V)\operatorname {Td} (TX)} 3281:, commuting with the elliptic operator, then one replaces ordinary K-theory with 1734:{\displaystyle \varphi :H^{k}(X;\mathbb {Q} )\to H^{n+k}(B(X)/S(X);\mathbb {Q} )} 289:, gave a short proof of the local index theorem for operators that are locally 3686: 3351: 2649:

of compact stably almost complex manifolds, then there is a commutative diagram

2254: 7641:(1966), "A Lefschetz Fixed Point Formula for Elliptic Differential Operators", 6514:)) (clutching functions) and symbols of elliptic pseudodifferential operators. 3103:), are the weakest analytical structures on topological manifolds of dimension 2989:), the extension of Atiyah–Singer's signature operator to Lipschitz manifolds ( 2769: 2759: 1620:{\displaystyle \varphi ^{-1}(\operatorname {ch} (d(p^{*}E,p^{*}F,\sigma (D))))} 8992: 8927:(1956), "Les classes caractéristiques de Pontrjagin de variétés triangulées", 8713:, Annals of Mathematics Studies in Mathematics, vol. 70, pp. 171–185 8357:

Getzler, E. (1988), "A short proof of the local Atiyah–Singer index theorem",

8164: 7733: 7668: 7638: 6636: 6464: 6340: 3390: 3262:. This gives a little extra information, as the map from the real K-theory of 2594:

integrality properties, as it implies that the topological index is integral.

1469: 968: 910: 252: 172: 8953: 988:

are both compact operators. An important consequence is that the kernel of

3696:

The concrete computation goes as follows: according to one variation of the

2943: 2292:

and dividing by the Euler class, the topological index may be expressed as

192: 8655:, Annals of Mathematics Studies, vol. 57, S.l.: Princeton Univ Press, 8568:

Foundational Essays on Topological Manifolds, Smoothings and Triangulations

7404:

Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974)

1037:(the constraints on the right-hand-side of an inhomogeneous equation like 8790:

Teleman, N. (1980), "Combinatorial Hodge theory and signature operator",

8625: 8601: 7406:, Asterisque, vol. 32–33, Soc. Math. France, Paris, pp. 43–72, 6566: 3377:

The topological interpretation of this invariant and its relation to the

3218: 1030: 361: 272:

establishes his theorem on the existence and uniqueness of Lipschitz and

196: 8832:

Publications Mathématiques de l'Institut des Hautes Études Scientifiques

8756:

Publications Mathématiques de l'Institut des Hautes Études Scientifiques

8380: 8093:

Publications Mathématiques de l'Institut des Hautes Études Scientifiques

7402:(1976), "Elliptic operators, discrete groups and von Neumann algebras", 7385:

Proc. Int. Conf. on Functional Analysis and Related Topics (Tokyo, 1969)

8909: 8880: 8843: 8811: 8767: 8752:"An analytic proof of Novikov's theorem on rational Pontrjagin classes" 8689: 8342: 8258: 8104: 7997: 7893: 7849: 7805: 7761: 7721: 7692: 7626: 7588: 7550: 7512: 7446: 7240: 6271: 3489:

to be the sum of the even exterior powers of the cotangent bundle, and

1359:

in other words the value of the top dimensional component of the mixed

1045:, or equivalently the kernel of the adjoint operator). In other words, 8487: 5292:

be the sums of the bundles of differential forms with coefficients in

2834: 9022:"The Atiyah–Singer Index Theorem: What it is and why you should care" 8470: 7671:(1967), "A Lefschetz Fixed Point Formula for Elliptic Complexes: I", 6604:

embeds in, this reduces the index theorem to the case of spheres. If

3226:) do not admit local boundary conditions. To handle these operators, 2808: 2689: 119:(related to the dimension of the space of solutions) is equal to the 9081: 8709:(1971), "Future extensions of index theory and elliptic operators", 8537:(1969), "On the triangulation of manifolds and the Hauptvermutung", 7713: 7684: 7618: 7580: 7542: 7504: 7438: 2693: 209:

published his results on the topological invariance of the rational

143:

The index problem for elliptic differential operators was posed by

8479: 8421: 8406: 8382:

Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem

5351:{\displaystyle {\overline {\partial }}+{\overline {\partial }}^{*}} 1017:, defined as the difference between the (finite) dimension of the 8976:(2nd ed.), Somerville, Mass.: International Press of Boston, 8866:

Teleman, N. (1984), "The index theorem on topological manifolds",

6876: 6553:

is a smooth vector bundle on the compact smooth oriented manifold

3689:

over the manifold. The index formula for this operator yields the

2605:

elliptic operators whose index does not vanish in odd dimensions.

276:

structures on topological manifolds of dimension different from 4.

8165:"Cyclic cohomology, the Novikov conjecture and hyperbolic groups" 7461:(1963), "The Index of Elliptic Operators on Compact Manifolds", 951:) is invertible for all non-zero cotangent vectors at any point 3289:, with terms coming from fixed-point submanifolds of the group 305:

Teleman establishes the index theorem on topological manifolds.

4208:{\textstyle e(TM)=\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )} 3072:

Sullivan's result on Lipschitz and quasiconformal structures (

8302:

reprinted in volume 1 of his collected works, p. 65–75,

8072:, École Normale Supérieure, Secrétariat mathématique, Paris, 8896:

Teleman, N. (1985), "Transversality and the index theorem",

5525:

th cohomology group is just the coherent cohomology group H(

2540:

is an elliptic differential operator between vector bundles

1402:{\displaystyle \operatorname {ch} (D)\operatorname {Td} (X)} 770:, and so is elliptic as this is nonzero whenever any of the 8626:"Topological invariance of the rational Pontrjagin classes" 6904: 6902: 3254:. In this case the index is an element of the K-theory of 8828:"The index of signature operators on Lipschitz manifolds" 8674:, Lecture Notes in Mathematics, vol. 638, Springer, 7225:, Lecture Notes in Mathematics, vol. 638, Springer, 6262:

states that the signature of a compact oriented manifold

3960:{\displaystyle x_{i}(E\otimes \mathbb {C} )=c_{1}(l_{i})} 2253:-dimensional orientable (compact) manifold with non-zero 882:, as the symbol vanishes for some non-zero values of the 417:

is a differential operator on a Euclidean space of order

8020:

Berline, Nicole; Getzler, Ezra; Vergne, Michèle (1992),

7516:

This gives a proof using K-theory instead of cohomology.

7295:"algebraic topology - How to understand the Todd class?" 6881:

Pages displaying short descriptions of redirect targets

3346:

to derive this index theorem on spaces equipped with a

1079:

Suppose that the manifold is the circle (thought of as

179:(which was rediscovered by Atiyah and Singer in 1961). 7783:

Atiyah, M.; Bott, R.; Patodi, V. K. (1975), "Errata",

7740:(1973), "On the heat equation and the index theorem", 5669: 4141: 2909:

is a point, then we recover the statement above. Here

7871:

Atiyah, Michael; Schmid, Wilfried (1979), "Erratum",

7208: 6704: 6359: 6303: 6164: 5744: 5550: 5500: 5377: 5317: 4913: 4224: 4098: 3973: 3900: 3803: 3755: 3726: 3706: 3671: 3638: 3615: 3588: 3561: 3522: 3495: 3468: 3439: 3419: 3135: 3080: 3046: 3042: 3017: 2915: 2889: 2658: 2623: 2497:

then there is a pushforward (or "shriek") map from K(

2466: 2421: 2301: 2262: 2236: 2216: 2191: 2162: 2131: 2102: 2073: 2043: 2013: 1958: 1888: 1810: 1751: 1635: 1532: 1500: 1478: 1442: 1419: 1368: 1225: 1202: 1182: 1162: 1142: 1122: 862: 803: 776: 720: 678: 640: 605: 567: 550:{\displaystyle x_{1},\dots ,x_{k},y_{1},\dots ,y_{k}} 485: 431: 6652: 2559:

