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Lectures on Cyclic Cohomology

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Lectures on Cyclic Cohomology Lectures on Cyclic Cohomology Masoud Khalkhali Mathematics Department, University of Western Ontario London ON, Canada Lectures at IMPAN, Warsaw, September 2016 Simons Semester on Noncommutative Geometry 2 Contents 0.1 Index theory in a noncommutative setting . .7 0.2 Elliptic operators . .7 0.3 The Gauss-Bonnet-Chern theorem as an index theorem . .8 0.4 The signature theorem . 11 0.5 Riemann-Roch-Hirzebruch theorem . 12 0.6 An odd index theorem . 12 0.7 Connes' noncommutative index theorem . 13 0.8 Hochschild cohomology . 15 0.9 Hochschild cohomology as a derived functor . 20 0.10 Deformation theory . 27 0.11 Topological algebras . 36 0.12 Hochschild (co)homology computations . 38 0.13 Cyclic Cohomology . 49 0.14 Connes' long exact sequence . 60 0.15 Connes' spectral sequence . 64 0.16 Cyclic cohomology computations . 66 0.17 What is the Chern character? . 79 0.18 Connes{Chern character in K-theory . 80 0.19 Connes{Chern character in K-homology . 90 0.20 Connes' index theorem . 101 Appendices 106 A Compact operators, Fredholm operators, and abstract index the- ory 107 B K-theory 115 Bibliography 119 3 4 Introduction Introduction In September 2016 I gave 5 introductory lectures on cyclic cohomology and some of its applications in IMPAN Warsaw, during the Simons Semester in Noncommu- tative Geometry. The audience consisted of graduate students and postdocs and my task was to introduce them to the subject. The following text is an expanded version of my lectures. In producing these lecture notes I have freely used mate- rial from my book, Basic Noncommutative Geometry. I would like to thank Piotr Hajac for the invitation to give these lectures. 5 6 Introduction Lecture 1: Index theory in a noncommutative setting 0.1 Index theory in a noncommutative setting Lecture 1, September 12, 2016. In this first lecture, we try to answer a fun- damental question: what is index theory and how it relates to noncommutative geometry? We shall see that the need to extend the Atiyah-Singer index theorem beyond its original domain had much influence on the development of noncom- mutative geometry and its basic tools like cyclic cohomology and Connes- Chern character maps. We shall then formulate the first index theorem in a noncom- mutative setting due to Connes [19] that exhibits this point in a very clar way . Most of what we say in these five lectures are geared towards understanding the statement of this noncommutative index theorem and its proof. In the last lecture we shall give some applications of this index theorem. We shall freely use concepts of functional analysis like Fredholm operators, compact operators and p-summable operators, and notions of differential and al- gebraic topology, topological K-theory, and differential geometry. The reader can consult the appendix for abstract Fredholm theory and K-theory. 0.2 Elliptic operators Elliptic differential operators acting on smooth sections of vector bundles over closed manifolds define Fredholm operators on the corresponding Sobolev spaces of sections. Computing the index of such Fredholm operators in terms of topological data is what the index theorem of Atiyah{Singer achieves. Let M be a smooth manifold and let E and F be smooth complex vector bundles on M. Let D : C1(E) ! C1(F ) be a liner differential operator. This means that D is a C-linear map which is locally expressible by a matrix of differential operators. This matrix of course depends on the choice of local coordinates on M and local frames for E and F . The principal symbol of D is defined by replacing differentiation by covectors in the 7 8 Introduction leading order terms of D. The resulting `matrix-valued function' on the cotangent bundle' 1 ∗ ∗ σD 2 C (Hom(π E; π F )) can be shown to be invariantly defined. Here π : T ∗M ! M is the natural projec- tion map of the cotangent bundle. To compute the principal symbol we can proceed as follows. Let D = [Dij], denote the matrix of the differential operators D in some local coordinate system P k for M, and local frames for E and F . Thus Dij = jk|≤d aij(x)@k with d = degree of D. Then σD(x; ξ) = σDij (x; ξ) ; where X 1 σ (x; ξ) = ak (x)( )jkjξk Dij ij i jkj=d 1 jkj k is