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Allanyashinski-Dissertation.Pdf The Pennsylvania State University The Graduate School PERIODIC CYCLIC HOMOLOGY AND SMOOTH DEFORMATIONS A Dissertation in Mathematics by Allan Yashinski c 2013 Allan Yashinski Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013 The dissertation of Allan Yashinski was reviewed and approved∗ by the following: Nigel Higson Evan Pugh Professor of Mathematics Dissertation Advisor, Chair of Committee Nathanial Brown Associate Professor of Mathematics John Roe Professor of Mathematics Radu Roiban Professor of Physics Yuxi Zheng Professor of Mathematics Department Head ∗Signatures are on file in the Graduate School. Abstract Given a formal deformation of an algebra, Getzler defined a connection on the periodic cyclic homology of the deformation, which he called the Gauss-Manin connection. We define and study this connection for smooth one-parameter defor- mations. Our main example is the smooth noncommutative n-torus AΘ, viewed as a deformation of the algebra C1(Tn) of smooth functions on the n-torus. In this case, we use the Gauss-Manin connection to give a parallel translation argument that shows that the periodic cyclic homology groups HP•(AΘ) are independent of the parameter Θ. As a consequence, we obtain differentiation formulas relating various cyclic cocycles on AΘ. We generalize this to a larger class of deformations, including nontrivial crossed product algebras by the group R. The algebras of such a deformation extend nat- urally to differential graded algebras, and we show that they are fiberwise isomor- phic as A1-algebras. In particular, periodic cyclic homology is preserved under this type of deformation. This clarifies and strengthens the periodic cyclic ho- mology isomorphism for noncommutative tori, and gives another proof of Connes' Thom isomorphism in cyclic homology. iii Table of Contents Acknowledgments vii Chapter 1 Introduction 1 1.1 Motivation and background . 1 1.1.1 Noncommutative tori . 1 1.1.2 Deformation theory . 3 1.2 Main results . 8 1.2.1 Integrating rGM ........................ 8 1.2.2 An A1-isomorphism . 10 Chapter 2 Preliminaries 14 2.1 Locally convex topological vector spaces . 14 2.1.1 Basics . 14 2.1.2 Constructions in LCTVS . 17 2.1.3 Locally convex algebras and modules . 25 2.1.4 Spaces of smooth functions . 30 2.2 Connections and parallel transport . 35 2.2.1 Parallel transport in free modules . 36 2.2.2 Integrability for Banach spaces . 40 2.2.3 Nilpotent perturbations of integrable connections . 42 2.3 Algebraic objects in LCTVS . 44 2.3.1 Graded locally convex modules . 44 2.3.2 Locally convex chain complexes . 45 2.3.3 Locally convex differential graded algebras . 47 2.3.4 Locally convex differential graded Lie algebras . 48 iv 2.3.5 Locally convex differential graded coalgebras . 49 2.4 Hochschild and cyclic homology . 50 2.4.1 Hochschild cochains . 51 2.4.2 Hochschild homology . 53 2.4.3 Cyclic homology . 53 2.4.4 Dual cohomology theories . 54 2.4.5 Noncommutative geometry dictionary . 55 2.5 Operations on the cyclic complex . 56 2.5.1 Lie derivatives, contractions, and the Cartan homotopy for- mula . 57 2.5.2 Some higher operations . 59 2.6 Cyclic homology of differential graded algebras . 60 2.7 A1-algebras . 67 2.7.1 Tensor coalgebra . 67 2.7.2 Locally convex A1-algebras . 71 2.7.3 Cyclic homology of A1-algebras . 77 Chapter 3 Smooth one-parameter deformations 80 3.1 Smooth deformations of algebras . 80 3.1.1 Smooth noncommutative tori . 83 3.1.2 Crossed products by R ..................... 87 3.1.3 Compatible connections . 92 3.2 Smooth deformations of cochain complexes . 95 3.2.1 Compatible connections . 96 3.3 The Gauss-Manin connection . 97 3.3.1 Gauss-Manin connection in periodic cyclic homology . 97 3.3.2 Dual Gauss-Manin connection . 101 3.3.3 Integrating rGM . 101 3.4 Smooth deformations of A1-algebras . 103 3.4.1 Compatible connections . 106 Chapter 4 Rigidity of periodic cyclic homology of noncommutative tori 111 4.1 The g-invariant complex . 112 4.1.1 g-invariant chains and cochains . 112 4.1.2 The abelian case . 113 4.1.3 Connections on the g-invariant complex . 115 4.2 Integrating rGM for noncommutative tori . 117 4.3 Interaction with the Chern character . 125 v 4.3.1 Chern character . 125 4.3.2 Chern character and rGM . 126 4.4 Differentiation formulas for cyclic cocycles in noncommutative tori . 128 4.4.1 Cyclic cocycles, characteristic maps, and cup products . 128 4.4.2 Noncommutative tori . 132 Chapter 5 Rigidity of A1-algebras 137 5.1 The Chevalley-Eilenberg differential graded algebra . 138 5.2 A1-deformations and cyclic homology . 