is the following: compute the (analytical) index of

2509:) is defined to be the image of this operation with 3685:, and the topological index is the integral of the 67: 59: 45: 35: 9098:"Interview with Michael Atiyah and Isadore Singer" 9078:"Lecture notes on the Atiyah–Singer Index Theorem" 8721:(1979), "Hyperbolic geometry and homeomorphisms", 8131: 6839: 6647:) gave a new proof of the index theorem using the 6413: 6331: 6231: 6140: 5727: 5652: 5533:), so the analytical index of this complex is the 5513: 5483: 5350: 5304:even or odd, and we let the differential operator 5245: 4893: 4207: 4124: 4084: 3959: 3886: 3789: 3741: 3712: 3677: 3653: 3621: 3601: 3574: 3547: 3508: 3481: 3454: 3425: 3205: 2930: 2901: 2872: 2641: 2478: 2449: 2404: 2280: 2245: 2222: 2197: 2177: 2146: 2117: 2088: 2059: 2029: 1999: 1944: 1869: 1795: 1733: 1619: 1518: 1484: 1460: 1425: 1401: 1348: 1208: 1188: 1168: 1148: 1128: 874: 848: 789: 762: 691: 664: 618: 591: 549: 463: 7184: 4215:. As for the Chern character and the Todd class, 3035: 3013: 557:, given by dropping all terms of order less than 9045:: CS1 maint: bot: original URL status unknown ( 7527:(1968b), "The Index of Elliptic Operators III", 7088: 6644: 3266:to the complex K-theory is not always injective. 2517:) can be naturally identified with the integers 889:The symbol of a differential operator of order 626:. So the symbol is homogeneous in the variables 259:gave a new proof of the index theorem using the 7565:(1971a), "The Index of Elliptic Operators IV", 7425:(1968), "The Index of Elliptic Operators: II", 7383:(1970), "Global Theory of Elliptic Operators", 7196: 7028: 6968: 6565:Atiyah and Singer's first published proof used 3008:Connes–Donaldson–Sullivan–Teleman index theorem 2998: 921:is a section of the pullback of the bundle Hom( 703:if the symbol is nonzero whenever at least one 151:. Some of the motivating examples included the 8725:, New York: Academic Press, pp. 543–595, 7603:(1971b), "The Index of Elliptic Operators V", 7491:(1968a), "The Index of Elliptic Operators I", 6451:, and its topological index is the L genus of 3116:The Atiyah–Singer theorem applies to elliptic 3107:for which the index theorem is known to hold. 3030:This theory is based on a signature operator 9027:. Archived from the original on June 24, 2006 7004: 7000: 6996: 6992: 6980: 6858:, which can be used to evaluate the limit as 3894:. Therefore, we can consider the Chern roots 3285:. Moreover, one gets generalizations of the 2950:Extensions of the Atiyah–Singer index theorem 2505:). The topological index of an element of K( 797:'s are nonzero. The wave operator has symbol 8: 9055:Voitsekhovskii, M.I.; Shubin, M.A. (2001) , 8448:Hilsum, M. (1999), "Structures riemaniennes 6944: 6908: 3433:is a compact oriented manifold of dimension 3316: 2586:that we can usually at least evaluate their 849:{\displaystyle -y_{1}^{2}+\cdots +y_{k}^{2}} 331:defined on differential forms of degree two. 131:, as special cases, and has applications to 30: 27:Mathematical result in differential geometry 6573:is any inclusion of compact manifolds from 6526:The initial proof was based on that of the 2533:under this map "is" the topological index. 1945:{\displaystyle d(p^{*}E,p^{*}F,\sigma (D))} 763:{\displaystyle y_{1}^{2}+\cdots +y_{k}^{2}} 8939:(1982), "Supersymmetry and Morse theory", 8652:Seminar on the Atiyah–Singer Index Theorem 6151:Applying the index theorem, we obtain the 5728:{\textstyle e(TX)=\prod _{i}^{n}x_{i}(TX)} 388:is an elliptic differential operator from 29: 9096:Raussen, Martin; Skau, Christian (2005), 8952: 8879: 8679: 8550: 8469: 8420: 8405: 8370: 8257: 8220: 8183: 8055: 7892: 7804: 7654: 7474: 7230: 6825: 6811: 6779: 6768: 6742: 6703: 6581:, they defined a 'pushforward' operation 6405: 6393: 6388: 6380: 6369: 6358: 6308: 6302: 6190: 6163: 6111: 6106: 6088: 6078: 6072: 6067: 6052: 6044: 6027: 6021: 6015: 6010: 5980: 5942: 5941: 5894: 5889: 5868: 5863: 5817: 5810: 5804: 5788: 5769: 5764: 5745: 5743: 5707: 5697: 5692: 5668: 5608: 5592: 5573: 5549: 5501: 5499: 5460: 5444: 5416: 5400: 5376: 5342: 5332: 5318: 5316: 5222: 5208: 5207: 5189: 5179: 5174: 5164: 5145: 5135: 5112: 5111: 5088: 5080: 5063: 5057: 5051: 5046: 5033: 5023: 5018: 4999: 4991: 4970: 4965: 4958: 4952: 4942: 4912: 4880: 4879: 4856: 4848: 4831: 4825: 4819: 4814: 4796: 4795: 4766: 4765: 4740: 4732: 4711: 4706: 4685: 4684: 4664: 4656: 4641: 4633: 4623: 4594: 4593: 4573: 4565: 4553: 4545: 4529: 4515: 4514: 4494: 4486: 4476: 4471: 4440: 4439: 4427: 4417: 4407: 4406: 4394: 4361: 4360: 4348: 4338: 4328: 4327: 4311: 4310: 4298: 4259: 4246: 4236: 4235: 4225: 4223: 4198: 4197: 4179: 4169: 4164: 4140: 4118: 4111: 4097: 4075: 4074: 4059: 4032: 4026: 4021: 4020: 4014: 4000: 3999: 3978: 3972: 3948: 3935: 3921: 3920: 3905: 3899: 3873: 3867: 3858: 3837: 3831: 3822: 3811: 3810: 3802: 3781: 3776: 3769: 3760: 3754: 3725: 3705: 3670: 3637: 3614: 3593: 3587: 3566: 3560: 3539: 3521: 3500: 3494: 3473: 3467: 3438: 3418: 3338:, at the suggestion of his Ph.D. advisor 3191: 3172: 3159: 3146: 3134: 2914: 2888: 2852: 2851: 2827: 2810: 2809: 2803: 2778: 2768: 2767: 2758: 2757: 2751: 2734: 2733: 2711: 2694: 2690: 2684: 2659: 2657: 2622: 2465: 2438: 2420: 2331: 2325: 2315: 2300: 2261: 2235: 2215: 2190: 2161: 2130: 2101: 2072: 2048: 2042: 2018: 2012: 1977: 1957: 1915: 1899: 1887: 1860: 1859: 1844: 1833: 1832: 1809: 1770: 1750: 1724: 1723: 1703: 1676: 1662: 1661: 1646: 1634: 1584: 1568: 1537: 1531: 1499: 1477: 1441: 1418: 1367: 1310: 1300: 1239: 1224: 1201: 1181: 1161: 1141: 1121: 966:on a compact manifold has a (non-unique) 861: 840: 835: 816: 811: 802: 781: 775: 754: 749: 730: 725: 719: 683: 677: 656: 644: 639: 634:. The symbol is well defined even though 610: 604: 583: 571: 566: 541: 522: 509: 490: 484: 455: 436: 430: 7945:Collected works. Vol. 4. Index theory: 2 7915:Collected works. Vol. 3. Index theory: 1 7112: 7076: 6932: 6670:is a differential operator with adjoint 3516:to be the sum of the odd powers, define 3073: 2994: 1029:= 0), and the (finite) dimension of the 665:{\displaystyle \partial /\partial x_{i}} 592:{\displaystyle \partial /\partial x_{i}} 285:motivated by ideas of Edward Witten and 9134:Elliptic partial differential equations 8089:"Non-commutative differential geometry" 7371: 7367: 7160: 7148: 7124: 7100: 7016: 6898: 6656: 5514:{\displaystyle {\overline {\partial }}} 3790:{\displaystyle l_{1},\,\ldots ,\,l_{r}} 3240: 3061: 2990: 2986: 2982: 2964: 2960: 2493:is a compact submanifold of a manifold 293:; this covers many of the useful cases. 9038: 8898:Integral Equations and Operator Theory 8603:The Atiyah–Patodi–Singer Index Theorem 8244:(1989), "Quasiconformal 4-manifolds", 7172: 7136: 7064: 7052: 6956: 6920: 6660: 3575:{\displaystyle \Lambda ^{\text{even}}} 3482:{\displaystyle \Lambda ^{\text{even}}} 3300: 3100: 3057: 2973: 2415:where division makes sense by pulling 1519:{\displaystyle \operatorname {ch} (D)} 1472:of the complexified tangent bundle of 1461:{\displaystyle \operatorname {Td} (X)} 1005:As the elliptic differential operator 933:. The differential operator is called 7982:"Riemann-Roch for singular varieties" 7480:An announcement of the index theorem. 3720:is a real vector bundle of dimension 3602:{\displaystyle \Lambda ^{\text{odd}}} 3509:{\displaystyle \Lambda ^{\text{odd}}} 2609:Relation to Grothendieck–Riemann–Roch 2457:back from the cohomology ring of the 1116:of an elliptic differential operator 7: 8504:Math. USSR Izvestija (Engl. Transl.) 7329:Some Remarks on the Paper of Callias 7040: 6879: – Term in quantum field theory 6653:Berline, Getzler & Vergne (1992) 6596:that preserves the index. By taking 3065: 2972:For any abstract elliptic operator ( 2601:has odd dimension, though there are 1095:) if λ is an integral multiple of 2π 7344:, Institute of Physics Publishing, 7209:Connes, Sullivan & Teleman 1994 6888: – Modified partition function 6655:. The proof is also published in ( 6471:In dimension 4 this result implies 3081:Connes, Sullivan & Teleman 1994 3047:Connes, Sullivan & Teleman 1994 3043:Connes, Sullivan & Teleman 1994 3018:Connes, Sullivan & Teleman 1994 314:publishes his fundamental paper on 6366: 5801: 5503: 5441: 5397: 5334: 5320: 4414: 4335: 4256: 4243: 3590: 3563: 3497: 3470: 3245:Atiyah–Patodi–Singer index theorem 2580:is equal to its topological index. 2513:some Euclidean space, for which K( 1433:up to a difference of sign. Here, 649: 641: 576: 568: 464:{\displaystyle x_{1},\dots ,x_{k}} 263:, described in a paper by Melrose. 