UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE CLASSIFICATION OF CODIMENSION ONE BIQUOTIENT FOLIATIONS IN LOW DIMENSIONS A DISSERTATION SUBMITTED TO THE GRADUATE FACULTY in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY By ANDREW LUTZ Norman, Oklahoma 2021 CLASSIFICATION OF CODIMENSION ONE BIQUOTIENT FOLIATIONS IN LOW DIMENSIONS A DISSERTATION APPROVED FOR THE DEPARTMENT OF MATHEMATICS BY THE COMMITTEE CONSISTING OF Dr. Krishnan Shankar, Chair Dr. Jason DeVito, Co-Chair Dr. Lucy Lifschitz Dr. Justin Malestein Dr. Ricardo Mendes Dr. Stephen Weldon © Copyright by ANDREW LUTZ 2021 All Rights Reserved. Acknowledgements There are so many people whom I owe a debt of gratitude that I hardly know where to begin. I suppose it is proper to start by thanking the primary mathematical contribu- tors. I will start by thanking my advisors Dr. Krishnan Shankar and Dr. Jason DeVito. I would especially like to thank Dr. Shankar for helping me when I needed it the most, and always making sure that I had the resources I needed to succeed. I also want to thank Dr. DeVito for being particularly quick at responding to my questions – seem- ingly at any time of the day or night. You have both been truly fantastic and I am truly blessed to have had such wonderful advisors. I would also like to thank the remainder of my committee: Dr. Lucy Lifschitz, Dr. Justin Malestein, Dr. Ricardo Mendes, and Dr. Stephen Weldon. I would, in particular, like to thank Professors Lifschitz, Malestein, and Mendes for answering my questions when I would drop by your offices and never making me feel unwelcome, as well as our conversations not related to mathematics. I would also like to especially thank Dr. Weldon, my committee member from outside of the department, for taking such an interest in my dissertation topic, even though your expertise is not mathematics. Next I would like to thank Dr. Deborah Moore-Russo and Dr. Sepideh Stewart, each of which I consider a mentor of mine. In particular, you have both provided me with excellent advice when it comes to mathematics pedagogy and I feel I have become a better mathematics instructor after working with you both. I would particularly like to thank Dr. Deborah Moore-Russo for spending a semester working with me when I was the only one who was attending the RUME seminar. This was a privilege that was very helpful and I learned a lot. I would also like to thank Dr. Sheldon Davis and Dr. Regan Beckham at the University of Texas at Tyler for encour- aging me to pursue a graduate degree in mathematics and for all of the exciting visits that we have had since then. Finally, I would like to thank Dr. Lee Kennard for his con- iv tributions. Next I would like to thank my family. To my parents, William and Diane and my sister Shelly, you have all encouraged me to pursue my education and have been there to support me throughout this long journey that I had never imagined that I would go on. It means a lot to me that you have always been there for me and I am truly blessed to have the best family that anyone could ask for. I would also like to thank Ling, who has been the closest by during my hard times and good times. I did not feel at home here until you came into my life. I could have never made it this far without you. To my uncle, Pete, I would like to thank you for your support as well. I am thankful that we have become closer in recent years and I have really enjoyed our conversations. I would also like to thank our cats Kitcat, Little Paw, and Oreo for adding more happi- ness to my life and relieving so much stress. It is a wonder how I made it through five years of grad school without you. I would like to thank my friends who have been supportive. I will start by thanking the office staff. To the late Lorraine Inkster, you were always very inspirational and encouraging. It was nice that whenever I was struggling, I could come to talk to you in your office. I’m sure I speak for many people when I say that you will be missed. I’m also sure that you have been reunited with your husband and found happiness in heaven. To Mike Donlon, Mike Hillerby, and Cristin Sloan, I have enjoyed getting to know all of you during the COVID pandemic.The social isolation from the pandemic would have been much worse if it were not for all of you. I would also like to thank Kyle, who was my officemate for six years. I really enjoyed our conversations in our office about mathematics and other things, and I could not have asked for a better v officemate. Finally, I would like to thank Sabrina for being one of my dearest friends who has always encouraged me. vi Contents 0 Introduction 1 1 Preliminaries 9 1.1 Transformation Groups . .9 1.2 Homogeneous Spaces . 