140 Appendix A Single variable calculus in locally convex vector spaces 144 A.1 Differentiation . 144 A.2 Integration . 145 Appendix B Noncommutative calculus proofs 149 B.1 Lie derivatives . 149 B.2 Cartan homotopy formula . 165 B.3 Some higher operations . 173 B.4 Commuting connections . 180 Bibliography 185 vi Acknowledgments This research was partially supported by the grant DMS1101382 from the National Science Foundation. I would like to thank my advisor, Nigel Higson, for his suggestion of this project and his constant support, encouragement, and enthusiasm throughout. Thank you to the members of the Student GFA/NCGMP Seminar, past and present. I have thoroughly enjoyed the opportunity to learn from all of you, and I hope you can say the same about me. Thank you to all my brilliant professors at Penn State who have exposed me to more mathematics than I could possibly absorb. Going back further, I would like to thank my professors at Bucknell University, especially George Exner, who taught me how to do rigorous mathematics, and Pamela Gorkin, who guided me in my transition to graduate school. I have been blessed with amazing family and friends who have shown me un- conditional support all my life. I'd especially like to thank my mother, Margaret Yashinski, for raising me to be who I am today, and encouraging me to enter college as a math major. Lastly, I thank my wife and best friend, Melisa, for traveling this journey with me. I truly would be lost without you. vii Dedication To my parents. viii Chapter 1 Introduction 1.1 Motivation and background 1.1.1 Noncommutative tori The motivating examples for this work are noncommutative tori [30]. The non- ∗ commutative 2-torus with parameter θ 2 T = R=Z is the universal C -algebra Aθ generated by two unitary elements u and v subject to the relation vu = e2πiθuv: ∼ 2 ∗ When θ = 0, A0 = C(T ), the C -algebra of continuous functions on the 2- torus. In this way, one views Aθ as the algebra of continuous functions on some \noncommutative space," as in the philosophy of Connes [6]. The algebra Aθ is also called the rotation algebra with angle θ, because it can be expressed as a crossed product ∼ Aθ = C(T) oRθ Z; where Rθ is the automorphism of C(T) induced by a rotation of angle θ. The noncommutative tori form an interesting family of algebras due to their erratic dependence on θ. For example, Aθ1 is isomorphic to Aθ2 if and only if ∗ θ1 = θ2 or θ1 = 1 − θ2. The C -algebra Aθ is simple if and only if θ is irrational, and Aθ has a unique tracial state if and only if θ is irrational. Another striking difference is the embeddability of Aθ into an AF -algebra when θ is irrational [27]. 2 Rieffel [29] proved that for every α 2 (Z+Zθ)\[0; 1], there is a projection in Aθ of ∼ 2 canonical trace α. This is in stark constrast to the commutative torus A0 = C(T ), which contains no nontrivial projections. K-theory for C∗-algebras played a large role in the classification of noncommu- tative tori. The six-term exact sequence of Pimsner and Voiculescu [26] allows one to calculate the K-theory groups of Aθ by taking advantage of the crossed product structure. For any θ, we have ∼ Ki(Aθ) = Z ⊕ Z; i = 0; 1; so that the K-theory of a noncommutative torus is the same as the K-theory of the commutative torus T2. So we know that the K-theory of Aθ is independent of θ because it has been calculated separately for each θ. This is somewhat unsatisfactory, because the col- ∗ lection of C -algebras fAθg vary continuously in θ in some sense, and one would like to take advantage of this continuity to argue that K-theory is rigid as θ varies. ∗ Indeed, the collection fAθg form a continuous field of C -algebras, and if AJ de- notes the sections of this field over a contractible subset J ⊂ T, then the evaluation homomorphisms AJ ! Aθ induces isomorphisms at the level of K-theory [8]. This is an improvement in the sense that it gives canonical isomorphisms ∼ K•(Aθ1 ) = K•(Aθ2 ); for any θ1; θ2 2 T. However, the proof still relies on the crossed product structure, and it is not clear how one could generalize to other deformations that lack this structure. Our investigation into a more satisfactory explanation of this rigidity shall be in the context of cyclic homology, rather than K-theory. Cyclic homology was discovered independently by Connes [5] and Tsygan [34], see also [22]. It can be viewed as a noncommutative analogue of de Rham cohomology. As de Rham cohomology is defined in terms of a smooth structure, one does not typically consider a C∗-algebra here, but rather a certain dense subalgebra which is thought of as the algebra of smooth functions on the noncommutative space.
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