171:had proved the integrality of the 25: 9139:Theorems in differential geometry 8430:Higson, Nigel; Roe, John (2000), 8277:(1960), "On elliptic equations", 6447:is the signature of the manifold 5280:with a holomorphic vector bundle 4125:{\displaystyle i=1,\,\ldots ,\,r} 2007:associated to two vector bundles 710:Example: The Laplace operator in 409:Symbol of a differential operator 76:Grothendieck–Riemann–Roch theorem 8022:Heat Kernels and Dirac Operators 6616:, then any elliptic operator on 6394: 6390: 6381: 5535:holomorphic Euler characteristic 3609:. Then the analytical index of 3099:+1)/2, introduced by M. Hilsum ( 1796:{\displaystyle p:B(X)/S(X)\to X} 8672:The Atiyah-Singer Index Theorem 8552:10.1090/S0002-9904-1969-12271-8 8524:10.1070/IM1975v009n04ABEH001497 8299:10.1070/rm1960v015n03ABEH004094 7656:10.1090/S0002-9904-1966-11483-0 7476:10.1090/S0002-9904-1963-10957-X 7223:The Atiyah-Singer Index Theorem 6528:Hirzebruch–Riemann–Roch theorem 6153:Hirzebruch-Riemann-Roch theorem 5494:with the differential given by 5264:Hirzebruch–Riemann–Roch theorem 3321:discrete series representations 3079:The quasiconformal structures ( 1952:is the "difference element" in 1013:. Any Fredholm operator has an 195:theory of the first proof with 157:Hirzebruch–Riemann–Roch theorem 9111:R. R. Seeley and other (1999) 8586:, Princeton University Press, 7342:Geometry, topology and physics 7274:, Princeton University Press, 7089:Atiyah, Bott & Patodi 1973 6748: 6735: 6717: 6711: 6324: 6312: 6226: 6217: 6208: 6202: 6180: 6168: 6131: 6122: 5967: 5958: 5946: 5929: 5916: 5907: 5856: 5850: 5785: 5775: 5722: 5713: 5682: 5673: 5647: 5635: 5626: 5614: 5589: 5579: 5563: 5557: 5475: 5472: 5466: 5431: 5428: 5422: 5387: 5381: 5240: 5231: 5212: 5195: 5161: 5151: 5132: 5122: 5116: 5099: 4939: 4929: 4923: 4917: 4884: 4867: 4800: 4783: 4770: 4753: 4689: 4672: 4620: 4610: 4598: 4581: 4519: 4502: 4391: 4381: 4315: 4291: 4202: 4185: 4154: 4145: 4079: 4065: 4004: 3990: 3954: 3941: 3925: 3911: 3648: 3642: 3319:to rederive properties of the 3197: 3184: 3178: 3165: 3152: 3139: 2925: 2919: 2856: 2842: 2821: 2815: 2798: 2792: 2738: 2724: 2705: 2699: 2679: 2673: 2633: 2435: 2425: 2399: 2393: 2381: 2372: 2364: 2358: 2346: 2340: 2312: 2302: 2275: 2266: 2172: 2166: 2141: 2135: 2112: 2106: 2083: 2077: 1994: 1991: 1985: 1974: 1968: 1962: 1939: 1936: 1930: 1892: 1864: 1850: 1837: 1826: 1820: 1787: 1784: 1778: 1767: 1761: 1728: 1717: 1711: 1700: 1694: 1688: 1669: 1666: 1652: 1614: 1611: 1608: 1605: 1599: 1561: 1555: 1546: 1513: 1507: 1455: 1449: 1396: 1390: 1381: 1375: 1343: 1337: 1328: 1322: 1297: 1287: 1281: 1275: 1272: 1266: 1257: 1251: 1236: 1226: 1196:-dimensional compact manifold 1136:between smooth vector bundles 109:elliptic differential operator 1: 8138:, San Diego: Academic Press, 7318:Index Theorems on Open Spaces 7185:Donaldson & Sullivan 1989 6459:Â genus and Rochlin's theorem 4904:Applying the index theorem, 3287:Lefschetz fixed-point theorem 3036:Donaldson & Sullivan 1989 3014:Donaldson & Sullivan 1989 2997:) and topological cobordism ( 2125:between them on the subspace 1065:This is sometimes called the 1009:has a pseudoinverse, it is a 8974:The Founders of Index Theory 8606:, Wellesley, Mass.: Peters, 8600:Melrose, Richard B. (1993), 8372:10.1016/0040-9383(86)90008-X 8222:10.1016/0040-9383(94)90003-5 8185:10.1016/0040-9383(90)90003-3 8057:10.1016/0022-1236(84)90101-0 6536:pseudodifferential operators 6484:Pseudodifferential operators 6260:Hirzebruch signature theorem 6254:Hirzebruch signature theorem 5990: 5827: 5506: 5337: 5323: 5284:. We let the vector bundles 4038: 3879: 3843: 3374:is even, it is always zero. 3370:−1)-sphere at infinity. If 3118:pseudodifferential operators 2000:{\displaystyle K(B(X)/S(X))} 929:) to the cotangent space of 161:Hirzebruch signature theorem 80:Hirzebruch signature theorem 9062:Encyclopedia of Mathematics 8929:Symp. Int. Top. Alg. Mexico 8434:, Oxford University Press, 7197:Connes & Moscovici 1990 7029:Kirby & Siebenmann 1969 7005:Atiyah & Singer (1971b) 7001:Atiyah & Singer (1971a) 6997:Atiyah & Singer (1968b) 6993:Atiyah & Singer (1968a) 6490:pseudodifferential operator 3555:, considered as a map from 2999:Kirby & Siebenmann 1977 1053:) = dim Ker(D) − dim Coker( 995:pseudodifferential operator 856:, which is not elliptic if 155:and its generalization the 107:(1963), states that for an 97:Atiyah–Singer index theorem 31:Atiyah–Singer index theorem 9155: 8634:Doklady Akademii Nauk SSSR 7299:Mathematics Stack Exchange 6691:. Therefore, the index of 6612:is some point embedded in 6569:rather than cobordism. If 6487: 6332:{\displaystyle i^{k(k-1)}} 3691:Chern–Gauss–Bonnet theorem 3409:Chern-Gauss-Bonnet theorem 3317:Atiyah & Schmid (1977) 2993:), Kasparov's K-homology ( 2450:{\displaystyle e(TX)^{-1}} 2178:{\displaystyle \sigma (D)} 2118:{\displaystyle \sigma (D)} 1411:fundamental homology class 223:'s results, combined with 125:Chern–Gauss–Bonnet theorem 72:Chern–Gauss–Bonnet theorem 8379:Gilkey, Peter B. (1994), 6981:Atiyah & Singer 1968a 6592:to elliptic operators of 6588:on elliptic operators of 3548:{\displaystyle D=d+d^{*}} 3295:equivariant index theorem 3083:) and more generally the 2615:Grothendieck–Riemann–Roch 1057:) = dim Ker(D) − dim Ker( 8792:Inventiones Mathematicae 8711:Prospects in Mathematics 8580:Michelsohn, Marie-Louise 8163:; Moscovici, H. (1990), 7340:Nakahara, Mikio (2003), 7268:Michelsohn, Marie-Louise 6909:Atiyah & Singer 1963 3654:{\displaystyle \chi (M)} 3277:on the compact manifold 3054:Prospects in Mathematics 2642:{\displaystyle f:X\to Y} 2576:The analytical index of 2548:over a compact manifold 905:, homogeneous of degree 8987:- Personal accounts on 8134:Noncommutative Geometry 7387:, University of Tokio, 6600:to be some sphere that 6458: 5276:of (complex) dimension 5258:Chern-Weil homomorphism 875:{\displaystyle k\geq 2} 316:noncommutative geometry 9129:Differential operators 8954:10.4310/jdg/1214437492 8750:; Teleman, N. (1983), 8539:Bull. Amer. Math. Soc. 7463:Bull. Amer. Math. Soc. 6841: 6455:, so these are equal. 6443:The analytic index of 6415: 6333: 6233: 6142: 6077: 6020: 5873: 5774: 5729: 5702: 5654: 5515: 5485: 5352: 5247: 5184: 5056: 5028: 4975: 4895: 4824: 4716: 4481: 4209: 4174: 4126: 4086: 3961: 3888: 3791: 3743: 3714: 3679: 3655: 3623: 3603: 3576: 3549: 3510: 3483: 3456: 3427: 3207: 2932: 2903: 2874: 2835: 2717: 2643: 2563:using only the symbol 2480: 2451: 2406: 2282: 2247: 2224: 2199: 2179: 2148: 2119: 2090: 2061: 2060:{\displaystyle p^{*}F} 2031: 2030:{\displaystyle p^{*}E} 2001: 1946: 1871: 1797: 1745:for the sphere bundle 1735: 1621: 1520: 1486: 1462: 1427: 1403: 1350: 1210: 1190: 1170: 1150: 1130: 937:if the element of Hom( 876: 850: 791: 764: 693: 672:does not commute with 666: 620: 593: 551: 465: 400:to smooth sections of 149:topological invariants 8670:Shanahan, P. (1978), 8617:Free online textbook. 8458:Annals of Mathematics 7701:Annals of Mathematics 7673:Annals of Mathematics 7606:Annals of Mathematics 7568:Annals of Mathematics 7530:Annals of Mathematics 7493:Annals of Mathematics 7427:Annals of Mathematics 7221:Shanahan, P. (1978), 6842: 6530:(1954), and involved 6416: 6334: 6234: 6143: 6063: 6006: 5859: 5760: 5730: 5688: 5655: 5516: 5486: 5353: 5248: 5170: 5042: 5014: 4961: 4896: 4810: 4702: 4467: 4210: 4160: 4127: 4087: 3962: 3889: 3792: 3744: 3715: 3680: 3656: 3624: 3604: 3577: 3550: 3511: 3484: 3457: 3428: 3332:Callias index theorem 3325:semisimple Lie groups 3208: 2955:Teleman index theorem 2933: 2904: 2875: 2804: 2685: 2644: 2481: 2452: 2407: 2283: 2281:{\displaystyle e(TX)} 2248: 2225: 2200: 2180: 2149: 2120: 2091: 2062: 2032: 2002: 1947: 1872: 1798: 1736: 1622: 1521: 1487: 1463: 1428: 1404: 1351: 1211: 1191: 1171: 1151: 1131: 893:on a smooth manifold 877: 851: 792: 790:{\displaystyle y_{i}} 765: 714:variables has symbol 694: 692:{\displaystyle x_{i}} 667: 621: 619:{\displaystyle y_{i}} 594: 552: 466: 221:Laurent C. Siebenmann 93:differential geometry 40:Differential geometry 18:Atiyah–Singer theorem 8826:Teleman, N. (1983), 8024:, Berlin: Springer, 7643:Bull. Am. Math. Soc. 6969:Cartan-Schwartz 1965 6702: 6357: 6301: 6162: 5742: 5667: 5548: 5498: 5375: 5315: 4911: 4222: 4139: 4096: 3971: 3898: 3801: 3753: 3742:{\displaystyle n=2r} 3724: 3704: 3669: 3636: 3631:Euler characteristic 3613: 3586: 3559: 3520: 3493: 3466: 3455:{\displaystyle n=2r} 3437: 3417: 3395:Robert Thomas Seeley 3385:, as generalized by 3283:equivariant K-theory 3133: 3068:) and index theory. 2931:{\displaystyle K(X)} 2913: 2887: 2656: 2621: 2529:), and the image in 2464: 2419: 2299: 2288:, then applying the 2260: 2234: 2214: 2189: 2160: 2147:{\displaystyle S(X)} 2129: 2100: 2089:{\displaystyle B(X)} 2071: 2041: 2011: 1956: 1886: 1808: 1749: 1633: 1530: 1498: 1476: 1440: 1417: 1366: 1223: 1200: 1180: 1160: 1140: 1120: 860: 801: 774: 718: 676: 638: 603: 565: 483: 475:is the function of 2 429: 213:on smooth manifolds. 189:Princeton University 165:Friedrich Hirzebruch 153:Riemann–Roch theorem 129:Riemann–Roch theorem 9090:Links of interviews 9014:Links on the theory 8804:1980InMat..61..227T 8516:1975IzMat...9..751K 8432:Analytic K-homology 8335:1983CMaPh..92..163G 8323:Commun. Math. Phys. 8291:1960RuMaS..15..113G 8040:Bismut, Jean-Michel 7885:1979InMat..54..189A 7842:1977InMat..42....1A 7797:1975InMat..28..277A 7754:1973InMat..19..279A 6620:is the image under 6434:exterior derivative 6341:Hodge star operator 3698:splitting principle 3389:, was published by 3336:Constantine Callias 3306:von Neumann algebra 2902:{\displaystyle Y=*} 2833: 2716: 2692: 2479:{\displaystyle BSO} 2096:and an isomorphism 845: 821: 759: 735: 364:(without boundary). 133:theoretical physics 32: 9107:, pp. 223–231 9084:on March 29, 2017. 9074:Wassermann, Antony 8910:10.1007/BF01201710 8881:10.1007/BF02392376 8844:10.1007/BF02953772 8812:10.1007/BF01390066 8768:10.1007/BF02953773 8762:, Paris: 291–293, 8690:10.1007/BFb0068264 8647:Palais, Richard S. 8343:10.1007/BF01210843 8259:10.1007/BF02392736 8105:10.1007/BF02698807 7998:10.1007/BF02684299 7894:10.1007/BF01408936 7850:10.1007/BF01389783 7806:10.1007/BF01425562 7762:10.1007/BF01425417 7601:Singer, Isadore M. 7597:Atiyah, Michael F. 7563:Singer, Isadore M. 7559:Atiyah, Michael F. 7525:Singer, Isadore M. 7521:Atiyah, Michael F. 7489:Singer, Isadore M. 7485:Atiyah, Michael F. 7459:Singer, Isadore M. 7455:Atiyah, Michael F. 7241:10.1007/BFb0068264 6837: 6496:Fourier transforms 6440:* is its adjoint. 