The index of elliptic operators. I.

*(English, Russian)*Zbl 0164.24001
Ann. Math. (2) 87, 484-530 (1968); Russian translation in Usp. Mat. Nauk 23, No. 5 (143), 99-142 (1968).

The earlier proof of the Atiyah-Singer index theorem as given in the book by R. S. Palais [Seminar on the Atiyah-Singer index theorem. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. Princeton, N. J.: Princeton University Press (1965, Zbl 0137.17002)] used cobordism theory and was in this respect modelled on the proof of the Riemann-Roch theorem due to the reviewer [New topological methods in algebraic geometry. Berlin-Göttingen-Heidelberg: Springer-Verlag (1956; Zbl 0070.16302)]. This proof did not lend itself to certain generalizations (for example in the equivariant case) because the corresponding cobordism theories are not known.

The index theorem of the paper under review includes the equivariant case: Suppose a compact Lie group \(G\) operates differentiably on a compact differentiable manifold \(X\) such that the action is compatible with a linear elliptic problem an \(X\). Then the index of this elliptic problem is an element of the representation ring \(R(G)\). In the simplest case the elliptic problem is given by an elliptic linear differential operator \(D:C^\infty(E)\to C^\infty(F)\) where \(E\), \(F\) are complex vector bundles over \(X\). The group \(G\) operates on the kernel and on the cokernel of \(D\). Thus we have two finite-dimensional representations of \(G\). The difference of the two representations as elements of \(R(G)\) is the index of the elliptic problem. The symbol of such an elliptic problem is an element of \(K(BX,SX)\) if one forgets about the \(G\)-action. (For the notation see the review of the book of Palais, loc. cit.) If a \(G\)-action is given, then equivariant \(K\)-theory must be used and the symbol is an element of \(K_G(BX,SX)\). The authors define \(K\) and \(K_6\) for a locally compact space as the corresponding reduced groups of the one-point-cpactification of the space. With this understanding we have \(K_G(BX,SX)= K_G(TX)\) where \(TX\) is the total space of the covariant tangent bundle. Using “enough operators” (pseudo-differential operators) and homotopy properties which ensure that the index depends only on the symbol the authors define the analytical index \(a\text{-ind}: K_G(TX)\to R(G)\). This is a homomorphism of \(R(G)\)-modules and has by its very definition the property that the index of an elliptic problem equals \(a\)-ind of its symbol. The definition of an \(R(G)\)-homomorphism \(t\text{-ind}: K_G(TX)\to R(G)\) in topological terms is given in §3. The “index theorem” is the main theorem 6.7 of the paper and asserts that \(a\)-ind and the topological index \(t\)-ind coincide. For the definition of \(t\)-ind and for the investigation of properties of \(a\)-ind and \(t\)-ind the following construction is fundamental. Suppose \(X,Y\) are \(G\)-manifolds with \(X\subset Y\) and \(X\) compact. Then \(TX\subset TY\) and the normal bundle \(W\) of \(TX\subset TY\) equals \(N\oplus N\) lifted to \(TX\) where \(N\) is the normal bundle of \(X\) in \(Y\). Since \(N\oplus N\) carries a complex structure, so also does \(W\). The de Rham complex of the exterior powers of \(W\) can be tensoredith any complex of vector bundles over \(TX\) lifted to \(W\) which has compact support in \(TX\). The result is a complex of veetor bundles over \(W\) with compact support, because the de Rham complex is canonically trivialized outside a zero-section of \(W\). In this way we get a homomorphism \(K_G(TX)\to K_G(W)\). Since \(W\) may be regarded as a tubular neighborhood of \(TX\) in \(TY\) and since the one-point-compactification of \(TY\) maps onto the one-point-compactification of \(W\), we have a homomorphism \(K_G(W)\to K_G(TY)\). The composition is an \(R(G)\)-homomorphism \(i_!: K_G(TX)\to K_G(TY)\) where \(i:X\to Y\) denotes the embedding.

