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Algebraic K-Theory, Periodic Cyclic Homology, and the Connes-Moscovici Index Theorem

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Algebraic K-Theory, Periodic Cyclic Homology, and the Connes-Moscovici Index Theorem Algebraic K-theory, periodic cyclic homology, and the Connes-Moscovici Index Theorem Master's thesis by Bram Mesland under supervision of prof dr. N.P. Landsman University of Amsterdam, Faculty of Science Korteweg-de Vries Institute for Mathematics Plantage Muidergracht 24, 1018 TV Amsterdam June 20th, 2005 2 3 Does it matter that this waste of time is what makes a life for you? -Frank Zappa 4 5 Algebraic K-theory, cyclic homology, and the Connes-Moscovici Index Theorem Abstract. We develop algebraic K-theory and cyclic homology from scratch. The boundary map in periodic cyclic cohomology is shown to be well-behaved with respect to the external product. Then we prove that the Chern-Connes character induces a natural transformation from the exact sequence in lower algebraic K-theory to the exact sequence in periodic cyclic homology. Using this, the Gohberg-Krein index theorem is easily derived. Finally, we prove the Connes-Moscovici index theorem, closely following Nistor in [20]. Keywords: Algebraic K-theory, cyclic homology, Chern-Connes char- acter, index theorem, noncommutative geometry. 6 Contents Introduction 9 1 Lower algebraic K-theory 15 1.1 Projective modules . 15 1.2 Grothendieck's K0 . 18 1.3 Idempotents . 20 1.4 Whitehead's K1 . 23 1.5 Relative K-theory . 25 1.6 Excision . 30 1.7 Topological K-theory . 33 1.8 C¤-algebras and index theorems . 38 2 Cyclic homology 41 2.1 The simplicial and cyclic categories . 41 2.2 Cyclic modules . 47 2.3 Hochschild homology . 49 2.4 Cyclic homology . 54 2.5 Periodic and negative cyclic homology . 58 2.6 Normalization and excision . 62 2.7 Di®erential forms . 66 n 2.8 Cyclic cohomology and Ext¤ . 70 2.9 Products . 74 2.10 Discrete groups . 82 3 The Index Formula 87 3.1 Pairings and the Fredholm index . 87 3.2 The boundary map . 91 3.3 The Chern Character . 93 3.4 The universal extension . 99 3.5 Some remarks on the topological case . 104 3.6 The Gohberg-Krein theorem . 106 7 8 CONTENTS 4 The Connes-Moscovici theorem 109 4.1 The Atiyah-Singer index theorem . 109 4.2 Covering spaces . 114 4.3 Etale¶ groupoids . 117 4.4 The Mischenko idempotent . 121 4.5 Four preparatory lemmas . 123 4.6 The index of ¡-invariant elliptic operators . 127 A Locally convex algebras 129 B Homological algebra 133 B.1 Double complexes . 133 B.2 Yoneda's Ext . 136 C Failure of excision for K1 141 D Fibre bundles 145 Bibliography 149 Introduction Index theory and noncommutative geometry This paper contains an exposition of two of the basic tools of what is commonly called "Noncommutative Geometry". This branch of mathematics studies al- gebras (usually over C, but even over Z) using tools and ideas originating in geometry. The philossophy of noncommutative geometry is close to that of al- gebraic geometry, in which the interplay between algebra and geometry can be illustrated by the equivalence of categories a±ne algebraic varieties over C () ¯nitely generated domains over C; implemented by associating to such a variety its algebra of complex valued regular functions. This basically states that all the information in a certain geometric structure is encoded in an algebraic structure associated to it. Of course, in proving such a statement, the direction ( is the more interesting one. In topology, there is a similar theorem, due to Gel'fand and Naimark, stating the equivalence of categories ¤ locally compact Hausdor® spaces () commutative C -algebras over C; implemented by associating to such a space its algebra of complex valued con- tinuous functions vanishing at in¯nity. The important thing about the above equivalences is that the algebras involved are commutative. Apparently, noncommutative algebras do not correspond to any concrete geometric structure. However, the study of spaces involves such constructions and objects as vector bundles, connections, di®erential forms and integration and the above equivalences allow one to carry out these construc- tions using the function algebra, without any reference to the space itself. For some of these constructions, the commutativity of the function algebra does not have any particular signi¯cance. It is at this point that noncommutative geometry starts to deviate from algebraic geometry. For in this case, one can use geometric constructions and intuition, although there is no geometric object to refer to. Enlarging the scope from commutative to noncommutative algebras at times greatly simpli¯es