6411: 6329: 6276:signature operator 6246:complex manifolds 6229: 6138: 6136: 5725: 5650: 5578: 5511: 5481: 5348: 5243: 4891: 4889: 4540: 4205: 4122: 4082: 3957: 3884: 3787: 3739: 3710: 3675: 3651: 3619: 3599: 3572: 3545: 3506: 3479: 3452: 3423: 3315:, and was used by 3224:signature operator 3203: 2940:Grothendieck group 2928: 2899: 2870: 2868: 2639: 2603:pseudodifferential 2476: 2447: 2402: 2278: 2246:{\displaystyle 2m} 2243: 2220: 2195: 2175: 2144: 2115: 2086: 2057: 2027: 1997: 1942: 1867: 1793: 1731: 1617: 1516: 1482: 1458: 1423: 1399: 1346: 1206: 1186: 1166: 1146: 1126: 872: 846: 831: 807: 787: 760: 745: 721: 689: 662: 616: 589: 547: 461: 325:Simon K. Donaldson 287:Luis Alvarez-Gaume 211:Pontryagin classes 9051:Pdf presentation. 8732:978-0-12-158860-1 8699:978-0-387-08660-6 8662:978-0-691-08031-4 8613:978-1-56881-002-7 8392:978-0-8493-7874-4 8279:Russ. Math. Surv. 8145:978-0-12-185860-5 8031:978-3-540-53340-5 7954:978-0-19-853278-1 7925:978-0-19-853277-4 7704:, Second Series, 7675:, Second series, 7609:, Second Series, 7571:, Second Series, 7533:, Second Series, 7429:, Second Series, 7250:978-0-387-08660-6 6850:for any positive 6473:Rochlin's theorem 6120: 6061: 5993: 5830: 5569: 5509: 5340: 5326: 5097: 5040: 4865: 4525: 4262: 4249: 4041: 3882: 3846: 3713:{\displaystyle E} 3678:{\displaystyle M} 3622:{\displaystyle D} 3596: 3569: 3503: 3476: 3426:{\displaystyle M} 2830: 2813: 2714: 2697: 2459:classifying space 2385: 2223:{\displaystyle X} 2198:{\displaystyle D} 2185:is the symbol of 1485:{\displaystyle X} 1426:{\displaystyle X} 1209:{\displaystyle X} 1189:{\displaystyle n} 1169:{\displaystyle F} 1149:{\displaystyle E} 1129:{\displaystyle D} 1114:topological index 1108:Topological index 1011:Fredholm operator 207:Sergey P. Novikov 121:topological index 89: 88: 84:Rokhlin's theorem 16:(Redirected from 9146: 9108: 9102: 9085: 9080:. Archived from 9069: 9057:"Index formulas" 9050: 9044: 9036: 9034: 9032: 9026: 8986: 8965: 8956: 8932: 8931:, pp. 54–67 8920: 8892: 8883: 8868:Acta Mathematica 8862: 8822: 8786: 8743: 8714: 8702: 8683: 8665: 8642: 8630: 8616: 8596: 8576:Lawson, H. Blane 8571: 8564:Siebenmann, L.C. 8555: 8554: 8535:Siebenmann, L.C. 8526: 8498: 8473: 8464:(3): 1007–1022, 8444: 8426: 8424: 8411: 8409: 8395: 8375: 8374: 8353: 8301: 8270: 8261: 8246:Acta Mathematica 8233: 8224: 8196: 8187: 8169: 8156: 8137: 8123: 8080: 8060: 8059: 8034: 8016: 7986:Acta Mathematica 7965: 7936: 7905: 7896: 7868: 7826:Schmid, Wilfried 7817: 7808: 7780: 7724: 7695: 7659: 7658: 7629: 7591: 7553: 7515: 7479: 7478: 7449: 7414: 7395: 7355: 7354: 7337: 7331: 7326: 7320: 7315: 7309: 7308: 7306: 7305: 7291: 7285: 7284: 7264:Lawson, H. Blane 7260: 7254: 7253: 7234: 7218: 7212: 7206: 7200: 7194: 7188: 7182: 7176: 7170: 7164: 7158: 7152: 7146: 7140: 7134: 7128: 7122: 7116: 7110: 7104: 7098: 7092: 7086: 7080: 7074: 7068: 7062: 7056: 7050: 7044: 7038: 7032: 7026: 7020: 7014: 7008: 6990: 6984: 6978: 6972: 6966: 6960: 6954: 6948: 6942: 6936: 6930: 6924: 6918: 6912: 6906: 6882: 6846: 6844: 6843: 6838: 6836: 6832: 6831: 6830: 6829: 6793: 6789: 6788: 6784: 6783: 6747: 6746: 6608:is a sphere and 6532:cobordism theory 6479:Proof techniques 6420: 6418: 6417: 6412: 6410: 6409: 6404: 6400: 6399: 6398: 6397: 6384: 6373: 6338: 6336: 6335: 6330: 6328: 6327: 6270:is given by the 6238: 6236: 6235: 6230: 6195: 6194: 6147: 6145: 6144: 6139: 6137: 6121: 6119: 6118: 6117: 6116: 6115: 6094: 6093: 6092: 6079: 6076: 6071: 6062: 6060: 6059: 6058: 6057: 6056: 6032: 6031: 6022: 6019: 6014: 5998: 5994: 5989: 5981: 5945: 5906: 5902: 5901: 5900: 5899: 5898: 5872: 5867: 5836: 5832: 5831: 5826: 5822: 5821: 5811: 5809: 5808: 5793: 5792: 5773: 5768: 5734: 5732: 5731: 5726: 5712: 5711: 5701: 5696: 5659: 5657: 5656: 5651: 5613: 5612: 5597: 5596: 5577: 5520: 5518: 5517: 5512: 5510: 5502: 5490: 5488: 5487: 5482: 5465: 5464: 5455: 5454: 5421: 5420: 5411: 5410: 5357: 5355: 5354: 5349: 5347: 5346: 5341: 5333: 5327: 5319: 5274:complex manifold 5252: 5250: 5249: 5244: 5227: 5226: 5211: 5194: 5193: 5183: 5178: 5169: 5168: 5150: 5149: 5140: 5139: 5115: 5098: 5096: 5095: 5094: 5093: 5092: 5068: 5067: 5058: 5055: 5050: 5041: 5039: 5038: 5037: 5027: 5022: 5012: 5011: 5007: 5006: 5005: 5004: 5003: 4974: 4969: 4959: 4957: 4956: 4947: 4946: 4900: 4898: 4897: 4892: 4890: 4883: 4866: 4864: 4863: 4862: 4861: 4860: 4836: 4835: 4826: 4823: 4818: 4799: 4769: 4752: 4748: 4747: 4746: 4745: 4744: 4715: 4710: 4695: 4688: 4671: 4670: 4669: 4668: 4648: 4647: 4646: 4645: 4628: 4627: 4597: 4580: 4579: 4578: 4577: 4560: 4559: 4558: 4557: 4539: 4518: 4501: 4500: 4499: 4498: 4480: 4475: 4454: 4450: 4449: 4448: 4444: 4443: 4432: 4431: 4422: 4421: 4399: 4398: 4371: 4370: 4369: 4365: 4364: 4353: 4352: 4343: 4342: 4314: 4303: 4302: 4271: 4270: 4269: 4265: 4264: 4263: 4260: 4251: 4250: 4247: 4214: 4212: 4211: 4206: 4201: 4184: 4183: 4173: 4168: 4131: 4129: 4128: 4123: 4091: 4089: 4088: 4083: 4078: 4064: 4063: 4048: 4047: 4046: 4042: 4037: 4036: 4027: 4019: 4018: 4003: 3989: 3988: 3966: 3964: 3963: 3958: 3953: 3952: 3940: 3939: 3924: 3910: 3909: 3893: 3891: 3890: 3885: 3883: 3878: 3877: 3868: 3863: 3862: 3847: 3842: 3841: 3832: 3827: 3826: 3814: 3796: 3794: 3793: 3788: 3786: 3785: 3765: 3764: 3748: 3746: 3745: 3740: 3719: 3717: 3716: 3711: 3684: 3682: 3681: 3676: 3663:Hodge cohomology 3660: 3658: 3657: 3652: 3628: 3626: 3625: 3620: 3608: 3606: 3605: 3600: 3598: 3597: 3594: 3581: 3579: 3578: 3573: 3571: 3570: 3567: 3554: 3552: 3551: 3546: 3544: 3543: 3515: 3513: 3512: 3507: 3505: 3504: 3501: 3488: 3486: 3485: 3480: 3478: 3477: 3474: 3461: 3459: 3458: 3453: 3432: 3430: 3429: 3424: 3348:Hermitian matrix 3212: 3210: 3209: 3204: 3196: 3195: 3177: 3176: 3164: 3163: 3151: 3150: 3127:elliptic complex 3111:Other extensions 2937: 2935: 2934: 2929: 2908: 2906: 2905: 2900: 2879: 2877: 2876: 2871: 2869: 2866: 2865: 2864: 2863: 2860: 2855: 2836: 2832: 2831: 2828: 2814: 2811: 2787: 2783: 2782: 2773: 2772: 2765: 2763: 2762: 2756: 2755: 2745: 2742: 2737: 2718: 2715: 2712: 2698: 2695: 2691: 2668: 2665: 2664: 2663: 2662: 2648: 2646: 2645: 2640: 2485: 2483: 2482: 2477: 2456: 2454: 2453: 2448: 2446: 2445: 2411: 2409: 2408: 2403: 2386: 2384: 2367: 2332: 2330: 2329: 2320: 2319: 2290:Thom isomorphism 2287: 2285: 2284: 2279: 2252: 2250: 2249: 2244: 2229: 2227: 2226: 2221: 2204: 2202: 2201: 2196: 2184: 2182: 2181: 2176: 2153: 2151: 2150: 2145: 2124: 2122: 2121: 2116: 2095: 2093: 2092: 2087: 2066: 2064: 2063: 2058: 2053: 2052: 2036: 2034: 2033: 2028: 2023: 2022: 2006: 2004: 2003: 1998: 1981: 1951: 1949: 1948: 1943: 1920: 1919: 1904: 1903: 1876: 1874: 1873: 1868: 1863: 1849: 1848: 1836: 1802: 1800: 1799: 1794: 1774: 1743:Thom isomorphism 1740: 1738: 1737: 1732: 1727: 1707: 1687: 1686: 1665: 1651: 1650: 1626: 1624: 1623: 1618: 1589: 1588: 1573: 1572: 1545: 1544: 1525: 1523: 1522: 1517: 1491: 1489: 1488: 1483: 1467: 1465: 1464: 1459: 1432: 1430: 1429: 1424: 1413:of the manifold 1408: 1406: 1405: 1400: 1361:cohomology class 1355: 1353: 1352: 1347: 1315: 1314: 1305: 1304: 1244: 1243: 1215: 1213: 1212: 1207: 1195: 1193: 1192: 1187: 1175: 1173: 1172: 1167: 1155: 1153: 1152: 1147: 1135: 1133: 1132: 1127: 1067:analytical index 1001:Analytical index 899:cotangent bundle 881: 879: 878: 873: 855: 853: 852: 847: 844: 839: 820: 815: 796: 794: 793: 788: 786: 785: 769: 767: 766: 761: 758: 753: 734: 729: 698: 696: 695: 690: 688: 687: 671: 669: 668: 663: 661: 660: 648: 625: 623: 622: 617: 615: 614: 598: 596: 595: 590: 588: 587: 575: 556: 554: 553: 548: 546: 545: 527: 526: 514: 513: 495: 494: 470: 468: 467: 462: 460: 459: 441: 440: 117:analytical index 113:compact manifold 33: 21: 9154: 9153: 9149: 9148: 9147: 9145: 9144: 9143: 9119: 9118: 9100: 9095: 9092: 9072: 9054: 9037: 9030: 9028: 9024: 9019: 9016: 9011: 9006: 8984: 8972:, ed. (2009) , 8968: 8935: 8923: 8895: 8865: 8825: 8789: 8746: 8733: 8717: 8705: 8700: 8681:10.1.1.193.9222 8669: 8663: 8645: 8628: 8620: 8614: 8599: 8594: 8574: 8558: 8529: 8501: 8447: 8442: 8429: 8414: 8399: 8393: 8378: 8356: 8313: 8275:Gel'fand, I. M. 8273: 8238:Donaldson, S.K. 8236: 8199: 8167: 8159: 8146: 8126: 8083: 8066:Cartan-Schwartz 8064: 8044:J. Funct. Anal. 8038: 8032: 8019: 7968: 7955: 7941:Atiyah, Michael 7939: 7926: 7910:Atiyah, Michael 7908: 7870: 7822:Atiyah, Michael 7820: 7782: 7728: 7714:10.2307/1970721 7697: 7685:10.2307/1970694 7663: 7633: 7619:10.2307/1970757 7595: 7581:10.2307/1970756 7557: 7543:10.2307/1970717 7519: 7505:10.2307/1970715 7483: 7453: 7439:10.2307/1970716 7417: 7398: 7379: 7364: 7359: 7358: 7352: 7339: 7338: 7334: 7327: 7323: 7316: 7312: 7303: 7301: 7293: 7292: 7288: 7282: 7262: 7261: 7257: 7251: 7232:10.1.1.193.9222 7220: 7219: 7215: 7207: 7203: 7195: 7191: 7183: 7179: 7171: 7167: 7159: 7155: 7147: 7143: 7135: 7131: 7123: 7119: 7111: 7107: 7099: 7095: 7087: 7083: 7075: 7071: 7063: 7059: 7051: 7047: 7039: 7035: 7027: 7023: 7015: 7011: 6991: 6987: 6979: 6975: 6967: 6963: 6955: 6951: 6943: 6939: 6931: 6927: 6919: 6915: 6907: 6900: 6895: 6880: 6873: 6821: 6807: 6803: 6775: 6764: 6760: 6738: 6700: 6699: 6633: 6626: 6587: 6563: 6524: 6492: 6486: 6481: 6461: 6389: 6379: 6375: 6374: 6355: 6354: 6349:Hodge Laplacian 6343:. The operator 6304: 6299: 6298: 6293:, that acts on 6256: 6186: 6160: 6159: 6135: 6134: 6107: 6102: 6095: 6084: 6080: 6048: 6040: 6033: 6023: 5999: 5982: 5976: 5920: 5919: 5890: 5885: 5878: 5874: 5837: 5813: 5812: 5800: 5784: 5759: 5755: 5740: 