(Of course, many details are omitted in this review. Constructions using the alternating sum of the exterior powers occur in several earlier papers of Atiyah and others.) To define \(t\)-ind for a compact differentiable \(G\)-manifold \(X\) we embed \(X\) in a real representation space \(E\) of \(G\). This is always possible [R. S. Palais, J. Math. Mech. 6, 73–678 (1957; Zbl 0086.02603)]. Let \(i\) be the embedding. We have \(i_!: K_G(TX)\to K_G(TE)\). Under the embedding \(j\) of the origin \(P\) in \(E\) we have \(j_!:K_G(TP)= K_G(P)= R(G)\to K_G(TE)\) \(j_!\) is an isomorphism. This is a special rase of the equivariant form of the Bott periodicity theorem [M. F. Atiyah and it D. W. Anderson, K-theory. With reprints of M. F. Atiyah: Power operations in K-theory. New York-Amsterdam: W. A. Benjamin, Inc. (1967; Zbl 0159.53302); M. F. Atiyah, Q. J. Math., Oxf. II. Ser. 19, 113–140 (1968; Zbl 0159.53501)]. \(t\)-ind is defined by \(j_!\cdot(t\text{-ind})=i_!\). The authors show that the definiton is independent of the choice of the embedding. \(t\)-ind is the identity of \(R(G)\) if \(X\) is a point (A1) and the diagram

\[ \begin{matrix} K_G(TX) &\overset {i_!}\longrightarrow &K_G(T_Y)\\ \underset{t\text{-ind}}{\qquad \searrow} && \underset{t\text{-ind}}{\swarrow\qquad }\\ &R(G) \end{matrix} \tag{A2} \]

commutes for any inclusion \(i:X\to Y\) with \(X,Y\) compact \(G\)-manifolds. An index function ind is given if we have for every compact differentiable manifold \(X\) an \(R(G)\)-homomorphism \(K_G(TX)@>\text{ind}>>R(G)\). If such an index function ind satisfies (A1) and (A2) then \(\text{ind}=t\text{-ind}\) (proposition 4.1). For the analytical index (A1) is trivial. To prove the main theorem, axiom (A2) has to be checked for the analytical index. This is not easy because for an operator \(D\) an \(X\) with symbol \(\gamma(D)\) we have to construct an operator on \(Y\) with symbol \(i_!\gamma(D)\) and show that the index of this new operator equals the index of \(D\). This construction is the essential analytical part of the proof of the main theorem. “Once this has been done, we can take \(Y\) to be a sphere, and the general index theorem is reduced to the case of operators on the sphere. For these the problem is easily solved.” At this point one recognizes that the whole proof has “in spirit, at least” much in common with Grothendieck’s proof of the Riemann-Roch theorem. Since it is difficult to verify (A2) directly it is shown that certain axioms (B1), (B2”) and (B3) imply (A2) for any index function. In §8 the axioms (B1) and (B2”) are proved for the analytical index; in §9 the axiom (B3) is proved. A special case of (B3) is the behaviour of the index function if one takes the cartesian product of two \(G\)-manifolds with elliptic problems. The analytical index behaves multiplicatively in this case. (B3) generalizes this multiplicative property to differentiable fibre bundles. In this case the bundle of “indices along the fibres” may not be trivial over the bare and enters essentially in the formulation. (B1) is an excision axiom: Let \(U\) be a (non-compact) \(G\)-manifold and \(j:U\to X\), \(j':U\to X'\) two open \(G\)-embeddings into compact \(G\)-manifolds \(X,X'\). Then the following diagram commutes.

\[ \begin{matrix} && K_G(TX)\\ &\overset {j^*}{\;\nearrow} && \overset\text{ind}{\searrow\;\;}\\ K_G(TU) & & & & R(G)\\ &\underset {j^*}{\;\searrow} && \underset\text{ind}{\nearrow\;\;}\\ && K_G(TX') \end{matrix} \]