14 1.3 Biquotients . 15 1.4 Cohomogeneity One Manifolds . 22 1.5 Codimension One Biquotient Foliations . 27 1.6 Additional Results About Codimension-One Biquotient Foliations . 43 2 Low-Dimensional Examples of Non-negative Curvature 50 2.1 Examples in Dimensions 4 and 5 . 53 2.2 Examples in Dimension 6 . 60 3 Classification of C1BFs 70 3.1 Classification in Dimension 4 . 71 3.2 Classification in Dimension 5 . 76 3.3 Classification in Dimension 6 . 93 4 Models of C1BFs in Detail 111 4.1 C1BFs in Dimension 4: . 111 4.2 C1BFs in Dimension 5 . 121 4.3 C1BFs in Dimension 6 With Simply Connected Principal Leaf . 157 vii Abstract In this thesis we study compact simply connected C1BFs (manifolds which arise as quotients of cohomogeneity one manifolds). In particular, we study various elemen- tary properties of C1BFs including their topological and curvature properties. More- over, we give a classification of their structures in low dimensions and also show that all simply connected manifolds which admit non-negative curvature in low dimensions admit a C1BF structure. viii Chapter 0 Introduction A very interesting class of Riemannian manifolds are those which admit metrics of non-negative sectional curvature. Elementary examples of such manifolds are the Eu- clidean spaces, spheres, and Lie groups equipped with a bi-invariant metric. Non- negatively curved manifolds have been of great interest in Riemannian geometry and a great deal of effort has been put into finding examples of manifolds of non-negative curvature. Aside from taking products, the easiest way of producing new examples is by taking quotients of manifolds with non-negative sectional curvature, which again have non-negative curvature by O’Neill’s formula [O’N66]. In particular, a very large class of non-negatively curved manifolds are the biquotient manifolds. A biquotient manifold is any manifold which can be expressed as the quotient of a homogeneous space M G/H by a free isometric group action, where G is assumed to be compact. Æ Note that by taking the free isometric group action to be the trivial action by the trivial group, we see that the class of biquotients contains all homogeneous spaces as a spe- cial case. Biquotients first appeared in Riemannian geometry shortly after Milnor’s discovery 1 of the so called exotic spheres (manifolds which are homeomorphic but not diffeomor- phic to the standard 7-dimensional sphere S7) [Mil56]. Milnor’s original construction of the exotic spheres realized them as S3- bundles over S4. Later, it was shown by Mil- nor, Smale, Kervaire–Milnor and Eells–Kuiper [KM63, Sma61, EK62] that there are in fact 28 possible oriented differentiable structures on S7 and that 20 of these are dif- feomorphic to S3-bundles over S4. With the discovery of the exotic spheres, a natu- ral question was whether the exotic spheres admit metrics of non-negative sectional curvature. This question turned out to be quite difficult to answer. The first progress toward answering this question came when Gromoll and Meyer showed that one of the exotic spheres in dimension 7 is a biquotient, hence admits non-negative sectional curvature [GM74]. This is particularly interesting because a result due to Borel says that any homogeneous space that is homeomorphic to a sphere is necessarily diffeo- morphic to a sphere. This in particular shows that the class of biquotients is strictly larger than the class of homogeneous spaces. It was later shown [Tot02, KZ04] that the Gromoll-Meyer sphere is the only exotic sphere which can be written as a biquotient. In particular, new techniques would need to be used to have any hope of putting met- rics of non-negative curvature on the remaining exotic spheres. Nearly three decades after the Gromoll-Meyer sphere was shown to admit non- negative curvature, it was shown by Grove-Ziller [GZ00] that all of the so called Milnor spheres (i.e. exotic spheres that are also S3-bundles over S4) in dimension 7 admit met- rics of non-negative sectional curvature. In particular, they showed that any cohomo- geneity one manifold with singular orbits of codimension at most two admits a metric of non-negative sectional curvature. They were then able to associate to each Milnor sphere a certain principal SO(3)-bundle over S4 that is cohomogeneity one and sat- 2 isfying this property and deduce that the Milnor spheres have non-negative sectional curvature. After this groundbreaking result, the question had been answered for all but 8 of the exotic spheres in dimension 7.