and clari¯es results in geometry and topology. Two nice ex- amples of this are provided by the extension of Atiyah-Hirzebruch topological K- theory from the category of spaces (or, equivalently, commutative C ¤-algebras) 9 10 CONTENTS to that of Banach algebras. This theory associates to a space X a sequence of abelian groups Ki(X), i 2 Z, such that for closed subspaces Y ½ X, there is a long exact sequence ::: - Ki(X=Y ) - Ki(X) - Ki(Y ) - Ki¡1(X=Y ) - ::: The central result in topological K-theory is Bott periodicity, stating that Ki(X) =» Ki+2(X) for all i. Bott's original proof of this remarkable fact is far more complicated then the short and elegant proof given by Cuntz in [8]. This proof takes place almost entirely in the noncommutative category. The second example is the C¤-algebraic proof of the Atiyah-Singer index theo- rem. This celebrated theorem, for which Atiyah and Singer received the Abel Prize in 2004, states that the analytic index of an elliptic pseudo di®erential operator P on a compact manifold M of dimension n can be computed from the topological formula Ind(P ) = (¡1)n ch(σ(P )) ^ Td(M): ¤ ZT M Elliptic pseudo di®erential operators are Fredholm operators when viewed as op- erators in the Hilbert space L2(M), meaning that both dim ker P and dim ker P ¤ are ¯nite, where P ¤ is the formal adjoint of P . The analytic index of P is just the Fredholm index Ind(P ) = dim ker P ¡ dim ker P ¤: The right hand side of the Atiyah-Singer formula, as an integral of di®erential forms over a manifold, is a topological invariant. The form Td(M) is an invariant of the manifold M, whereas ch(σ(P )), the Chern character of the principal symbol of P , can vary with P . Classically, the Chern character on a manifold M is a homomorphism 1 i 2j+i ch : K (M) ! HDR (M); j=0 M from the topological K-theory of M to its DeRham cohomology made 2-periodic. The current paper treats a generalization of this homomorphism. One modern proof of the index theorem is related to the existence of a short exact sequence σ 0 - K(L2(M)) - ª0(M) - C(S¤M) - 0 of C¤-algebras. Here K(L2(M)) is the ideal of compact operators on the sepa- rable Hilbert space L2(M), ª0(M) is the completion of the algebra of order at most 0 pseudodi®erential operators on M when viewed as an algebra of opera- tors on L2(M), and σ is the principal symbol map. An operator P is elliptic if its principal symbol σ(P ) is invertible, and this im- top ¤ plies that σ(P ) de¯nes a class in the group K1 (C(S M)). The superscript CONTENTS 11 ¤ 'top' stresses that this is the C -algebraic K1-group, as opposed to the purely algebraic one we will discuss later and mainly work with in this paper. The boundary map top ¤ top » K1 (C(S M)) ! K0 (K) = Z maps the K1 class of the symbol σ(P ) to its Fredholm index. This map is therefore denoted Ind. This shows that the index is actually a K-theoretic quantity, and is starting point for a more general, algebraically flavoured index theory, which is the subject of this paper. Non-compact index theory The name most commonly associated with noncommutative geometry is that of Alain Connes. He initiated the subject in the late 1970's. In 1990, Henri Moscovici and Connes published the paper [7], in which they proved an index theorem for a class of elliptic operators on noncompact manifolds M~ , equipped with a free action of a discrete group ¡, such that the quotient M~ =¡ is compact. Such spaces are just normal covering spaces of M with ¡ the group of covering transformations. Normal covering spaces are classi¯ed by homotopy classes of maps à : M ! B¡, where B¡ is the classifying space of ¡. In the noncompact setting, problems arise when one tries to de¯ne the index of an elliptic operator. These are in general not Fredholm, so the Fredholm index does not make sense. One could try to de¯ne the index as a K-theory element. This is impossible however, since the topological K-theory of a non- ¤ compact manifold M~ is the K-theory of the C -algebra C0(M~ ) of functions on M vanishing at in¯nity. This is a nonunital algebra, and therefore does not posses any invertible elements. The symbol σ(P ) is an invertible element in the algebra C(M), so we cannot associate an element in K 1(M~ ) to it. It is possible to de¯ne a K-theoretic index for more restrictive classes of ellip- tic operators. If the operator P is ¡-invariant, then it has a principal symbol σ(P ) 2 C1(S¤M) and an index ^ Ind[σ(P )] 2 K0(K ­ C[¡]): Here K^ is the ideal of smooth compact operators on L2M and C[¡] is the group algebra of ¡. These are not C¤-algebras, and the K-group in which the index lives, is de¯ned purely algebraically.
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