Observe that the one-point-compactifications of \(TX\) and \(TX'\) map onto the one-point conpactification of \(TU\). By these maps \(j^*\) and \(j^{\prime*}\) are induced. The excision property of the analytical index was observed by R. T. Seeley [Trans. Am. Math. Soc. 117, 167–204 (1965; Zbl 0135.37102)]. The axiom (B2”) is a normalisation axiom for certain operators on \(S^1\) and \(S^2\). Information on operators on other spheres follows by using excision and the multiplicative property. The idea of proof for the essential property (A2) of the analytical index is sketched by the authors as follows (§1): let \(i:X\to Y\) be an inclusion of compact manifolds (we forget the \(G\)-action). Let \(U\) be a tubular neighborhood of \(X\) and \(Z\) its double. Then the excision property (B1) shows that \(\operatorname{ind} i_! A= \operatorname{ind} k_! A\) where \(A\in(TX)\) and \(k:X\to Z\) is the inclusion in the double. \(Z\) is fibred over \(X\) by spheres. The multiplicative property and the information on spheres gives the desired result \(\operatorname{ind} i_! A= \operatorname{ind} A\). This paper contains an impressive amount of analysis. The theory of pseudo-differential operators (Hörmander, Kohn-Nirenberg, Seeley) is essential to have “enough operators” to realize all elements of \(K_G(TX)\) as symbols and to carry through all constructions. The analytical index was “calculated” in this paper by topological terms (topological index defined by \(K\)-theory). In the following papers II and III this topological index will be interpreted in two steps. In II the topological index is expressed in terms of fixed-point sets of \(G\). This is done in \(K\)-theory. It leads to a general “Lefschetz fixed-point theorem” where the fixed-point set of an element of \(G\) is a disjoint union of submanifolds. For isolated fixed points this is contained in the fixed point theorem of M. F. Atiyah and R. Bott [Ann. Math. (2) 86, 374–407 (1967; Zbl 0161.43201)], a theorem on differentiable maps \(g\) with isolated simple fixed points, where \(g\) need not be invertible. In III the result is reformulated in cohomological terms. If \(G\) is the identity, this gives the index theorem in its well-known cohomological form (Palais, loc. cit.). In general, it gives the cohomological form of the fixed-point theorem.

The index theorem of the paper under review includes the equivariant case: Suppose a compact Lie group \(G\) operates differentiably on a compact differentiable manifold \(X\) such that the action is compatible with a linear elliptic problem an \(X\). Then the index of this elliptic problem is an element of the representation ring \(R(G)\). In the simplest case the elliptic problem is given by an elliptic linear differential operator \(D:C^\infty(E)\to C^\infty(F)\) where \(E\), \(F\) are complex vector bundles over \(X\). The group \(G\) operates on the kernel and on the cokernel of \(D\). Thus we have two finite-dimensional representations of \(G\). The difference of the two representations as elements of \(R(G)\) is the index of the elliptic problem. The symbol of such an elliptic problem is an element of \(K(BX,SX)\) if one forgets about the \(G\)-action. (For the notation see the review of the book of Palais, loc. cit.) If a \(G\)-action is given, then equivariant \(K\)-theory must be used and the symbol is an element of \(K_G(BX,SX)\). The authors define \(K\) and \(K_6\) for a locally compact space as the corresponding reduced groups of the one-point-cpactification of the space. With this understanding we have \(K_G(BX,SX)= K_G(TX)\) where \(TX\) is the total space of the covariant tangent bundle. Using “enough operators” (pseudo-differential operators) and homotopy properties which ensure that the index depends only on the symbol the authors define the analytical index \(a\text{-ind}: K_G(TX)\to R(G)\). This is a homomorphism of \(R(G)\)-modules and has by its very definition the property that the index of an elliptic problem equals \(a\)-ind of its symbol. The definition of an \(R(G)\)-homomorphism \(t\text{-ind}: K_G(TX)\to R(G)\) in topological terms is given in §3. The “index theorem” is the main theorem 6.7 of the paper and asserts that \(a\)-ind and the topological index \(t\)-ind coincide. For the definition of \(t\)-ind and for the investigation of properties of \(a\)-ind and \(t\)-ind the following construction is fundamental. Suppose \(X,Y\) are \(G\)-manifolds with \(X\subset Y\) and \(X\) compact. Then \(TX\subset TY\) and the normal bundle \(W\) of \(TX\subset TY\) equals \(N\oplus N\) lifted to \(TX\) where \(N\) is the normal bundle of \(X\) in \(Y\). Since \(N\oplus N\) carries a complex structure, so also does \(W\). The de Rham complex of the exterior powers of \(W\) can be tensoredith any complex of vector bundles over \(TX\) lifted to \(W\) which has compact support in \(TX\). The result is a complex of veetor bundles over \(W\) with compact support, because the de Rham complex is canonically trivialized outside a zero-section of \(W\). In this way we get a homomorphism \(K_G(TX)\to K_G(W)\). Since \(W\) may be regarded as a tubular neighborhood of \(TX\) in \(TY\) and since the one-point-compactification of \(TY\) maps onto the one-point-compactification of \(W\), we have a homomorphism \(K_G(W)\to K_G(TY)\). The composition is an \(R(G)\)-homomorphism \(i_!: K_G(TX)\to K_G(TY)\) where \(i:X\to Y\) denotes the embedding.

