NATURAL STRUCTURES IN DIFFERENTIAL GEOMETRY Lucas Mason-Brown A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in mathematics Trinity College Dublin March 2015 Declaration I declare that the thesis contained herein is entirely my own work, ex- cept where explicitly stated otherwise, and has not been submitted for a degree at this or any other university. I grant Trinity College Dublin per- mission to lend or copy this thesis upon request, with the understanding that this permission covers only single copies made for study purposes, subject to normal conditions of acknowledgement. Contents 1 Summary 3 2 Acknowledgment 6 I Natural Bundles and Operators 6 3 Jets 6 4 Natural Bundles 9 5 Natural Operators 11 6 Invariant-Theoretic Reduction 16 7 Classical Results 24 8 Natural Operators on Differential Forms 32 II Natural Operators on Alternating Multivector Fields 38 9 Twisted Algebras in VecZ 40 10 Ordered Multigraphs 48 11 Ordered Multigraphs and Natural Operators 52 12 Gerstenhaber Algebras 57 13 Pre-Gerstenhaber Algebras 58 irred 14 A Lie-admissable Structure on Graacyc 63 irred 15 A Co-algebra Structure on Graacyc 67 16 Loday's Rigidity Theorem 75 17 A Useful Consequence of K¨unneth'sTheorem 83 18 Chevalley-Eilenberg Cohomology 87 1 Summary Many of the most important constructions in differential geometry are functorial. Take, for example, the tangent bundle. The tangent bundle can be profitably viewed as a functor T : Manm ! Fib from the category of m-dimensional manifolds (with local diffeomorphisms) to 3 the category of fiber bundles (with locally invertible bundle maps). This functor takes every m-dimensional manifold M to its tangent bundle TM := tm2M TmM and every local diffeomor- phism f : M ! N to its pushforward T f := f∗ : TM ! TN. The statement that T is a functor is essentially the chain rule. Functors of the general type Manm ! Fib are ubiquitous in differential geometry. Other examples include: the cotangent bundle T ∗, tensor bundles and their various sub-bundles (most notably the exterior bundles ^nT and ^nT ∗), frame bundles, jet bundles, and others. Over the years such functors have been variously termed \bundle functors," \natural bundles," and \geometric objects." In this thesis, we use the term \natural bundle" throughout. If we view natural bundles as functors, certain constructions play the role of natural transfor- mations. Other constructions play the role of natural operators, transforming smooth sections of one bundle into smooth sections of another, through a rule or recipe which respects the local action of diffeomorphisms. Classical examples include: the exterior derivative, the wedge prod- uct, the contraction (i.e. the evaluation of a form at a vector field), the Lie bracket, and the much-more-general Schouten-Nijenhuis bracket of alternating multivector fields. The functorial approach to natural bundles and operators was first introduced by A. Nijenhuis in his talk at the 1958 International Congress of Mathematicians. The foundations of a general theory were later developed by C.L. Terng and R. Palais ([1], [2]) with important contributions from D.B.A. Epstein, W.P. Thurston ([3]), and others. Following in the early footsteps of R. Palais, who demonstrated in 1959 that the exterior derivative is, apart from the identity, the only unary natural operator on differential forms ([4]), recent work in the field of \natural differential geometry" has largely been devoted to the task of classifying natural bundles and operators of a prescribed type. Much of this work is summarized in the foundational monograph of I. Kolar, P. Michor, and J. Slovak ([5]), which was the primary reference for the first part of this thesis. However, since the publication of this monograph, a few major holes have been filled. Over the past decade, M. Markl has published a string of papers, culminating in an exhaustive classification of natural n-ary operators on vector fields. He demonstrates, not surprisingly, that over high-dimensional manifolds the only such operators are given by linear combinations of various iterations of the Lie bracket. His work draws heavily on Operad theory, particularly on the work of F. Chapoton, M. Livernet ([6]), and J.