obtained by replacing the partial derivatives by covectors: @k ! ( i ) ξ : Definition 0.2.1. A differential operator D is called an elliptic operator if for all ∗ x 2 M and all ξ 6= 0 in Tx M, the matrix σD(x; ξ) is invertible. Theorem 0.2.1. (Finiteness theorem) Let M be a smooth and closed manifold and E and F smooth complex vector bundles over M. If D : C1(E) ! C1(F ) is an elliptic operator, then dim(ker D) < 1 and dim(coker D) < 1 In particular, we can define the index of D by index(D) = dim(ker D) − dim(coker D): Let W s(E) denote the Sobolev space of sections of E (roughly speaking, it consists of sections whose `derivatives of order s' are square integrable). The main results of the theory of linear elliptic PDE's show that for each s 2 R, D has a unique extension to a bounded and Fredholm operator D : W s(E) ! W s−d(F ): Moreover the Fredholm index of D is independent of s and coincides with the index defined using smooth sections. We start with a few classic examples of elliptic operators and their indices. 0.3 The Gauss-Bonnet-Chern theorem as an in- dex theorem Let M be a Riemannian manifold, and let d :Ωp(M) ! Ωp+1(M); where Ωp(M) := C1(Vp T ∗M) is the space of p-forms on M, denote the de Rham differential. It is a differential operator but is not elliptic. Let d∗ :Ωp(M) ! Ωp−1(M) denote the adjoint of d. Here d∗ = − ∗ d∗, with ∗ :Ωp(M) ! Ωn−p(M) the Hodge ∗ operator. Consider the differential operator d + d∗ :Ωev(M) −! Ωodd(M); 0.3 The Gauss-Bonnet-Chern theorem as an index theorem 9 where Ωev(M) = ⊕Ω2p(M):This is an elliptic operator. In fact it is easy to find the symbol of d + d∗ : ev ∗ odd ∗ σd+d∗ (x; ξ): ^ (Tx M) ! ^ (Tx M) is given by p σd+d∗ (x; ξ) = −1(ξ ± iξ); where iξ is the interior multiplication by ξ. A simple computation then shows ∗ that σd+d∗ (x; ξ) is invertible for ξ 6= 0 and therefore d + d is an elliptic operator. Consequently, index(d + d∗) is defined. Using Hodge theory, one shows that Proposition 0.3.1. With the notations and assumptions as above index(d + d∗) = χ(M), where χ(M) is the Euler characteristic of M. In fact, with regard to ker(d + d∗):Ωev ! Ωodd, we have (d + d∗)w = 0 if and only if dw = 0 and d∗w = 0 (we can assume ! is homogeneous). Consider the Laplacian on forms ∆ := (d + d∗)2 = dd∗ + d∗d: We have ∗ L 2p ker(d + d )jΩev = ker ∆ = p H (M), where H2p(M) is the space of harmonic 2p forms on M. We need the following theorem. Theorem 0.3.1. (Hodge) Under the previous assumptions, the map w 7! [w] defines an isomorphism of vector spaces p p H (M) ! HdR(M) Remark 1. The space Hp(M) depends on the metric we choose. However, the dimension of this space is a topological invariant and independent of the choice of the metric. Thus we get ∗ P 2p dim ker(d + d )jΩev = p dim HdR(M). Similarly we have: ∗ P 2p+1 dim ker(d + d )jΩodd = p dim HdR (M). This shows that ∗ P p p index(d + d ) = p(−1) dim HdR(M) = χ(M). 10 Introduction Now, we can consider the following question: is there a local formula for χ(M)? More precisely we want to know if there is a naturally defined cohomology class whose integral over M gives the Euler characteristic of M: An affirmative answer to this question is given by the celebrated s Gauss-Bonnet-Chern theorem: Theorem 0.3.2. Let M be a closed oriented Riemannian manifold of dimension 2n. Its Euler characteristic can be expressed as Z χ(M) = (2π)−2n P f(Ω): M Here, working in a coordinate system, Ω = (wij) is an skew symmetric matrix of two forms (the curvature matrix of M), and the Pfaffian P f is an invariant polynomial on the space of skew symmetric 2n × 2n matrices. The Pfaffian is an square root of the determinant in the sense that P f(Ω)2 = det(Ω): The first few are 0 a pf = a: −a 0 2 0 a b c3 6−a 0 d e7 pf 6 7 = af − be + dc: 4−b −d 0 f5 −c −e −f 0 2 3 0 a1 0 0 6−a1 0 b1 0 7 6 7 6 0 −b1 0 a2 7 6 7 6 .. .. 7 pf 6 0 0 −a2 . 7 = a1a2 ··· an: 6 7 6 .. .. 7 6 . bn−1 7 6 7 4 −bn−1 0 an5 −an 0 In general, if A = (aij) is an skew symmetric 2n×2n matrix, its Pfaffian is defined by n 1 X Y pf(A) = sgn(σ) a 2nn! σ(2i−1),σ(2i) σ2S2n i=1 The Gauss-Bonnet-Chern theorem is a prototype of an index theorem. An invariant of a manifold defined as the index of an elliptic operator, the analytic index, is expressed in terms of the integral of a differential form, called a topological index.
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