(Of course, many details are omitted in this review. Constructions using the alternating sum of the exterior powers occur in several earlier papers of Atiyah and others.) To define \(t\)-ind for a compact differentiable \(G\)-manifold \(X\) we embed \(X\) in a real representation space \(E\) of \(G\). This is always possible [R. S. Palais, J. Math. Mech. 6, 73–678 (1957; Zbl 0086.02603)]. Let \(i\) be the embedding. We have \(i_!: K_G(TX)\to K_G(TE)\). Under the embedding \(j\) of the origin \(P\) in \(E\) we have \(j_!:K_G(TP)= K_G(P)= R(G)\to K_G(TE)\) \(j_!\) is an isomorphism. This is a special rase of the equivariant form of the Bott periodicity theorem [M. F. Atiyah and it D. W. Anderson, K-theory. With reprints of M. F. Atiyah: Power operations in K-theory. New York-Amsterdam: W. A. Benjamin, Inc. (1967; Zbl 0159.53302); M. F. Atiyah, Q. J. Math., Oxf. II. Ser. 19, 113–140 (1968; Zbl 0159.53501)]. \(t\)-ind is defined by \(j_!\cdot(t\text{-ind})=i_!\). The authors show that the definiton is independent of the choice of the embedding. \(t\)-ind is the identity of \(R(G)\) if \(X\) is a point (A1) and the diagram

\[ \begin{matrix} K_G(TX) &\overset {i_!}\longrightarrow &K_G(T_Y)\\ \underset{t\text{-ind}}{\qquad \searrow} && \underset{t\text{-ind}}{\swarrow\qquad }\\ &R(G) \end{matrix} \tag{A2} \]