5739: 5703: 5665: 5664: 5604: 5588: 5546: 5545: 5496: 5495: 5456: 5440: 5412: 5396: 5373: 5372: 5331: 5313: 5312: 5266: 5218: 5185: 5160: 5141: 5131: 5084: 5076: 5069: 5059: 5029: 5013: 4995: 4987: 4980: 4976: 4960: 4948: 4938: 4909: 4908: 4888: 4887: 4852: 4844: 4837: 4827: 4803: 4774: 4773: 4736: 4728: 4721: 4717: 4693: 4692: 4660: 4652: 4637: 4629: 4619: 4569: 4561: 4549: 4541: 4490: 4482: 4452: 4451: 4423: 4413: 4412: 4408: 4390: 4344: 4334: 4333: 4329: 4294: 4272: 4255: 4242: 4241: 4237: 4220: 4219: 4175: 4137: 4136: 4094: 4093: 4055: 4028: 4022: 4010: 3974: 3969: 3968: 3944: 3931: 3901: 3896: 3895: 3869: 3854: 3833: 3818: 3799: 3798: 3777: 3756: 3751: 3750: 3722: 3721: 3702: 3701: 3667: 3666: 3634: 3633: 3611: 3610: 3589: 3584: 3583: 3562: 3557: 3556: 3535: 3518: 3517: 3496: 3491: 3490: 3469: 3464: 3463: 3435: 3434: 3415: 3414: 3411: 3406: 3379:Hörmander index 3187: 3168: 3155: 3142: 3131: 3130: 3113: 3010: 2957: 2952: 2911: 2910: 2885: 2884: 2867: 2861: 2859: 2837: 2801: 2785: 2784: 2774: 2764: 2747: 2743: 2741: 2719: 2682: 2666: 2654: 2653: 2619: 2618: 2611: 2462: 2461: 2434: 2417: 2416: 2368: 2333: 2321: 2311: 2297: 2296: 2258: 2257: 2232: 2231: 2212: 2211: 2187: 2186: 2158: 2157: 2127: 2126: 2098: 2097: 2069: 2068: 2044: 2039: 2038: 2014: 2009: 2008: 1954: 1953: 1911: 1895: 1884: 1883: 1879:Chern character 1840: 1806: 1805: 1747: 1746: 1672: 1642: 1631: 1630: 1580: 1564: 1533: 1528: 1527: 1496: 1495: 1474: 1473: 1438: 1437: 1415: 1414: 1364: 1363: 1306: 1296: 1235: 1221: 1220: 1198: 1197: 1178: 1177: 1158: 1157: 1138: 1137: 1118: 1117: 1110: 1003: 949: 942: 858: 857: 799: 798: 777: 772: 771: 716: 715: 679: 674: 673: 652: 636: 635: 606: 601: 600: 579: 563: 562: 537: 518: 505: 486: 481: 480: 451: 432: 427: 426: 411: 350: 291:Dirac operators 270:Dennis Sullivan 145:Israel Gel'fand 141: 82: 78: 74: 28: 23: 22: 15: 12: 11: 5: 9152: 9150: 9142: 9141: 9136: 9131: 9121: 9120: 9117: 9116: 9109: 9105:Notices of AMS 9091: 9088: 9087: 9086: 9070: 9052: 9020:Mazzeo, Rafe. 9015: 9012: 9010: 9009:External links 9007: 9005: 9004: 8983:978-1571461377 8982: 8970:Shing-Tung Yau 8966: 8947:(4): 661–692, 8941:J. Diff. Geom. 8937:Witten, Edward 8933: 8921: 8904:(5): 693–719, 8893: 8863: 8823: 8798:(3): 227–249, 8787: 8744: 8731: 8715: 8703: 8698: 8667: 8661: 8643: 8618: 8612: 8597: 8592: 8572: 8556: 8545:(4): 742–749, 8527: 8510:(4): 751–792, 8499: 8480:10.2307/121079 8445: 8440: 8427: 8412: 8397: 8391: 8376: 8354: 8329:(2): 163–178, 8311: 8285:(3): 113–123, 8271: 8234: 8215:(4): 663–681, 8197: 8178:(3): 345–388, 8157: 8144: 8124: 8081: 8062: 8036: 8030: 8017: 7978:Macpherson, R. 7966: 7953: 7937: 7924: 7906: 7879:(2): 189–192, 7818: 7791:(3): 277–280, 7748:(4): 279–330, 7726: 7708:(3): 451–491, 7679:(2): 374–407, 7661: 7631: 7613:(1): 139–149, 7593: 7575:(1): 119–138, 7555: 7537:(3): 546–604, 7517: 7499:(3): 484–530, 7481: 7469:(3): 422–433, 7451: 7433:(3): 531–545, 7415: 7396: 7376: 7363: 7360: 7357: 7356: 7350: 7332: 7321: 7310: 7286: 7280: 7255: 7249: 7213: 7201: 7189: 7177: 7165: 7153: 7141: 7129: 7117: 7105: 7093: 7081: 7069: 7057: 7045: 7033: 7021: 7009: 6985: 6973: 6961: 6949: 6937: 6925: 6913: 6897: 6896: 6894: 6891: 6890: 6889: 6883: 6872: 6869: 6848: 6847: 6835: 6828: 6824: 6820: 6817: 6814: 6810: 6806: 6802: 6799: 6796: 6792: 6787: 6782: 6778: 6774: 6771: 6767: 6763: 6759: 6756: 6753: 6750: 6745: 6741: 6737: 6734: 6731: 6728: 6725: 6722: 6719: 6716: 6713: 6710: 6707: 6639:, and 6632: 6629: 6624: 6585: 6562: 6559: 6523: 6520: 6488:Main article: 6485: 6482: 6480: 6477: 6460: 6457: 6432:is the Cartan 6424:restricted to 6422: 6421: 6408: 6403: 6396: 6392: 6387: 6383: 6378: 6372: 6368: 6365: 6362: 6326: 6323: 6320: 6317: 6314: 6311: 6307: 6266:of dimension 4 6255: 6252: 6240: 6239: 6228: 6225: 6222: 6219: 6216: 6213: 6210: 6207: 6204: 6201: 6198: 6193: 6189: 6185: 6182: 6179: 6176: 6173: 6170: 6167: 6149: 6148: 6133: 6130: 6127: 6124: 6114: 6110: 6105: 6101: 6098: 6091: 6087: 6083: 6075: 6070: 6066: 6055: 6051: 6047: 6043: 6039: 6036: 6030: 6026: 6018: 6013: 6009: 6005: 6002: 6000: 5997: 5992: 5988: 5985: 5979: 5975: 5972: 5969: 5966: 5963: 5960: 5957: 5954: 5951: 5948: 5944: 5940: 5937: 5934: 5931: 5928: 5925: 5922: 5921: 5918: 5915: 5912: 5909: 5905: 5897: 5893: 5888: 5884: 5881: 5877: 5871: 5866: 5862: 5858: 5855: 5852: 5849: 5846: 5843: 5840: 5838: 5835: 5829: 5825: 5820: 5816: 5807: 5803: 5799: 5796: 5791: 5787: 5783: 5780: 5777: 5772: 5767: 5763: 5758: 5754: 5751: 5748: 5747: 5724: 5721: 5718: 5715: 5710: 5706: 5700: 5695: 5691: 5687: 5684: 5681: 5678: 5675: 5672: 5661: 5660: 5649: 5646: 5643: 5640: 5637: 5634: 5631: 5628: 5625: 5622: 5619: 5616: 5611: 5607: 5603: 5600: 5595: 5591: 5587: 5584: 5581: 5576: 5572: 5568: 5565: 5562: 5559: 5556: 5553: 5508: 5505: 5492: 5491: 5480: 5477: 5474: 5471: 5468: 5463: 5459: 5453: 5450: 5447: 5443: 5439: 5436: 5433: 5430: 5427: 5424: 5419: 5415: 5409: 5406: 5403: 5399: 5395: 5392: 5389: 5386: 5383: 5380: 5361:restricted to 5359: 5358: 5345: 5339: 5336: 5330: 5325: 5322: 5265: 5262: 5254: 5253: 5242: 5239: 5236: 5233: 5230: 5225: 5221: 5217: 5214: 5210: 5206: 5203: 5200: 5197: 5192: 5188: 5182: 5177: 5173: 5167: 5163: 5159: 5156: 5153: 5148: 5144: 5138: 5134: 5130: 5127: 5124: 5121: 5118: 5114: 5110: 5107: 5104: 5101: 5091: 5087: 5083: 5079: 5075: 5072: 5066: 5062: 5054: 5049: 5045: 5036: 5032: 5026: 5021: 5017: 5010: 5002: 4998: 4994: 4990: 4986: 4983: 4979: 4973: 4968: 4964: 4955: 4951: 4945: 4941: 4937: 4934: 4931: 4928: 4925: 4922: 4919: 4916: 4902: 4901: 4886: 4882: 4878: 4875: 4872: 4869: 4859: 4855: 4851: 4847: 4843: 4840: 4834: 4830: 4822: 4817: 4813: 4809: 4806: 4804: 4802: 4798: 4794: 4791: 4788: 4785: 4782: 4779: 4776: 4775: 4772: 4768: 4764: 4761: 4758: 4755: 4751: 4743: 4739: 4735: 4731: 4727: 4724: 4720: 4714: 4709: 4705: 4701: 4698: 4696: 4694: 4691: 4687: 4683: 4680: 4677: 4674: 4667: 4663: 4659: 4655: 4651: 4644: 4640: 4636: 4632: 4626: 4622: 4618: 4615: 4612: 4609: 4606: 4603: 4600: 4596: 4592: 4589: 4586: 4583: 4576: 4572: 4568: 4564: 4556: 4552: 4548: 4544: 4538: 4535: 4532: 4528: 4524: 4521: 4517: 4513: 4510: 4507: 4504: 4497: 4493: 4489: 4485: 4479: 4474: 4470: 4466: 4463: 4460: 4457: 4455: 4453: 4447: 4442: 4438: 4435: 4430: 4426: 4420: 4416: 4411: 4405: 4402: 4397: 4393: 4389: 4386: 4383: 4380: 4377: 4374: 4368: 4363: 4359: 4356: 4351: 4347: 4341: 4337: 4332: 4326: 4323: 4320: 4317: 4313: 4309: 4306: 4301: 4297: 4293: 4290: 4287: 4284: 4281: 4278: 4275: 4273: 4268: 4258: 4254: 4245: 4240: 4234: 4231: 4228: 4227: 4204: 4200: 4196: 4193: 4190: 4187: 4182: 4178: 4172: 4167: 4163: 4159: 4156: 4153: 4150: 4147: 4144: 4121: 4117: 4114: 4110: 4107: 4104: 4101: 4081: 4077: 4073: 4070: 4067: 4062: 4058: 4054: 4051: 4045: 4040: 4035: 4031: 4025: 4017: 4013: 4009: 4006: 4002: 3998: 3995: 3992: 3987: 3984: 3981: 3977: 3956: 3951: 3947: 3943: 3938: 3934: 3930: 3927: 3923: 3919: 3916: 3913: 3908: 3904: 3881: 3876: 3872: 3866: 3861: 3857: 3853: 3850: 3845: 3840: 3836: 3830: 3825: 3821: 3817: 3813: 3809: 3806: 3784: 3780: 3775: 3772: 3768: 3763: 3759: 3738: 3735: 3732: 3729: 3709: 3674: 3650: 3647: 3644: 3641: 3618: 3592: 3565: 3542: 3538: 3534: 3531: 3528: 3525: 3499: 3472: 3451: 3448: 3445: 3442: 3422: 3410: 3407: 3405: 3402: 3401: 3400: 3399: 3398: 3387:Lars Hörmander 3328: 3298: 3269:If there is a 3267: 3248: 3241:Melrose (1993) 3215: 3202: 3199: 3194: 3190: 3186: 3183: 3180: 3175: 3171: 3167: 3162: 3158: 3154: 3149: 3145: 3141: 3138: 3122: 3112: 3109: 3087:-structures, 3045:)). The work ( 3028: 3027: 3009: 3006: 2979: 2978: 2956: 2953: 2951: 2948: 2927: 2924: 2921: 2918: 2898: 2895: 2892: 2881: 2880: 2862: 2858: 2854: 2850: 2847: 2844: 2841: 2838: 2826: 2823: 2820: 2817: 2807: 2802: 2800: 2797: 2794: 2791: 2788: 2786: 2781: 2777: 2771: 2766: 2761: 2754: 2750: 2746: 2744: 2740: 2736: 2732: 2729: 2726: 2723: 2720: 2710: 2707: 2704: 2701: 2688: 2683: 2681: 2678: 2675: 2672: 2669: 2667: 2661: 2638: 2635: 2632: 2629: 2626: 2610: 2607: 2583: 2582: 2475: 2472: 2469: 2444: 2441: 2437: 2433: 2430: 2427: 2424: 2413: 2412: 2401: 2398: 2395: 2392: 2389: 2383: 2380: 2377: 2374: 2371: 2366: 2363: 2360: 2357: 2354: 2351: 2348: 2345: 2342: 2339: 2336: 2328: 2324: 2318: 2314: 2310: 2307: 2304: 2277: 2274: 2271: 2268: 2265: 2242: 2239: 2219: 2208: 2207: 2206: 2205: 2194: 2174: 2171: 2168: 2165: 2155: 2143: 2140: 2137: 2134: 2114: 2111: 2108: 2105: 2085: 2082: 2079: 2076: 2056: 2051: 2047: 2026: 2021: 2017: 1996: 1993: 1990: 1987: 1984: 1980: 1976: 1973: 1970: 1967: 1964: 1961: 1941: 1938: 1935: 1932: 1929: 1926: 1923: 1918: 1914: 1910: 1907: 1902: 1898: 1894: 1891: 1881: 1866: 1862: 1858: 1855: 1852: 1847: 1843: 1839: 1835: 1831: 1828: 1825: 1822: 1819: 1816: 1813: 1803: 1792: 1789: 1786: 1783: 1780: 1777: 1773: 1769: 1766: 1763: 1760: 1757: 1754: 1730: 1726: 1722: 1719: 1716: 1713: 1710: 1706: 1702: 1699: 1696: 1693: 1690: 1685: 1682: 1679: 1675: 1671: 1668: 1664: 1660: 1657: 1654: 1649: 1645: 1641: 1638: 1616: 1613: 1610: 1607: 1604: 1601: 1598: 1595: 1592: 1587: 1583: 1579: 1576: 1571: 1567: 1563: 1560: 1557: 1554: 1551: 1548: 1543: 1540: 1536: 1515: 1512: 1509: 1506: 1503: 1493: 1481: 1457: 1454: 1451: 1448: 1445: 1422: 1398: 1395: 1392: 1389: 1386: 1383: 1380: 1377: 1374: 1371: 1357: 1356: 1345: 1342: 1339: 1336: 1333: 1330: 1327: 1324: 1321: 1318: 1313: 1309: 1303: 1299: 1295: 1292: 1289: 1286: 1283: 1280: 1277: 1274: 1271: 1268: 1265: 1262: 1259: 1256: 1253: 1250: 1247: 1242: 1238: 1234: 1231: 1228: 1205: 