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Morse Theory and Flow Categories Morse Theory and Flow Categories A Thesis Presented to The Division of Mathematical and Natural Sciences Reed College In Partial Fulfillment of the Requirements for the Degree Bachelor of Arts Maxine Elena Calle May 2020 Approved for the Division (Mathematics) Kyle M. Ormsby Acknowledgements First and foremost, I must thank Kyle Ormsby all his help. He got stuck as my advisor and I imagine he's gotten rather sick of me at this point, but he has been so helpful throughout this process. I'm very grateful to him for a number of things, including saying lots of things that make sense, putting up with me saying lots of things that make no sense, and preventing me from naming this thesis \Go with the Flow." It is thanks to Kyle that I know more about homotopy theory than I did before. I would also like to thank Ang´elicaOsorno, who I have learned a lot from both in and out of mathematics. There are many things in this thesis that I understand better because of conversations with her. She has been a wonderful mentor and inspiration throughout my time at Reed. There are many other people without whom this thesis would not have been. I am indebted to Corey Dunn for telling me that I might like Morse theory, and to Frederico Marques's lecture which reminded me what Corey had said. I am profoundly grateful to Ralph Cohen for sharing insights on the material in Chapter 4. Without his generosity, I would have had far less to say. Many thanks also goes to David Ayala, whose helpful observations were crucial to both the successful completion of this thesis and the preservation of my sleep schedule within the last week. To the faculty and staff that have impacted my time at Reed, I extend heartfelt gratitude. I would especially like to thank Paul Hovda, for enlightening me as to the importance of giving things names; Jerry Shurman, for teaching my most beloved math class at Reed; and Irena Swanson, for encouraging me to stick with math in the first place. I'm grateful to the entire Mathematics Department for their support, and in particular Lisa Mackola has always been a great help with the logistical side of things. On a more personal note, I blame my mom for getting me hooked on math in the first place, and Nannie for encouraging me with multiplication problems over the phone. I greatly appreciate Laura and Dennis for their hospitality; they did not have to host me here in Portland, but they did, and that is very sweet of them. Maia Houck has known me through all my phases in life, including the middle school years, and still wants to live with me; I love her for that (among so many other things). Lucas Williams has been a great friend to me over the years, even if he refused to speak to me when we took Physics together. To be fair, I also refused to speak to him. I'm very glad we got over that. I'm also very glad that he is willing to go to Tom Yum with me so often. There are many things that I want to thank him for, but I'm planning to spread them out over the years, so I'm going to stop here. I could not have made it through my four years here at Reed without my wonderful friends. Big love goes to Laurel Schuster, for being my day-one; to Zia Pollis, for her warmth and incredible mochas; to Lauren Chacon, for her understanding and excellent sense of fashion; to Sadie Baker-Wacks, for understanding when I don't text back; to Ema Chomsky, for flipping me upside down; to Elena Turner, for all our strange connections; to Robert Irving and Tobias Rubel Janssen, for carving out a slice of thesis heaven on the second floor of the library; to Alex Richter and Sophia Carson, for their sweetness; to Ryan Kobler and Marisa Agger, for the late nights in Eliot Hall; and to all the folks in the aerial acrobatics troupe, STEMGeMs, the 2019 CMRG, and the 2018 CSUSB REU, for the community we made. I thank you all for caring enough to ask about my thesis, and for being polite enough to listen to me talk about it. Finally, I do not gratefully thank COVID-19 for helping me to finish out my senior year from my bedroom. As much as I like my own room, there's nothing like basking in the sunlight on the great lawn. Reed College, I will remember you fondly. Index of Notation A list of some symbols used throughout this thesis, ordered more or less alphabetically: γ1 ◦ γ2 the concatenation of the curves γ1 and γ2, a broken flow. 32, 71 γ1 ◦s γ2 the flow that \stays s away from" the critical point that joins γ1 and γ2. 69 Cat the category of small categories and functors. 92 BC the classifying space of a category C . 