-L Loday ([7]), as well as a graphical approach to tensor calculus originally formalized by S.A. Merkulov in [8]. We see in the foundational work of R. Palais and the more recent work of M. Markl the contours of a much more general problem. Namely, what is the complete space of natural multilinear operators on differential forms and alternating multivector fields? More explicitly, if ^αi T ∗ is βi the natural bundle of differential forms of degree αi and ^ T is the natural bundle of alternating multivector fields of degree βi, can we classify the space of natural multilinear operators of the 4 general type: α ∗ α ∗ β β α0 ∗ α0 β0 β0 η : ^ 1 T × ::: × ^ p T × ^ 1 T × ::: × ^ q T ^ 1 T × ::: × ^ r × ^ 1 T × ::: × ^ s T (1) We conjecture that all such operators can be built up from 5 `elementary' operations. On differential forms, there is the exterior derivative and the wedge product of differential forms. On alternating multivector fields, there is the Schouten-Nijenhuis bracket and the wedge product of alternating multivector fields. Linking the two is the contraction. We believe that (over high- dimensional manifolds) all natural operators of type (1) can be realized as linear combinations of iterations of these 5 operations. A proof of this conjecture would have intrinsic value within the field of natural differential geometry. Such a proof would also be valuable in a much broader sense. Natural operators are precisely the kinds of operators that present themselves in mathematical formulations of fundamental physics. Furthermore, differential forms and alternating multivector fields are among the most pervasive mathematical objects in these fundamental laws. Thus, a proof of this conjecture would radically limit the range of plausible natural laws that physicists might be tempted to formulate. In this thesis, we make several steps towards proving this conjecture. In the first part, we develop the basic theory of natural differential operators, roughly following chapters IV, V, and VI of [5]. The key concept in the first part is the method of invariant- theoretic reduction, whereby the geometric problem of classifying natural differential operators is reduced to the much more tractable algebraic problem of classifying module homomorphisms of a particular source and target. We conclude part I with an original proof that all natural n-ary operators on differential forms are exhausted by iterations of the wedge product and exterior derivative (making no assumption of multilinearity). In the second and final part, we shift our attention to the more obstreperous case of alternating multivector fields. Generalizing the work of M. Markl, we introduce the notion of an ordered multigraph as a general framework for the study of natural operators on alternating multivec- tor fields. Inspired by Markl, we draw connections between the space of natural operators on alternating multivector fields and the cohomology of a particular complex of multigraphs. We then develop the notion of a pre-Gerstenhaber algebra, which combines the structure of a graded commutative algebra with the structure of a graded Lie-admissable algebra, and prove that these operads are compatible in a suitable sense. We immediately see that this is the correct algebraic structure for the geometric problem at hand. Finally, we apply Loday's rigidity theorem to deduce certain facts about the structure of the relevant algebras. 5 2 Acknowledgment Special thanks go to Vladimir Dotsenko. Without his patient support and insight, none of this would have been possible. Thanks also to my undergraduate mentors and advisors|Benedict Gross, Mike Rosen, Dan Abramovic, and Tom Goodwillie, just to name a few|who enabled and empowered me as a mathematician. Part I Natural Bundles and Operators 3 Jets Suppose f : R ! R is a smooth function and x0 2 R. By Taylor's theorem f (r)(x ) f(x) = f(x ) + f 0(x )(x − x ) + ::: + 0 (x − x )r + h (x)(x − x )r 0 0 0 r! 0 r 0 where limx!x0 hr(x) = 0. Definition 1. The r-jet of f : R ! R is the rth order Taylor polynomial (r) r 0 f (x0) r j (x) = f(x0) + f (x0)x + ::: + x x0 r! How can we extend this notion to smooth maps between arbitrary smooth manifolds? One idea is to pick chart maps near x and f(x) and consider multivariate Taylor polynomials in these chosen local coordinates. We wish to formalize this approach in a coordinate-free manner. We begin by defining jets of smooth curves. Let M be a manifold and x 2 M. By a smooth curve through x we mean a smooth map f : R ! M with f(0) = x. Two smooth curves f and g through x are equivalent to order r if for every smooth function : M ! R, we have r r j0 ( ◦ f) = j0 ( ◦ g) Note, these expressions make sense, since ◦ f and ◦ g are smooth functions from R to itself. 6 Definition 2. The r-jet of a smooth curve f through x is its equivalence class under the equiv- alence relation defined above. r At the risk of abusing notation, we will denote the r-jet of f by j0 (f). Two smooth maps f; g : M ! N are equivalent to order r at x if for every smooth curve φ : R ! M through x, we have r r j0 (f ◦ φ) = j0 (g ◦ φ) Definition 3. The r-jet of a smooth map f : M ! N at x is its equivalence class under the equivalence relation defined above. r We will denote the r-jet of f : M ! N at x by jx(f).