1185: 1165: 1145: 1125: 1109: 1106: 1063: 1062: 1025:(solutions of 1002: 999: 947: 940: 871: 868: 865: 843: 838: 834: 830: 827: 824: 819: 814: 810: 806: 784: 780: 757: 752: 748: 744: 741: 738: 733: 728: 724: 686: 682: 659: 655: 651: 647: 643: 613: 609: 586: 582: 578: 574: 570: 561:and replacing 544: 540: 536: 533: 530: 525: 521: 517: 512: 508: 504: 501: 498: 493: 489: 458: 454: 450: 447: 444: 439: 435: 410: 407: 406: 405: 383: 376:vector bundles 365: 349: 346: 345: 344: 338: 332: 319: 306: 300: 294: 277: 274:quasiconformal 264: 246: 240: 234: 228: 214: 185:Richard Palais 177:Dirac operator 140: 137: 105:Isadore Singer 101:Michael Atiyah 87: 86: 69: 65: 64: 61: 60:First proof in 57: 56: 54:Isadore Singer 50:Michael Atiyah 47: 46:First proof by 43: 42: 37: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 9151: 9140: 9137: 9135: 9132: 9130: 9127: 9126: 9124: 9114: 9110: 9106: 9099: 9094: 9093: 9089: 9083: 9079: 9075: 9071: 9068: 9064: 9063: 9058: 9053: 9048: 9042: 9023: 9018: 9017: 9013: 9008: 9002: 8998: 8994: 8990: 8985: 8979: 8975: 8971: 8967: 8964: 8960: 8955: 8950: 8946: 8942: 8938: 8934: 8930: 8926: 8922: 8919: 8915: 8911: 8907: 8903: 8899: 8894: 8891: 8887: 8882: 8877: 8873: 8869: 8864: 8861: 8857: 8853: 8849: 8845: 8841: 8837: 8833: 8829: 8824: 8821: 8817: 8813: 8809: 8805: 8801: 8797: 8793: 8788: 8785: 8781: 8777: 8773: 8769: 8765: 8761: 8757: 8753: 8749: 8745: 8742: 8738: 8734: 8728: 8724: 8720: 8716: 8712: 8708: 8704: 8701: 8695: 8691: 8687: 8682: 8677: 8673: 8668: 8664: 8658: 8654: 8653: 8648: 8644: 8640: 8636: 8635: 8627: 8623: 8622:Novikov, S.P. 8619: 8615: 8609: 8605: 8604: 8598: 8595: 8593:0-691-08542-0 8589: 8585: 8584:Spin Geometry 8581: 8577: 8573: 8569: 8565: 8561: 8557: 8553: 8548: 8544: 8540: 8536: 8532: 8528: 8525: 8521: 8517: 8513: 8509: 8505: 8500: 8497: 8493: 8489: 8485: 8481: 8477: 8472: 8467: 8463: 8459: 8456:-homologie", 8455: 8451: 8446: 8443: 8441:9780191589201 8437: 8433: 8428: 8423: 8418: 8413: 8408: 8403: 8398: 8394: 8388: 8385:, CRC Press, 8384: 8383: 8377: 8373: 8368: 8364: 8360: 8355: 8352: 8348: 8344: 8340: 8336: 8332: 8328: 8324: 8320: 8316: 8312: 8309: 8308:0-387-13619-3 8305: 8300: 8296: 8292: 8288: 8284: 8280: 8276: 8272: 8269: 8265: 8260: 8255: 8251: 8247: 8243: 8239: 8235: 8232: 8228: 8223: 8218: 8214: 8210: 8206: 8202: 8198: 8195: 8191: 8186: 8181: 8177: 8173: 8166: 8162: 8158: 8155: 8151: 8147: 8141: 8136: 8135: 8129: 8125: 8122: 8118: 8114: 8110: 8106: 8102: 8098: 8094: 8090: 8086: 8082: 8079: 8075: 8071: 8067: 8063: 8058: 8053: 8049: 8045: 8041: 8037: 8033: 8027: 8023: 8018: 8015: 8011: 8007: 8003: 7999: 7995: 7991: 7987: 7983: 7979: 7975: 7971: 7967: 7964: 7960: 7956: 7950: 7946: 7942: 7938: 7935: 7931: 7927: 7921: 7917: 7916: 7911: 7907: 7904: 7900: 7895: 7890: 7886: 7882: 7878: 7874: 7873:Invent. Math. 7867: 7863: 7859: 7855: 7851: 7847: 7843: 7839: 7835: 7831: 7830:Invent. Math. 7827: 7823: 7819: 7816: 7812: 7807: 7802: 7798: 7794: 7790: 7786: 7785:Invent. Math. 7779: 7775: 7771: 7767: 7763: 7759: 7755: 7751: 7747: 7743: 7742:Invent. Math. 7739: 7738:Patodi, V. K. 7735: 7731: 7727: 7723: 7719: 7715: 7711: 7707: 7703: 7702: 7694: 7690: 7686: 7682: 7678: 7674: 7670: 7666: 7665:Atiyah, M. F. 7662: 7657: 7652: 7649:(2): 245–50, 7648: 7644: 7640: 7636: 7635:Atiyah, M. F. 7632: 7628: 7624: 7620: 7616: 7612: 7608: 7607: 7602: 7598: 7594: 7590: 7586: 7582: 7578: 7574: 7570: 7569: 7564: 7560: 7556: 7552: 7548: 7544: 7540: 7536: 7532: 7531: 7526: 7522: 7518: 7514: 7510: 7506: 7502: 7498: 7494: 7490: 7486: 7482: 7477: 7472: 7468: 7464: 7460: 7456: 7452: 7448: 7444: 7440: 7436: 7432: 7428: 7424: 7420: 7419:Atiyah, M. F. 7416: 7413: 7409: 7405: 7401: 7400:Atiyah, M. F. 7397: 7394: 7390: 7386: 7382: 7381:Atiyah, M. F. 7378: 7377: 7375: 7373: 7369: 7361: 7353: 7351:0-7503-0606-8 7347: 7343: 7336: 7333: 7330: 7325: 7322: 7319: 7314: 7311: 7300: 7296: 7290: 7287: 7283: 7281:0-691-08542-0 7277: 7273: 7272:Spin Geometry 7269: 7265: 7259: 7256: 7252: 7246: 7242: 7238: 7233: 7228: 7224: 7217: 7214: 7210: 7205: 7202: 7198: 7193: 7190: 7186: 7181: 7178: 7174: 7169: 7166: 7162: 7157: 7154: 7150: 7145: 7142: 7138: 7133: 7130: 7126: 7121: 7118: 7114: 7113:Sullivan 1979 7109: 7106: 7102: 7097: 7094: 7090: 7085: 7082: 7078: 7077:Kasparov 1972 7073: 7070: 7066: 7061: 7058: 7054: 7049: 7046: 7042: 7037: 7034: 7030: 7025: 7022: 7018: 7013: 7010: 7006: 7002: 6998: 6994: 6989: 6986: 6982: 6977: 6974: 6970: 6965: 6962: 6958: 6953: 6950: 6946: 6945:Gel'fand 1960 6941: 6938: 6935:, p. 11. 6934: 6933:Hamilton 2020 6929: 6926: 6922: 6917: 6914: 6910: 6905: 6903: 6899: 6892: 6887: 6884: 6878: 6875: 6874: 6870: 6868: 6865: 6861: 6857: 6853: 6833: 6826: 6822: 6818: 6815: 6812: 6808: 6804: 6800: 6797: 6794: 6790: 6785: 6780: 6776: 6772: 6769: 6765: 6761: 6757: 6754: 6751: 6743: 6739: 6732: 6729: 6726: 6723: 6720: 6714: 6708: 6705: 6698: 6697: 6696: 6695:is given by 6694: 6690: 6686: 6681: 6677: 6673: 6669: 6664: 6662: 6658: 6654: 6650: 6649:heat equation 6646: 6642: 6638: 6631:Heat equation 6630: 6628: 6623: 6619: 6615: 6611: 6607: 6603: 6599: 6595: 6591: 6584: 6580: 6576: 6572: 6568: 6560: 6558: 6556: 6552: 6548: 6544: 6539: 6537: 6533: 6529: 6521: 6519: 6515: 6513: 6509: 6505: 6499: 6497: 6491: 6483: 6478: 6476: 6474: 6469: 6466: 6456: 6454: 6450: 6446: 6441: 6439: 6435: 6431: 6427: 6406: 6401: 6385: 6376: 6370: 6363: 6360: 6353: 6352: 6351: 6350: 6346: 6342: 6321: 6318: 6315: 6309: 6305: 6296: 6292: 6288: 6284: 6279: 6277: 6273: 6269: 6265: 6261: 6253: 6251: 6249: 6245: 6223: 6220: 6214: 6211: 6205: 6199: 6196: 6191: 6187: 6183: 6177: 6174: 6171: 6165: 6158: 6157: 6156: 6154: 6128: 6125: 6112: 6108: 6103: 6099: 6096: 6089: 6085: 6081: 6073: 6068: 6064: 6053: 6049: 6045: 6041: 6037: 6034: 6028: 6024: 6016: 6011: 6007: 6003: 6001: 5995: 5986: 5983: 5977: 5973: 5970: 5964: 5961: 5955: 5952: 5949: 5938: 5935: 5932: 5926: 5923: 5913: 5910: 5903: 5895: 5891: 5886: 5882: 5879: 5875: 5869: 5864: 5860: 5853: 5847: 5844: 5841: 5839: 5833: 5823: 5818: 5814: 5805: 5797: 5794: 5789: 5781: 5778: 5770: 5765: 5761: 5756: 5752: 5749: 5738: 5737: 5736: 5719: 5716: 5708: 5704: 5698: 5693: 5689: 5685: 5679: 5676: 5670: 5644: 5641: 5638: 5632: 5629: 5623: 5620: 5617: 5609: 5605: 5601: 5598: 5593: 5585: 5582: 5574: 5570: 5566: 5560: 5554: 5551: 5544: 5543: 5542: 5540: 5536: 5532: 5528: 5524: 5478: 5469: 5461: 5457: 5451: 5448: 5445: 5437: 5434: 5425: 5417: 5413: 5407: 5404: 5401: 5393: 5390: 5384: 5378: 5371: 5370: 5369: 5366: 5364: 5343: 5328: 5311: 5310: 5309: 5307: 5303: 5299: 5295: 5291: 5287: 5283: 5279: 5275: 5271: 5263: 5261: 5259: 5237: 5234: 5228: 5223: 5219: 5215: 5204: 5201: 5198: 5190: 5186: 5180: 5175: 5171: 5165: 5157: 5154: 5146: 5142: 5136: 5128: 5125: 5119: 5108: 5105: 5102: 5089: 5085: 5081: 5077: 5073: 5070: 5064: 5060: 5052: 5047: 5043: 5034: 5030: 5024: 5019: 5015: 5008: 5000: 4996: 4992: 4988: 4984: 4981: 4977: 4971: 4966: 4962: 4953: 4949: 4943: 4935: 4932: 4926: 4920: 4914: 4907: 4906: 4905: 4876: 4873: 4870: 4857: 4853: 4849: 4845: 4841: 4838: 4832: 4828: 4820: 4815: 4811: 4807: 4805: 4792: 4789: 4786: 4780: 4777: 4762: 4759: 4756: 4749: 4741: 4737: 4733: 4729: 4725: 4722: 4718: 4712: 4707: 4703: 4699: 4697: 4681: 4678: 4675: 4665: 4661: 4657: 4653: 4649: 4642: 4638: 4634: 4630: 4624: 4616: 4613: 4607: 4604: 4601: 4590: 4587: 4584: 4574: 4570: 4566: 4562: 4554: 4550: 4546: 4542: 4536: 4533: 4530: 4526: 4522: 4511: 4508: 4505: 4495: 4491: 4487: 4483: 4477: 4472: 4468: 4464: 4461: 4458: 4456: 4445: 4436: 4433: 4428: 4424: 4418: 4409: 4403: 4400: 4395: 4387: 4384: 4378: 4375: 4372: 4366: 4357: 4354: 4349: 4345: 4339: 4330: 4324: 4321: 4318: 4307: 4304: 4299: 4295: 4288: 4285: 4282: 4279: 4276: 4274: 4266: 4252: 4238: 4232: 4229: 4218: 4217: 4216: 4194: 4191: 4188: 4180: 4176: 4170: 4165: 4161: 4157: 4151: 4148: 4142: 4133: 4119: 4115: 4112: 4108: 4105: 4102: 4099: 4071: 4068: 4060: 4056: 4052: 4049: 4043: 4033: 4029: 4023: 4015: 4011: 4007: 3996: 3993: 3985: 3982: 3979: 3975: 3949: 3945: 3936: 3932: 3928: 3917: 3914: 3906: 3902: 3874: 3870: 3864: 3859: 3855: 3851: 3848: 3838: 3834: 3828: 3823: 3819: 3815: 3807: 3804: 3782: 3778: 3773: 3770: 3766: 3761: 3757: 3736: 3733: 3730: 3727: 3707: 3699: 3694: 3692: 3688: 3672: 3664: 3645: 3639: 3632: 3616: 3540: 3536: 3532: 3529: 3526: 3523: 3462:. If we take 3449: 3446: 3443: 3440: 3420: 3413:Suppose that 3408: 3403: 3396: 3392: 3388: 3384: 3383:Boris Fedosov 3380: 3376: 3375: 3373: 3369: 3365: 3361: 3357: 3353: 3349: 3345: 3344:axial anomaly 3341: 3337: 3333: 3329: 3326: 3322: 3318: 3314: 3313:index theorem 3312: 3307: 3302: 3301:Atiyah (1976) 3299: 3296: 3292: 3288: 3284: 3280: 3276: 3272: 3268: 3265: 3261: 3257: 3253: 3249: 3246: 3242: 3237: 3233: 3229: 3225: 3220: 3216: 3200: 3192: 3188: 3181: 3173: 3169: 3160: 3156: 3147: 3143: 3136: 3129: 3128: 3123: 3119: 3115: 3114: 3110: 3108: 3106: 3102: 3098: 3094: 3090: 3086: 3082: 3077: 3075: 3074:Sullivan 1979 3069: 3067: 3063: 3059: 3055: 3050: 3048: 3044: 3039: 3037: 3033: 3026: 3023: 3022: 3021: 3019: 3015: 3007: 3005: 3002: 3000: 2996: 2995:Kasparov 1972 2992: 2988: 2984: 2977: 2975: 2970: 2969: 2968: 2966: 2962: 2954: 2949: 2947: 2945: 2941: 2922: 2916: 2896: 2893: 2890: 2848: 2845: 2839: 2824: 2818: 2805: 2795: 2789: 2779: 2775: 2752: 2748: 2730: 2727: 2721: 2708: 2702: 2686: 2676: 2670: 2652: 2651: 2650: 2636: 2630: 2627: 2624: 2616: 2608: 2606: 2604: 2600: 2595: 2591: 2589: 2581: 2579: 2574: 2573: 2572: 2570: 2566: 2562: 2558: 2557:index