commutes for any inclusion \(i:X\to Y\) with \(X,Y\) compact \(G\)-manifolds. An index function ind is given if we have for every compact differentiable manifold \(X\) an \(R(G)\)-homomorphism \(K_G(TX)@>\text{ind}>>R(G)\). If such an index function ind satisfies (A1) and (A2) then \(\text{ind}=t\text{-ind}\) (proposition 4.1). For the analytical index (A1) is trivial. To prove the main theorem, axiom (A2) has to be checked for the analytical index. This is not easy because for an operator \(D\) an \(X\) with symbol \(\gamma(D)\) we have to construct an operator on \(Y\) with symbol \(i_!\gamma(D)\) and show that the index of this new operator equals the index of \(D\). This construction is the essential analytical part of the proof of the main theorem. “Once this has been done, we can take \(Y\) to be a sphere, and the general index theorem is reduced to the case of operators on the sphere. For these the problem is easily solved.” At this point one recognizes that the whole proof has “in spirit, at least” much in common with Grothendieck’s proof of the Riemann-Roch theorem. Since it is difficult to verify (A2) directly it is shown that certain axioms (B1), (B2”) and (B3) imply (A2) for any index function. In §8 the axioms (B1) and (B2”) are proved for the analytical index; in §9 the axiom (B3) is proved. A special case of (B3) is the behaviour of the index function if one takes the cartesian product of two \(G\)-manifolds with elliptic problems. The analytical index behaves multiplicatively in this case. (B3) generalizes this multiplicative property to differentiable fibre bundles. In this case the bundle of “indices along the fibres” may not be trivial over the bare and enters essentially in the formulation. (B1) is an excision axiom: Let \(U\) be a (non-compact) \(G\)-manifold and \(j:U\to X\), \(j':U\to X'\) two open \(G\)-embeddings into compact \(G\)-manifolds \(X,X'\). Then the following diagram commutes.

\[ \begin{matrix} && K_G(TX)\\ &\overset {j^*}{\;\nearrow} && \overset\text{ind}{\searrow\;\;}\\ K_G(TU) & & & & R(G)\\ &\underset {j^*}{\;\searrow} && \underset\text{ind}{\nearrow\;\;}\\ && K_G(TX') \end{matrix} \]

Observe that the one-point-compactifications of \(TX\) and \(TX'\) map onto the one-point conpactification of \(TU\). By these maps \(j^*\) and \(j^{\prime*}\) are induced. The excision property of the analytical index was observed by R. T. Seeley [Trans. Am. Math. Soc. 117, 167–204 (1965; Zbl 0135.37102)]. The axiom (B2”) is a normalisation axiom for certain operators on \(S^1\) and \(S^2\). Information on operators on other spheres follows by using excision and the multiplicative property. The idea of proof for the essential property (A2) of the analytical index is sketched by the authors as follows (§1): let \(i:X\to Y\) be an inclusion of compact manifolds (we forget the \(G\)-action). Let \(U\) be a tubular neighborhood of \(X\) and \(Z\) its double. Then the excision property (B1) shows that \(\operatorname{ind} i_! A= \operatorname{ind} k_! A\) where \(A\in(TX)\) and \(k:X\to Z\) is the inclusion in the double. \(Z\) is fibred over \(X\) by spheres. The multiplicative property and the information on spheres gives the desired result \(\operatorname{ind} i_! A= \operatorname{ind} A\). This paper contains an impressive amount of analysis. The theory of pseudo-differential operators (Hörmander, Kohn-Nirenberg, Seeley) is essential to have “enough operators” to realize all elements of \(K_G(TX)\) as symbols and to carry through all constructions. The analytical index was “calculated” in this paper by topological terms (topological index defined by \(K\)-theory). In the following papers II and III this topological index will be interpreted in two steps. In II the topological index is expressed in terms of fixed-point sets of \(G\). This is done in \(K\)-theory. It leads to a general “Lefschetz fixed-point theorem” where the fixed-point set of an element of \(G\) is a disjoint union of submanifolds. For isolated fixed points this is contained in the fixed point theorem of M. F. Atiyah and R. Bott [Ann. Math. (2) 86, 374–407 (1967; Zbl 0161.43201)], a theorem on differentiable maps \(g\) with isolated simple fixed points, where \(g\) need not be invertible. In III the result is reformulated in cohomological terms. If \(G\) is the identity, this gives the index theorem in its well-known cohomological form (Palais, loc. cit.). In general, it gives the cohomological form of the fixed-point theorem.

Reviewer: Friedrich Hirzebruch (Bonn)

##### MSC:

58J20 | Index theory and related fixed-point theorems on manifolds |