Atiyah's L2-Index Theorem: Index of Elliptic Operators on Manifolds, Papers of Calculus

Download Atiyah's L2-Index Theorem: Index of Elliptic Operators on Manifolds and more Papers Calculus in PDF only on Docsity! ATIYAH’S L2-INDEX THEOREM INDIRA CHATTERJI AND GUIDO MISLIN 1. Introduction The L2-Index Theorem of Atiyah [1] expresses the index of an el- liptic operator on a closed manifold M in terms of the G-equivariant index of some regular covering M̃ of M , with G the group of covering transformations. Atiyah’s proof is analytic in nature. Our proof is algebraic and involves an embedding of a given group into an acyclic one, together with naturality properties of the indices. 2. Review of the L2-Index Theorem The main reference for this section is Atiyah’s paper [1]. All man- ifolds considered are smooth Riemannian, without boundary. Cov- ering spaces of manifolds carry the induced smooth and Riemannian structure. Let M be a closed manifold and let E, F denote two com- plex (Hermitian) vector bundles over M . Consider an elliptic pseudo- differential operator D : C∞(M, E)→ C∞(M, F ) acting on the smooth sections of the vector bundles. One defines its space of solutions SD = {s ∈ C∞(M, E) |Ds = 0}. The complex vector space SD has finite dimension (see [13]), and so has SD∗ the space of solutions of the adjoint D ∗ of D where D∗ : C∞(M, F )→ C∞(M, E) is the unique continuous linear map satisfying 〈Ds, s′〉 = ∫ M 〈Ds(m), s′(m)〉F dm = 〈s, D ∗s′〉 = ∫ M 〈s(m), D∗s′(m)〉E dm for all s ∈ C∞(M, E), s′ ∈ C∞(M, F ). One now defines the index of D as follows: Index(D) = dimC(SD)− dimC(SD∗) ∈ Z. Date: July 2, 2003. 1 2 INDIRA CHATTERJI AND GUIDO MISLIN An explicit formula for Index(D) is given by the famous Atiyah-Singer Theorem (cf. [2]). Consider a not necessarily connected, regular cover- ing π : M̃ →M with countable covering transformation group G. The projection π can be used to define an elliptic operator D̃ := π∗(D) : C∞c (M̃, π ∗E)→ C∞c (M̃, π∗F ). Denote by SD̃ the closure of {s ∈ C∞c (M̃, π∗E)|D̃s = 0} in L2(M̃, π∗E). Let D̃∗ denote the adjoint of D̃. The space SD̃ is not necessarily fi- nite dimensional, but being a closed G-invariant subspace of the L2- completion L2(M̃, π∗E) of the space of smooth sections with compact supports C∞c (M̃, π ∗E), its von Neumann dimension is therefore defined as follows. Write N(G) = {P : `2(G)→ `2(G) bounded and G-invariant} for the group von Neumann algebra of G, where G acts on `2(G) via the right regular representation. Then SD̃ is a finitely generated Hilbert G-module and hence can be represented by an idempotent matrix P = (pij) ∈ Mn(N(G)) (recall that a finitely generated Hilbert G-module is isometrically G-isomorphic to a Hilbert G-subspace of the Hilbert space `2(G)n for some n ≥ 1, see [9]). One then sets dimG(SD̃) = n∑ i=1 〈pii(e), e〉 = κ(P ) ∈ R, where by abuse of notation e denotes the element in `2(G) taking value 1 on the neutral element e ∈ G and 0 elsewhere (see Eckmann’s survey [9] on L2-cohomology for more on von Neumann dimensions). The map κ : Mn(N(G)) → C is the Kaplansky trace. One defines the L2-index of D̃ by IndexG(D̃) = dimG(SD̃)− dimG(SD̃∗). We can now state Atiyah’s L2-Index Theorem. Theorem 2.1 (Atiyah [1]). For D an elliptic pseudo-differential oper- ator on a closed Riemannian manifold M Index(D) = IndexG(D̃) for any countable group G and any lift D̃ of D to a regular G-cover M̃ of M . In particular, the L2-index of D̃ is always an integer, even though it is a priori given in terms of real numbers. The following serves as an illustration of the L2-Index Theorem. Example 2.2 (Atiyah’s formula [1]). Let Ω• be the de Rham complex of complex valued differential forms on the closed connected manifold M and consider the de Rham differential D = d+d∗ : Ωev → Ωodd. Let π : M̃ →M be the universal cover of M so that G = π1(M). Then L2-INDEX THEOREM 5 4. On K-homology Many ideas of this section go back to the seminal article by Baum and Connes [3], which has been circulating for many years and has only recently been published. An elliptic pseudo-differential operator D on the closed manifold M can also be used to define an element [D] ∈ K0(M), the K-homology of M , and according to Baum and Douglas [4], all elements of K0(M) are of the form [D]. The index defined in Section 2 extends to a well- defined homomorphism (cf. [4]) Index : K0(M)→ Z, such that Index([D]) = Index(D). On the other hand, the projection pr : M → {pt} induces, after identifying K0({pt}) with Z, a homomor- phism pr∗ : K0(M)→ Z, (∗) which, as explained in [4], satisfies pr∗([D]) = Index([D]). More generally (cf. [4]), for a not necessarily finite CW-complex X, every x ∈ K0(X) is of the form f∗[D] for some f : M → X, and K0(X) is obtained as a colimit over K0(Mα), where the Mα form a directed system consisting of closed Riemannian manifolds (these ho- mology groups K0(X) are naturally isomorphic to the ones defined us- ing the Bott