45, 53, 55 Crit(f) the set of critical points of a function f.8, 35 c an ordered sequence of critical points, c1 · · · ck. 31 c(a; b) an ordered sequence of critical points connecting a and b, a c1 · · · ck b. 31, 69 Diff the category of differentiable manifolds and smooth maps. 25, 90 M(a; b) compactified moduli space of flow lines from a to b. 32, 35, 68 T ∗M the cotangent bundle of M.9, 82 sd : sSet ! sSet, the edgewise subdivision functor. 46 ' a gradient flow line. 11, 30 Cf the flow category of a Morse function f. 35, 55 W (a; b) = W u(a) \ W s(b), the space of flow lines from a to b. 17, 31, 56, 72 µc the gluing map associated with an ordered sequence c. 70 γ a height-parameterized flow line. 30 2 (d f)a the Hessian of f at a.8 ' homotopy. 55, 86 ind(a) the Morse index of a critical point a. 10 I the unit interval [0; 1]. 57, 72, 86 ∼= isomorphism. 55, 90 M(a; b) = W (a; b)=R, the moduli space of flow lines from a to b. 17, 68 NC the nerve of a category C . 43, 45, 52 |−| : sSet ! Top, the geometric realization functor. 44 Set the category of sets and functions. 40, 90 j∆nj the standard n-simplex in Top. 42 ∆ the standard simplex category. 40, 46 sSet the category of simplicial sets and simplicial maps. 40 W s(a) the stable manifold of a critical point a. 14 Sn the topological n-sphere.9, 23, 27, 56, 89 TM the tangent bundle of M. 82 Top the category of topological spaces and continuous maps. 33, 90 TopCat the category of categories internal to Top and continuous functors. 34 t transverse. 16, 17 p γq = γqjdomγp , a truncated flow line. 65 tw : Cat ! Cat, the twisted arrow functor. 47 W u(a) the unstable manifold of a critical point a. 14 Table of Contents Introduction ................................... 1 Chapter 1: Morse Theory ........................... 7 1.1 Definitions and Basic Properties.....................8 1.1.1 Morse Functions.........................8 1.1.2 Gradient Flow........................... 11 1.2 The Morse-Smale Condition....................... 13 1.2.1 Stable and Unstable Manifolds.................. 14 1.2.2 The Morse-Smale Condition................... 16 1.3 A Survey of the Theory.......................... 20 1.3.1 Sublevel Sets........................... 20 1.3.2 Morse Homology......................... 21 1.3.3 The Morse Inequalities and Other Applications........ 25 Chapter 2: The Flow Category ........................ 29 2.1 Broken Flow................................ 29 2.1.1 Reparameterizing the Flow.................... 30 2.1.2 The Space of Broken Flow Lines................. 31 2.2 Topology of the Flow Category..................... 33 2.2.1 Categories Internal to Top .................... 33 2.2.2 The Flow Category as a Topological Category......... 35 Chapter 3: The Classifying Space of the Flow Category ........ 39 3.1 Some Simplicial Homotopy Theory................... 40 3.1.1 Simplicial Sets........................... 40 3.1.2 Classifying Spaces......................... 43 3.1.3 Edgewise Subdivision....................... 46 3.2 Homotopy Invariance........................... 48 3.2.1 Fibrations and Cofibrations................... 49 3.2.2 Homotopy Pullbacks....................... 50 3.2.3 Inducing Equivalence on Classifying Spaces........... 52 Chapter 4: The Cohen-Jones-Segal Theorem ............... 55 4.1 Examples................................. 56 4.1.1 The Sphere............................ 56 4.1.2 The Vertical Torus........................ 57 4.1.3 The Tilted Torus......................... 60 4.2 Proof of the Theorem........................... 61 4.2.1 Part 1: The General Case.................... 61 4.2.2 Part 2: The Morse-Smale Case................. 67 Suggested Further Reading .......................... 75 Appendix A: Some Things to Know ..................... 77 A.1 A Bit of Differential Topology...................... 77 A.1.1 Manifolds............................. 77 A.1.2 Tangent Vectors.......................... 80 A.1.3 Vector Bundles.......................... 81 A.1.4 Vector Fields........................... 83 A.1.5 Riemannian Metrics....................... 84 A.1.6 Flow................................ 85 A.2 A Bit of Algebraic Topology....................... 85 A.2.1 Homotopy............................. 86 A.2.2 Homology............................. 87 A.3 A Bit of Category Theory........................ 89 A.3.1 Basic Notions........................... 90 A.3.2 The Yoneda Lemma....................... 93 A.3.3 Limits and Colimits........................ 94 A.3.4 Adjunctions............................ 96 References ..................................... 97 List of Figures 1 A bug's path...............................2 2 Alternate sphere partitioned by flows..................3 3 The classifying space of the alternate sphere..............4 1.1 The height function on the n-sphere................... 10 1.2 Gradient flow lines on the sphere.................... 15 1.3 Examples of intersecting curves..................... 16 1.4 Height functions on the 2-dimensional torus.............
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