Derived Geometry and The Integrability Problem for G-Structures by Anthony Ryan McCormick A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master of Mathematics in Pure Mathematics Waterloo, Ontario, Canada, 2017 c Anthony Ryan McCormick 2017 AUTHOR'S DECLARATION I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract In this thesis, we study the integrability problem for G-structures. Broadly speaking, this is the problem of determining topological obstructions to the existence of principal G-subbundles of the frame bundle of a manifold, subject to certain differential equations. We begin this investigation by introducing general methods from homological algebra used to obtain cohomological obstructions to the existence of solutions to certain geometric problems. This leads us to a precise analogy be- tween deformation theory and the formal integrability properties of partial differential equations. Along the way, we prove the following differential-geometric analogue of a well-known result from derived algebraic geometry: H∗(N•(S1 ⊗ C¥(M))) =∼ W−∗(M) as well as the identification of the infinitesimal generator of the natural S1-action corresponding to the de Rham differential. As a short corollary we obtain a natural isomorphism ∗ • 1 ¥ ¥ ∼ −∗ H (Hom(N (S ⊗ C (M)), C (M))) = G(M, L TM) leading to a comparison between the standard Gerstenhaber bracket on the left-hand-side of the above equation and the Schouten bracket on the right. These two results are well-known in de- rived algebraic geometry and are folk-lore in differential geometry, where we were unable to find an explicit proof in the literature. In the end, this machinery is used to provide what the author believes is a new perspective on the integrability problem for G-structures. iii Acknowledgements First and foremost I would like to thank my supervisor Spiro Karigiannis for spending so much time answering my questions every week, for going through this monstrosity of a document with a fine-tooth comb, and for running the Geometric Analysis Working Seminar in which I learned so much. I also want to thank Anton Borissov, Henry Liu and Raymond Cheng for running the Geometry Working Seminar as well as the Mirror Symmetry Seminar in the summer of 2016. Much of this the- sis would have been far more difficult for me without some of the intuition for geometry I gained in that seminar. Furthermore, I extend my gratitude to my friends and family for bearing with me in this crazy year. Especially my parents Angelina and Jim, and my sister Michelle. A special thanks to Grant and Ian for organizing boardgame nights, Brandon for organizing trivia nights, Sam and Ilia for Friday, Parham for the office antics, and Osman and Danny for making time spent in some of the grossest apartments bearable, if not enjoyable. Finally, I’d like to thank Benoit Charbonneau for sparking my initial interest in differential geome- try, Jason Bell for teaching me most of the algebra I know, and Andu Nica for showing me how to integrate in infinite dimensions (something which blows my mind to this day). iv Table of Contents 1 Preliminaries 1 1.1 Introduction . .1 1.2 Topology and PDEs . 16 1.3 Homotopical Algebra . 26 1.4 Homological Algebra . 37 2 Algebraic Methods in Differential Geometry 51 2.1 C¥-Rings . 51 2.2 Cofiber and Fiber Sequences . 69 2.3 Chern-Weil Theory and the Derived Loop Space . 82 3 Applications 94 3.1 Prolongation . 94 3.2 Deformation Theory . 99 3.3 Formal Integrability and the Spencer Complex . 110 3.4 The Frölicher-Nijenhuis and Schouten Brackets . 118 Bibliography 123 A Category Theory Review 127 v Chapter 1 Preliminaries Throughout this thesis all of our complexes will be cohomological. Given a complex A, we’ll write A[n] for the complex with A[n]i = Ai+n and differential multiplied by (−1)n. Notational conven- tions are in the appendix. 1.1 Introduction In this section I’ll be describing the basic motivating problem for this thesis. Our main goal will be to obtain a thorough understanding of the integrability problem for G-structures. To do this, let’s begin by developing an intuitive understanding of what a G-structure is. Let’s think about the universe we live in. A hypothetical high-school student, when asked what an appropriate mathematical model for space is, might respond with R3. Indeed, the 3-dimensional space we live in is often how we first think of the vector space R3. But this is inaccurate. For ex- ample, one could ask this high-school student: where is the point (0, 0, 0) 2 R3 located in space? In which directions do the three standard coordinate axes point? This line of questioning leads us to posit that perhaps space is best represented by an affine R3. So one can choose any point in space and any three independent direction vector emanating from that point and from this one will obtain an identification of space with R3. But this description also falls short. Indeed, using the very same thought process, one could arrive at the conclusion that the surface of the earth is modelled by an affine R2, i.e. that the earth is flat. In this modern era we know this to be false. The point being that while locally the earth does indeed appear to be flat, globally this is not the case. For example, we know that the surface of the earth is compact (a topological property) while affine R2 is not. As such, we cannot assume that space is an affine R3 a priori. Locally it certainly seems to look like one but globally there may be topological non-trivialities. Another way in which space differs from affine R3 is curvature. One measures distance on an affine R3 using the standard Euclidean distance function. Returning to the example of the surface of the earth, we can recall that if one draws a suf- ficiently large triangle on the ground and sums the interior angles, it is possible to obtain a value larger than 180◦. This is not true for affine R3. Although, if one is only allowed to draw triangles on the ground in such a way that by standing at any corner one can see the whole triangle, then the sum of the interior angles should be a reasonably good approximation of 180◦. The point is, the distance function on the surface differs non-trivially from the usual Euclidean distance function. This difference is called the curvature of the distance function. In the same way that we should not have assumed the surface of the earth was flat (had no curvature) we shouldn’t assume space is flat. 1 So we now have two potential ways in which space could differ non-trivially from affine R3: the topology and the curvature. It turns out that these two properties are distinct but are related in subtle ways. Let’s now describe the type of mathematical object we will use to model space. Definition 1.1.1. A smooth manifold is a Hausdorff second countable topological space M together n with an open cover by Ui’s and continuous maps ji : Ui ! R i which are homeomorphisms onto their (open) images such that −1 ji ◦ jj : jj(Ui \ Uj) ! ji(Ui \ Uj) is a diffeomorphism for all i, j such that Ui \ Uj 6= Æ. Furthermore, we will assume that the collec- tion of such pairs (Ui, ji) is maximal and we call this collection a (maximal) atlas. The pairs (Ui, ji) themselves are called charts. A standard fact from differential geometry is that the ni’s appearing in the above definition are locally constant with respect to the topology on M. However, it is possible to have non-connected manifolds for which the ni’s tend to infinity since second-countability allows the number of con- nected components to be countably infinite. We should mention a couple of things regarding the above definition. The assumption that our space be Hausdorff can be thought of as saying that it is possible to distinguish between any two points in space. The second countability assumption is equivalent, for topological spaces satisfying all of the other assumptions in our definition, to having countably many connected components each of which is metrizable. The last point in our definition was that the transition functions between charts (i.e. changes of coordinates) were assumed to be diffeomorphisms. This is essentially saying that the rules of cal- culus work the same in any coordinate chart. An important advantage of this is that it allows one to perform multivariable calculus globally on manifolds by simply doing constructions in each chart. One thing worth observing is that we defined manifolds in dimensions other than just 3. This is because other objects than just space end up being modelled well by manifolds. For example, the surface of the earth is a 2-dimensional manifold, space-time is a 4-dimensional manifold, and the space of possible configurations of a rigid body in space looks like (at least locally) R3 o SO(3). This is a 6-dimensional manifold (and, in fact, a Lie group). The most important example of a higher dimensional manifold for us will be the frame bundle of a manifold. Recall that the whole point of thinking of space-time as a manifold is that it is, in some sense, the most general type of reasonable object for which one can choose a reference frame at each point.