spectrum; sometimes, they are referred to as K-homology groups with compact supports). The index map from above extends to a homomorphism Index : K0(X)→ Z, such that Index(x) = Index([D]) if x = f∗[D], with f : M → X. We now consider the case of X = BG, the classifying space of the discrete group G, and obtain thus for any f : M → BG a commutative diagram K0(M) Index−−−→ Z f∗ y ∥∥∥ K0(BG) Index−−−→ Z. Note that (∗) from above implies the following naturality property for the index homomorphism. Lemma 4.1. For any homomorphism ϕ : H → G one has a commu- tative diagram K0(BH) Index−−−→ Z (Bϕ)∗ y ∥∥∥ K0(BG) Index−−−→ Z. 6 INDIRA CHATTERJI AND GUIDO MISLIN  We now turn to the L2-index of Section 2. It extends to a homomor- phism IndexG : K0(BG)→ R as follows. Each x ∈ K0(BG) is of the form f∗(y) for some y = [D] ∈ K0(M), f : M → BG, M a closed smooth manifold and D an elliptic operator on M . Let D̃ be the lifted operator to M̃ , the G-covering space induced by f : M → BG. Then put IndexG(x) := IndexG(D̃). One checks that IndexG(x) is indeed well-defined, either by direct com- putation, or by identifying it with τ(x), where τ denotes the compos- ite of the assembly map K0(BG) → K0(C∗r G) with the natural trace K0(C ∗ r G)→ R (for this latter point of view, see Higson-Roe [10]; for a discussion of the assembly map see e.g. Kasparov [12], or Valette [14]). The following naturality property of this index map is a consequence of Lemma 3.1. Lemma 4.2. For H < G the following diagram commutes K0(BH) IndexH−−−−→ Ry ∥∥∥ K0(BG) IndexG−−−−→ R.  Atiyah’s L2-Index Theorem 2.1 for a given G can now be expressed as the statement (as already observed in [10]) IndexG = Index : K0(BG)→ R. 5. Algebraic proof of Atiyah’s L2-index theorem Recall that a group A is said to be acyclic if H∗(BA, Z) = 0 for ∗ > 0. For G a countable group, there exists an embedding G → AG into a countable acyclic group AG. There are many constructions of such a group AG available in the literature, see for instance Kan-Thurston [11, Proposition 3.5], Berrick-Varadarajan [5] or Berrick-Chatterji-Mislin [6]; these different constructions are to be compared in Berrick’s forth- coming work [7]. It follows that the suspension ΣBAG is contractible, and therefore the inclusion {e} → AG induces an isomorphism K0(B{e}) ∼=−→ K0(BAG). Our strategy is as follows. We show that the Atiyah L2-Index The- orem holds in the special case of acyclic groups, and finish the proof combining the above embedding of a group into an acyclic group. L2-INDEX THEOREM 7 Proof of Theorem 2.1. If a group A is acyclic, the equation IndexA = Index follows from the diagram K0(BA) IndexA−−−−→ R Index←−−− K0(BA) ∼= x x ∼=x K0(B{e}) Index{e}−−−−−→∼= Z Index←−−−∼= K0(B{e}) because Index{e} = Index on the bottom line. For a general group G, consider an embedding into an acyclic group AG and complete the proof by using Lemma 3.1, together with Lemmas 4.1 and 4.2.  References [1] M. F. Atiyah. Elliptic operators, discrete groups and von Neumann algebras. Astérisque 32-3 (1976), 43–72. [2] M. F. Atiyah and I. M. Singer. The index of elliptic operators III. Ann. of Math 87 (1968), 546–604. [3] P. Baum and A. Connes. K-theory for Lie groups and foliations. Enseign. Math. (2) 46 (2000), no. 1-2, 3–42. [4] P. Baum and R. Douglas. K-homology and index theory. Proceedings of Sym- posia in Pure Mathematics, 38, Part 1 (1982), 117–173. [5] A. J. Berrick and K. Varadarajan. Binate towers of groups. Arch. Math. 62 (1994), 97–111. [6] A. J. Berrick, I. Chatterji and G. Mislin. From acyclic groups to the Bass Conjecture for amenable groups. Submitted for publication 2002. [7] A. J. Berrick. The acyclic group dichotomy. Preprint in preparation. [8] W. Dicks and T. Schick. The spectral measure of certain elements of the com- plex group ring of a wreath product. Geometriae Dedicata 93 (2002), 121–137. [9] B. Eckmann. Introduction to l2-methods in topology: reduced l2-homology, har- monic chains, l2-Betti numbers. Notes prepared by Guido Mislin. Israel J. Math. 117 (2000), 183–219. [10] N. Higson and J. Roe. Analytic K-Homology. Oxford Mathematical Mono- graphs, Oxford University Press, 2000. [11] D. M. Kan and W. P. Thurston. Every connected space has the homology of a K(π, 1). Topology 15 (1976), 253–258. [12] G. Kasparov. K-theory, group C*-algebras, and higher signatures (Conspec- tus). Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), 101–146, London Math. Soc. Lecture Note Ser., 226, Cambridge Univ. Press, 1995. [13] Y. P. Solovyov and E. V. Troitsky. C∗-algebras and elliptic operators in differ- ential topology. Translated from the 1996 Russian original by Troitsky. Trans- lations of Mathematical Monographs, 192. American Mathematical Society, Providence, RI, 2001. x+213 pp. [14] A. Valette. Introduction to the Baum-Connes Conjecture. Notes taken by Indira Chatterji. With an appendix by Guido Mislin. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2002.