problem 2553: 2551: 2547: 2543: 2539: 2534: 2532: 2528: 2524: 2520: 2516: 2512: 2508: 2504: 2500: 2496: 2492: 2487: 2473: 2470: 2467: 2460: 2442: 2439: 2431: 2428: 2422: 2396: 2390: 2387: 2378: 2375: 2369: 2361: 2355: 2352: 2349: 2343: 2337: 2334: 2326: 2322: 2316: 2308: 2305: 2295: 2294: 2293: 2291: 2272: 2269: 2263: 2256: 2240: 2237: 2217: 2192: 2169: 2163: 2156: 2138: 2132: 2109: 2103: 2080: 2074: 2054: 2049: 2045: 2024: 2019: 2015: 1988: 1982: 1978: 1971: 1965: 1959: 1933: 1927: 1924: 1921: 1916: 1912: 1908: 1905: 1900: 1896: 1889: 1882: 1880: 1856: 1853: 1845: 1841: 1829: 1823: 1817: 1814: 1811: 1804: 1790: 1781: 1775: 1771: 1764: 1758: 1755: 1752: 1744: 1720: 1714: 1708: 1704: 1697: 1691: 1683: 1680: 1677: 1673: 1658: 1655: 1647: 1643: 1639: 1636: 1629: 1628: 1602: 1596: 1593: 1590: 1585: 1581: 1577: 1574: 1569: 1565: 1558: 1552: 1549: 1541: 1538: 1534: 1510: 1504: 1501: 1494: 1479: 1471: 1452: 1446: 1443: 1436: 1435: 1434: 1420: 1412: 1393: 1387: 1384: 1378: 1372: 1369: 1362: 1340: 1334: 1331: 1325: 1319: 1316: 1311: 1307: 1301: 1293: 1290: 1284: 1278: 1269: 1263: 1260: 1254: 1248: 1245: 1240: 1232: 1229: 1219: 1218: 1217: 1203: 1183: 1163: 1143: 1123: 1115: 1107: 1105: 1102: 1098: 1094: 1090: 1086: 1082: 1078: 1074: 1072: 1068: 1060: 1056: 1052: 1048: 1047: 1046: 1044: 1040: 1036: 1032: 1028: 1024: 1020: 1016: 1012: 1008: 1000: 998: 996: 991: 987: 983: 980:′ such that 979: 975: 974:pseudoinverse 971: 970: 965: 960: 958: 954: 950: 943: 936: 932: 928: 924: 920: 916: 912: 908: 904: 900: 896: 892: 887: 885: 869: 866: 863: 841: 836: 832: 828: 825: 822: 817: 812: 808: 804: 782: 778: 755: 750: 746: 742: 739: 736: 731: 726: 722: 713: 708: 706: 702: 684: 680: 657: 653: 645: 633: 629: 611: 607: 584: 580: 572: 560: 542: 538: 534: 531: 528: 523: 519: 515: 510: 506: 502: 499: 496: 491: 487: 478: 474: 456: 452: 448: 445: 442: 437: 433: 424: 420: 416: 408: 403: 399: 395: 391: 387: 384: 381: 377: 373: 369: 366: 363: 359: 355: 352: 351: 347: 342: 339: 336: 333: 330: 326: 323: 320: 317: 313: 310: 307: 304: 301: 298: 295: 292: 288: 284: 281: 278: 275: 271: 268: 265: 262: 261:heat equation 258: 254: 250: 247: 244: 241: 238: 235: 232: 229: 226: 222: 218: 215: 212: 208: 205: 202: 201: 200: 198: 194: 190: 186: 180: 178: 174: 170: 166: 162: 158: 154: 150: 146: 138: 136: 134: 130: 126: 122: 118: 114: 110: 106: 102: 98: 94: 85: 81: 77: 73: 70: 66: 62: 58: 55: 51: 48: 44: 41: 38: 34: 19: 9104: 9082:the original 9060: 9029:. Retrieved 8973: 8944: 8940: 8928: 8901: 8897: 8871: 8867: 8835: 8831: 8795: 8791: 8759: 8755: 8748:Sullivan, D. 8722: 8719:Sullivan, D. 8710: 8707:Singer, I.M. 8671: 8651: 8638: 8632: 8602: 8583: 8567: 8542: 8538: 8507: 8503: 8471:math/9905210 8461: 8457: 8453: 8449: 8431: 8381: 8362: 8358: 8326: 8322: 8282: 8278: 8249: 8245: 8242:Sullivan, D. 8212: 8208: 8205:Sullivan, D. 8175: 8171: 8133: 8096: 8092: 8069: 8047: 8043: 8021: 7989: 7985: 7944: 7914: 7876: 7872: 7833: 7829: 7788: 7784: 7745: 7741: 7705: 7699: 7676: 7672: 7646: 7642: 7610: 7604: 7572: 7566: 7534: 7528: 7496: 7492: 7466: 7462: 7430: 7426: 7423:Segal, G. B. 7403: 7384: 7365: 7341: 7335: 7324: 7313: 7302:. Retrieved 7298: 7289: 7271: 7258: 7222: 7216: 7204: 7192: 7180: 7168: 7161:Teleman 1984 7156: 7149:Teleman 1983 7144: 7132: 7125:Getzler 1983 7120: 7108: 7101:Melrose 1993 7096: 7084: 7072: 7060: 7048: 7036: 7024: 7017:Novikov 1965 7012: 6988: 6976: 6964: 6952: 6940: 6928: 6916: 6886:Witten index 6863: 6859: 6855: 6851: 6849: 6692: 6688: 6684: 6679: 6675: 6671: 6667: 6665: 6657:Melrose 1993 6635:Atiyah, 6634: 6621: 6617: 6613: 6609: 6605: 6601: 6597: 6593: 6589: 6582: 6578: 6574: 6570: 6564: 6554: 6550: 6546: 6542: 6540: 6525: 6516: 6511: 6507: 6503: 6500: 6493: 6470: 6462: 6452: 6448: 6444: 6442: 6437: 6429: 6425: 6423: 6344: 6294: 6290: 6286: 6282: 6281:The bundles 6280: 6267: 6263: 6257: 6247: 6243: 6241: 6150: 5662: 5538: 5530: 5526: 5522: 5493: 5367: 5362: 5360: 5305: 5301: 5297: 5296:of type (0, 5293: 5289: 5285: 5281: 5277: 5269: 5267: 5255: 4903: 4134: 3695: 3412: 3381:proposed by 3371: 3367: 3366:) over the ( 3363: 3359: 3355: 3340:Roman Jackiw 3310: 3309: 3293:. See also: 3290: 3278: 3274: 3271:group action 3263: 3259: 3255: 3251: 3125: 3104: 3096: 3092: 3088: 3084: 3078: 3070: 3062:Teleman 1985 3053: 3051: 3040: 3031: 3029: 3024: 3011: 3003: 2991:Teleman 1983 2987:Teleman 1983 2983:Teleman 1980 2980: 2971: 2965:Teleman 1984 2961:Teleman 1983 2958: 2882: 2612: 2602: 2598: 2596: 2592: 2587: 2584: 2577: 2575: 2568: 2564: 2560: 2556: 2554: 2549: 2545: 2541: 2537: 2535: 2530: 2526: 2522: 2518: 2514: 2510: 2506: 2502: 2498: 2494: 2490: 2488: 2414: 2209: 1526:is equal to 1358: 1216:is given by 1113: 1111: 1100: 1096: 1092: 1088: 1084: 1080: 1076: 1075: 1070: 1066: 1064: 1058: 1054: 1050: 1042: 1038: 1034: 1026: 1022: 1014: 1006: 1004: 989: 985: 981: 977: 973: 967: 963: 961: 956: 952: 945: 938: 934: 930: 926: 922: 918: 914: 906: 902: 894: 890: 888: 883: 711: 709: 707:is nonzero. 704: 700: 631: 630:, of degree 627: 558: 476: 422: 418: 414: 412: 401: 397: 393: 389: 385: 379: 371: 367: 353: 340: 334: 328: 321: 312:Alain Connes 308: 302: 296: 283:Ezra Getzler 279: 266: 257:Vijay Patodi 248: 242: 236: 230: 217:Robion Kirby 203: 181: 169:Armand Borel 142: 120: 116: 99:, proved by 96: 90: 68:Consequences 8874:: 117–152, 8838:: 251–290, 8365:: 111–117, 8315:Getzler, E. 8252:: 181–252, 8099:: 257–360, 7992:: 155–191, 7173:Connes 1986 7137:Witten 1982 7065:Singer 1971 7053:Atiyah 1970 6957:Palais 1965 6921:Kayani 2020 6661:Gilkey 1994 6651:, see e.g. 5521:. Then the 5308:be the sum 3687:Euler class 3352:Higgs field 3350:called the 3342:, used the 3273:of a group 3101:Hilsum 1999 3058:Singer 1971 2974:Atiyah 1970 2569:topological 2255:Euler class 471:, then its 374:are smooth 9123:Categories 9031:January 3, 8997:Hirzebruch 8890:0547.58036 8860:0531.58044 8784:0531.58045 8741:0478.57007 8422:1910.01080 8407:1512.02632 8268:0704.57008 8231:0840.57013 8201:Connes, A. 8194:0759.58047 8161:Connes, A. 8154:0818.46076 8128:Connes, A. 8121:0592.46056 8085:Connes, A. 8078:0149.41102 8014:0332.14003 7974:Fulton, W. 7730:Atiyah, M. 7393:0193.43601 7362:References 7304:2021-02-05 6339:times the 6297:-forms as 6244:projective 3797:such that 3391:Raoul Bott 2588:difference 2536:As usual, 1470:Todd class 969:parametrix 911:jet bundle 479:variables 425:variables 253:Raoul Bott 159:, and the 9067:EMS Press 8918:121137053 8852:121497293 8820:122247909 8676:CiteSeerX 8641:: 298–300 8560:Kirby, R. 8531:Kirby, R. 8496:119708566 8113:122740195 8050:: 56–99, 7943:(1988b), 7912:(1988a), 7866:189831012 7778:115700319 7227:CiteSeerX 7041:Thom 1956 6893:Citations 6827:∗ 6813:− 6801:⁡ 6795:− 6781:∗ 6770:− 6758:⁡ 6744:∗ 6733:⁡ 6727:⁡ 6709:⁡ 6522:Cobordism 6395:∗ 6367:Δ 6364:≡ 6319:− 6215:⁡ 6200:⁡ 6188:∫ 6166:χ 6100:− 6082:− 6065:∏ 6046:− 6038:− 6008:∏ 5991:¯ 5974:⁡ 5956:⁡ 5939:⊗ 5927:⁡ 5883:− 5861:∏ 5848:⁡ 5828:¯ 5819:∗ 5802:Λ 5798:⊗ 5779:− 5762:∑ 5753:⁡ 5690:∏ 5633:χ 5602:⁡ 5583:− 5571:∑ 5555:⁡ 5507:¯ 5504:∂ 5479:⋯ 5476:→ 5462:∗ 5442:Λ 5438:⊗ 5432:→ 5418:∗ 5398:Λ 5394:⊗ 5388:→ 5382:→ 5344:∗ 5338:¯ 5335:∂ 5324:¯ 5321:∂ 5220:∫ 5205:⊗ 5172:∏ 5155:− 5143:∫ 5126:− 5109:⊗ 5082:− 5074:− 5044:∏ 5016:∏ 4993:− 4985:− 4963:∏ 4950:∫ 4933:− 4915:χ 4877:⊗ 4850:− 4842:− 4812:∏ 4793:⊗ 4781:⁡ 4763:⊗ 4734:− 4726:− 4704:∏ 4682:⊗ 4658:− 4650:⋯ 4635:− 4614:− 4605:… 4591:⊗ 4567:− 4547:− 4527:∑ 4512:⊗ 4488:− 4469:∑ 4465:− 4437:⊗ 4429:∗ 4415:Λ 4404:⁡ 4385:− 4376:… 4373:− 4358:⊗ 4350:∗ 4336:Λ 4325:⁡ 4308:⊗ 4300:∗ 4289:⁡ 4283:− 4257:Λ 4253:− 4244:Λ 4233:⁡ 4195:⊗ 4162:∏ 4113:… 4072:⊗ 4053:− 4039:¯ 3997:⊗ 3918:⊗ 3880:¯ 3865:⊕ 3852:⋯ 3849:⊕ 3844:¯ 3829:⊕ 3808:⊗ 3771:… 3640:χ 3591:Λ 3564:Λ 3541:∗ 3498:Λ 3471:Λ 3198:→ 3185:→ 3182:⋯ 3179:→ 3166:→ 3153:→ 3140:→ 3066:Thom 1956 2959:Due to ( 2944:Chow ring 2897:∗ 2825:⋅ 2780:∗ 2753:∗ 2709:⋅ 2634:→ 2440:− 2391:⁡ 2356:⁡ 2350:− 2338:⁡ 2323:∫ 2306:− 2164:σ 2104:σ 2050:∗ 2020:∗ 1928:σ 1917:∗ 1901:∗ 1846:∗ 1838:→ 1830:⊗ 1788:→ 1670:→ 1637:φ 1597:σ 1586:∗ 1570:∗ 1553:⁡ 1539:− 1535:φ 1505:⁡ 1447:⁡ 1388:⁡ 1373:⁡ 1335:⁡ 1320:⁡ 1308:∫ 1291:− 1264:⁡ 1249:⁡ 1230:− 867:≥ 826:⋯ 805:− 740:⋯ 650:∂ 642:∂ 577:∂ 569:∂ 532:… 500:… 446:… 225:René Thom 193:cobordism 9041:cite web 8925:Thom, R. 8649:(1965), 8624:(1965), 8582:(1989), 8566:(1977), 8359:Topology 8351:55438589 8317:(1983), 8209:Topology 8172:Topology 8130:(1994), 8087:(1986), 8068:(1965), 8006:83458307 7980:(1979), 7970:Baum, P. 7836:: 1–62, 7734:Bott, R. 7669:Bott, R. 7639:Bott, R. 7270:(1989), 6871:See also 6567:K-theory 6561:K-theory 6549:) where 6428:, where 6347:is the 5272:to be a 3404:Examples 3219:manifold 3012:Due to ( 2806:→ 2770:↓ 2760:↓ 2687:→ 1627:, where 1077:Example: 1031:cokernel 935:elliptic 701:elliptic 362:manifold 348:Notation 251:Atiyah, 197:K-theory 8963:0683171 8800:Bibcode 8776:8348213 8512:Bibcode 8331:Bibcode 8287:Bibcode 7963:0951895 7934:0951894 7903:0550183 7881:Bibcode 7858:0463358 7838:Bibcode 7815:0650829 7793:Bibcode 7770:0650828 7750:Bibcode 7722:1970721 7693:1970694 7627:1970757 7589:1970756 7551:1970717 7513:1970715 7447:1970716 7412:0420729 6674:, then 6659:) and ( 6643: ( 6465:Â genus 6272:L genus 5300:) with 3661:of the 3629:is the 3243:of the 3217:If the 3121:easier. 2938:is the 2501:) to K( 1877:is the 1741:is the 1468:is the 1409:on the 1087:), and 982:DD′ -1 360:smooth 358:compact 173:Â genus 139:History 9001:Singer 8989:Atiyah 8980: 8961: 8916: 8888: 8858: 8850: 8818: 8782: 8774: 8739: 8729: 8696: 8678: 8659: 8610: 8590: 8494: 8488:121079 8486: 8438: 8389: 8349: 8306: 8266: 8229: 8192: 8152: 8142: 8119: 8111: 8076: 8028: 8012: 8004: 7961: 7951: 7932: 7922: 7901: 7864: 7856: 7813: 7776: 7768: 7720: 7691: 7625: 7587: 7549: 7511: 7445: 7410: 7391: 7348: 7278: 7247: 7229: 6641:Patodi 3236:Singer 3232:Patodi 3228:Atiyah 1176:on an 1049:Index( 1019:kernel 986:D′D -1 473:symbol 356:is a 255:, and 115:, the 95:, the 9101:(PDF) 9025:(PDF) 8914:S2CID 8848:S2CID 8816:S2CID 8772:S2CID 8629:(PDF) 8492:S2CID 8484:JSTOR 8466:arXiv 8417:arXiv 8402:arXiv 8347:S2CID 8168:(PDF) 8109:S2CID 8002:S2CID 7862:S2CID 7774:S2CID 7718:JSTOR 7689:JSTOR 7623:JSTOR 7585:JSTOR 7547:JSTOR 7509:JSTOR 7443:JSTOR 7372:1988b 7368:1988a 6877:(-1)F 6706:index 5552:index 5268:Take 3700:, if 3091:> 2230:is a 1015:index 378:over 341:1994: 335:1990: 322:1989: 309:1986: 303:1984: 297:1983: 280:1983: 267:1977: 249:1973: 243:1972: 237:1971: 231:1969: 204:1965: 111:on a 36:Field 9047:link 9033:2006 8999:and 8993:Bott 8978:ISBN 8727:ISBN 8694:ISBN 8657:ISBN 8608:ISBN 8588:ISBN 8436:ISBN 8387:ISBN 8304:ISBN 8140:ISBN 8026:ISBN 7949:ISBN 7920:ISBN 7696:and 7346:ISBN 7276:ISBN 7245:ISBN 6687:and 6678:and 6645:1973 6637:Bott 6534:and 6463:The 6436:and 6285:and 6258:The 5735:and 5288:and 4534:< 4248:even 3568:even 3475:even 3393:and 3330:The 3234:and 3038:)). 3016:), ( 2985:), ( 2963:), ( 2613:The 2567:and 2555:The 2544:and 2037:and 1156:and 1112:The 984:and 972:(or 917:and 370:and 219:and 167:and 127:and 103:and 63:1963 52:and 8949:doi 8906:doi 8886:Zbl 8876:doi 8872:153 8856:Zbl 8840:doi 8808:doi 8780:Zbl 8764:doi 8737:Zbl 8686:doi 8639:163 8547:doi 8520:doi 8476:doi 8462:149 8452:et 8367:doi 8339:doi 8295:doi 8264:Zbl 8254:doi 8250:163 8227:Zbl 8217:doi 8190:Zbl 8180:doi 8150:Zbl 8117:Zbl 8101:doi 8074:Zbl 8052:doi 8010:Zbl 7994:doi 7990:143 7889:doi 7846:doi 7801:doi 7758:doi 7710:doi 7681:doi 7651:doi 7615:doi 7577:doi 7539:doi 7501:doi 7471:doi 7435:doi 7389:Zbl 7237:doi 6730:Ker 6724:dim 6680:DD* 6676:D*D 6666:If 6663:). 6577:to 6506:), 5599:dim 5537:of 5365:. 5260:). 4261:odd 3665:of 3595:odd 3582:to 3502:odd 3323:of 3020:): 3001:). 2967:): 2883:if 2067:on 1069:of 1033:of 1021:of 997:.) 955:of 901:of 886:s. 599:by 421:in 413:If 392:to 187:at 91:In 9125:: 9103:, 9076:. 9065:, 9059:, 9043:}} 9039:{{ 8995:, 8991:, 8959:MR 8957:, 8945:17 8943:, 8912:, 8900:, 8884:, 8870:, 8854:, 8846:, 8836:58 8834:, 8830:, 8814:, 8806:, 8796:61 8794:, 8778:, 8770:, 8760:58 8758:, 8754:, 8735:, 8692:, 8684:, 8637:, 8631:, 8578:; 8562:; 8543:75 8541:, 8533:; 8518:, 8506:, 8490:, 8482:, 8474:, 8460:, 8363:25 8361:, 8345:, 8337:, 8327:92 8325:, 8321:, 8293:, 8283:15 8281:, 8262:, 8248:, 8240:; 8225:, 8213:33 8211:, 8203:; 8188:, 8176:29 8174:, 8170:, 8148:, 8115:, 8107:, 8097:62 8095:, 8091:, 8048:57 8046:, 8008:, 8000:, 7988:, 7984:, 7976:; 7972:; 7959:MR 7957:, 7930:MR 7928:, 7899:MR 7897:, 7887:, 7877:54 7875:, 7869:, 7860:, 7854:MR 7852:, 7844:, 7834:42 7832:, 7824:; 7811:MR 7809:, 7799:, 7789:28 7787:, 7781:. 7772:, 7766:MR 7764:, 7756:, 7746:19 7744:, 7736:; 7732:; 7716:, 7706:88 7687:, 7677:86 7667:; 7647:72 7645:, 7637:; 7621:, 7611:93 7599:; 7583:, 7573:93 7561:; 7545:, 7535:87 7523:; 7507:, 7497:87 7495:, 7487:; 7467:69 7465:, 7457:; 7441:, 7431:87 7421:; 7408:MR 7374:) 7370:, 7297:. 7266:; 7243:, 7235:, 7003:; 6999:; 6995:; 6901:^ 6798:Tr 6755:Tr 6689:D* 6672:D* 6545:, 6538:. 6371::= 6278:. 6250:. 6212:Td 6197:ch 6155:: 5971:Td 5953:Td 5924:Td 5845:ch 5750:ch 5541:: 5529:, 5523:i' 4778:Td 4401:ch 4322:ch 4286:ch 4230:ch 4132:. 4092:, 3967:, 3693:. 3364:dU 3230:, 2829:ch 2812:Td 2713:ch 2696:Td 2552:. 2527:TX 2515:TY 2507:TX 2503:TY 2499:TX 2486:. 2388:Td 2353:ch 2335:ch 1812:ch 1550:ch 1502:ch 1444:Td 1385:Td 1370:ch 1332:Td 1317:ch 1261:Td 1246:ch 1073:. 1061:). 1059:D* 1041:= 1039:Df 1027:Df 976:) 959:. 944:, 925:, 163:. 135:. 9049:) 9035:. 9003:. 8951:: 8908:: 8902:8 8878:: 8842:: 8810:: 8802:: 8766:: 8688:: 8549:: 8522:: 8514:: 8508:9 8478:: 8468:: 8454:K 8450:L 8425:. 8419:: 8410:. 8404:: 8369:: 8341:: 8333:: 8297:: 8289:: 8256:: 8219:: 8182:: 8103:: 8054:: 7996:: 7891:: 7883:: 7848:: 7840:: 7803:: 7795:: 7760:: 7752:: 7712:: 7683:: 7653:: 7617:: 7579:: 7541:: 7503:: 7473:: 7437:: 7307:. 7239:: 7211:. 7199:. 7187:. 7175:. 7163:. 7151:. 7139:. 7127:. 7115:. 7103:. 7091:. 7079:. 7067:. 7055:. 7043:. 7031:. 7019:. 7007:. 6983:. 6971:. 6959:. 6947:. 6923:. 6911:. 6864:t 6860:t 6856:t 6852:t 6834:) 6823:D 6819:D 6816:t 6809:e 6805:( 6791:) 6786:D 6777:D 6773:t 6766:e 6762:( 6752:= 6749:) 6740:D 6736:( 6721:= 6718:) 6715:D 6712:( 6693:D 6685:D 6668:D 6625:! 6622:i 6618:Y 6614:Y 6610:X 6606:Y 6602:X 6598:Y 6594:Y 6590:X 6586:! 6583:i 6579:Y 6575:X 6571:i 6555:X 6551:V 6547:V 6543:X 6512:X 6510:( 6508:S 6504:X 6453:X 6449:X 6445:D 6438:d 6430:d 6426:E 6407:2 6402:) 6391:d 6386:+ 6382:d 6377:( 6361:D 6345:D 6325:) 6322:1 6316:k 6313:( 6310:k 6306:i 6295:k 6291:X 6287:F 6283:E 6268:k 6264:X 6248:X 6227:) 6224:X 6221:T 6218:( 6209:) 6206:V 6203:( 6192:X 6184:= 6181:) 6178:V 6175:, 6172:X 6169:( 6132:) 6129:X 6126:T 6123:( 6113:j 6109:x 6104:e 6097:1 6090:j 6086:x 6074:n 6069:j 6054:i 6050:x 6042:e 6035:1 6029:i 6025:x 6017:n 6012:i 6004:= 5996:) 5987:X 5984:T 5978:( 5968:) 5965:X 5962:T 5959:( 5950:= 5947:) 5943:C 5936:X 5933:T 5930:( 5917:) 5914:X 5911:T 5908:( 5904:) 5896:j 5892:x 5887:e 5880:1 5876:( 5870:n 5865:j 5857:) 5854:V 5851:( 5842:= 5834:) 5824:X 5815:T 5806:j 5795:V 5790:j 5786:) 5782:1 5776:( 5771:n 5766:j 5757:( 5723:) 5720:X 5717:T 5714:( 5709:i 5705:x 5699:n 5694:i 5686:= 5683:) 5680:X 5677:T 5674:( 5671:e 5648:) 5645:V 5642:, 5639:X 5636:( 5630:= 5627:) 5624:V 5621:, 5618:X 5615:( 5610:p 5606:H 5594:p 5590:) 5586:1 5580:( 5575:p 5567:= 5564:) 5561:D 5558:( 5539:V 5531:V 5527:X 5473:) 5470:X 5467:( 5458:T 5452:2 5449:, 5446:0 5435:V 5429:) 5426:X 5423:( 5414:T 5408:1 5405:, 5402:0 5391:V 5385:V 5379:0 5363:E 5329:+ 5306:D 5302:i 5298:i 5294:V 5290:F 5286:E 5282:V 5278:n 5270:X 5241:) 5238:M 5235:T 5232:( 5229:e 5224:M 5216:= 5213:) 5209:C 5202:M 5199:T 5196:( 5191:i 5187:x 5181:r 5176:i 5166:r 5162:) 5158:1 5152:( 5147:M 5137:r 5133:) 5129:1 5123:( 5120:= 5117:) 5113:C 5106:M 5103:T 5100:( 5090:i 5086:x 5078:e 5071:1 5065:i 5061:x 5053:n 5048:i 5035:i 5031:x 5025:r 5020:i 5009:) 5001:i 4997:x 4989:e 4982:1 4978:( 4972:n 4967:i 4954:M 4944:r 4940:) 4936:1 4930:( 4927:= 4924:) 4921:M 4918:( 4885:) 4881:C 4874:M 4871:T 4868:( 4858:i 4854:x 4846:e 4839:1 4833:i 4829:x 4821:n 4816:i 4808:= 4801:) 4797:C 4790:M 4787:T 4784:( 4771:) 4767:C 4760:M 4757:T 4754:( 4750:) 4742:i 4738:x 4730:e 4723:1 4719:( 4713:n 4708:i 4700:= 4690:) 4686:C 4679:M 4676:T 4673:( 4666:n 4662:x 4654:e 4643:1 4639:x 4631:e 4625:n 4621:) 4617:1 4611:( 4608:+ 4602:+ 4599:) 4595:C 4588:M 4585:T 4582:( 4575:j 4571:x 4563:e 4555:i 4551:x 4543:e 4537:j 4531:i 4523:+ 4520:) 4516:C 4509:M 4506:T 4503:( 4496:i 4492:x 4484:e 4478:n 4473:i 4462:1 4459:= 4446:) 4441:C 4434:M 4425:T 4419:n 4410:( 4396:n 4392:) 4388:1 4382:( 4379:+ 4367:) 4362:C 4355:M 4346:T 4340:2 4331:( 4319:+ 4316:) 4312:C 4305:M 4296:T 4292:( 4280:1 4277:= 4267:) 4239:( 4203:) 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2796:Y 2793:( 2790:K 2776:f 2749:f 2739:) 2735:Q 2731:; 2728:X 2725:( 2722:H 2706:) 2703:X 2700:( 2680:) 2677:X 2674:( 2671:K 2637:Y 2631:X 2628:: 2625:f 2599:X 2578:D 2565:s 2561:D 2550:X 2546:F 2542:E 2538:D 2531:Z 2523:X 2519:Z 2511:Y 2495:Y 2491:X 2474:O 2471:S 2468:B 2443:1 2436:) 2432:X 2429:T 2426:( 2423:e 2400:) 2397:X 2394:( 2382:) 2379:X 2376:T 2373:( 2370:e 2365:) 2362:F 2359:( 2347:) 2344:E 2341:( 2327:X 2317:m 2313:) 2309:1 2303:( 2276:) 2273:X 2270:T 2267:( 2264:e 2241:m 2238:2 2218:X 2193:D 2173:) 2170:D 2167:( 2154:. 2142:) 2139:X 2136:( 2133:S 2113:) 2110:D 2107:( 2084:) 2081:X 2078:( 2075:B 2055:F 2046:p 2025:E 2016:p 1995:) 1992:) 1989:X 1986:( 1983:S 1979:/ 1975:) 1972:X 1969:( 1966:B 1963:( 1960:K 1940:) 1937:) 1934:D 1931:( 1925:, 1922:F 1913:p 1909:, 1906:E 1897:p 1893:( 1890:d 1865:) 1861:Q 1857:; 1854:X 1851:( 1842:H 1834:Q 1827:) 1824:X 1821:( 1818:K 1815:: 1791:X 1785:) 1782:X 1779:( 1776:S 1772:/ 1768:) 1765:X 1762:( 1759:B 1756:: 1753:p 1729:) 1725:Q 1721:; 1718:) 1715:X 1712:( 1709:S 1705:/ 1701:) 1698:X 1695:( 1692:B 1689:( 1684:k 1681:+ 1678:n 1674:H 1667:) 1663:Q 1659:; 1656:X 1653:( 1648:k 1644:H 1640:: 1615:) 1612:) 1609:) 1606:) 1603:D 1600:( 1594:, 1591:F 1582:p 1578:, 1575:E 1566:p 1562:( 1559:d 1556:( 1547:( 1542:1 1514:) 1511:D 1508:( 1492:. 1480:X 1456:) 1453:X 1450:( 1421:X 1397:) 1394:X 1391:( 1382:) 1379:D 1376:( 1344:) 1341:X 1338:( 1329:) 1326:D 1323:( 1312:X 1302:n 1298:) 1294:1 1288:( 1285:= 1282:] 1279:X 1276:[ 1273:) 1270:X 1267:( 1258:) 1255:D 1252:( 1241:n 1237:) 1233:1 1227:( 1204:X 1184:n 1164:F 1144:E 1124:D 1101:D 1097:i 1093:x 1089:D 1085:Z 1083:/ 1081:R 1071:D 1055:D 1051:D 1043:g 1035:D 1023:D 1007:D 990:D 978:D 964:D 957:X 953:x 948:x 946:F 941:x 939:E 931:X 927:F 923:E 919:F 915:E 907:n 903:X 895:X 891:n 884:y 870:2 864:k 842:2 837:k 833:y 829:+ 823:+ 818:2 813:1 809:y 783:i 779:y 756:2 751:k 747:y 743:+ 737:+ 732:2 727:1 723:y 712:k 705:y 685:i 681:x 658:i 654:x 646:/ 632:n 628:y 612:i 608:y 585:i 581:x 573:/ 559:n 543:k 539:y 535:, 529:, 524:1 520:y 516:, 511:k 507:x 503:, 497:, 492:1 488:x 477:k 457:k 453:x 449:, 443:, 438:1 434:x 423:k 419:n 415:D 404:. 402:F 398:E 394:F 390:E 386:D 382:. 380:X 372:F 368:E 354:X 329:S 318:. 20:)

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