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Construction of Maps by Postnikov Towers CONSTRUCTION OF MAPS BY POSTNIKOV TOWERS DISSERTATION Presented in Partial Fulllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Chris Kennedy, Graduate Program in Mathematics The Ohio State University 2018 Dissertation Committee: Jean-François R. Lafont, Advisor James Fowler John E. Harper c Copyright by Chris Kennedy 2018 ABSTRACT Using Postnikov towers, we investigate the possible degrees of self-maps of various spaces, including SU(3), Sp(2), SU(4), and the principal Sp(1)-bundles over S7. This investigation requires determining the Postnikov invariants for these spaces, which involves a detailed exploration of the secondary and tertiary cohomology operations that relate their cohomology classes. We also present some conditions for nding and lifting H-maps in Postnikov towers, and conjecture what corresponding results about SU(n) and Sp(n) should look like. ii For my parents iii ACKNOWLEDGMENTS This thesis would not have been possible without a lot of help. I am grateful to my advisor, Jean Lafont, for helping and encouraging me through the last several years, and dedicating countless hours to hearing me attempt to explain what I had read or done that week, even as my research strayed further and further from the original plan. I also thank Jim Fowler for giving me so much of his time to help unravel things, and the many helpful suggestions and clarifying remarks he made. Among my peers, special thanks to Kyle Parsons, Jake Blomquist, Tom Dinitz, and Mike Steward for being great friends, and also for providing a supportive en- vironment for learning how to play board games about Renaissance-era commerce. Jake deserves additional thanks for elding many questions about category and ho- motopy theory. Thanks also to the broader circle of math folks here, including but not limited to Laine Noble, Neal Edgren, Charles Baker, Mike Belfanti, Drew Meyer, and Stacy Kim (not math, but close enough), for helping me weather the storm of grad school. Outside OSU, thank you to Phil Engel, who enthusiastically heard at least four very dierent research shpiels from me at various points, and to Lily Berger, who put iv up with Phil and me talking about math more often than strictly necessary. Thanks to both for continuing to be great friends to me, as they have been for years. I am indebted to the grant Algebraic topology and its applications, NSF-DMS #1547357, which allowed me to focus on my research for a number of very productive semesters in the last couple of years. Finally, to my parents, thank you so much for your unlimited, unbelievable sup- port and love through everything. v VITA 2010 . B.Sc. in Chemistry, B.Sc in Physics, Massachusetts Institute of Technology 2011-Present . Graduate Teaching Associate, The Ohio State University FIELDS OF STUDY Major Field: Mathematics vi TABLE OF CONTENTS Abstract . ii Dedication . iii Acknowledgments . iv Vita......................................... vi CHAPTER PAGE 1 Introduction . .1 2 Postnikov Towers . .4 2.1 The Postnikov tower of a space . .4 2.2 Maps of Postnikov systems . .8 2.3 The Whitehead exact sequence . 12 3 H-Spaces . 18 3.1 H-spaces and algebraic structures . 18 3.2 Homotopy properties of H-maps . 21 3.3 Fibrations . 23 3.4 Postnikov stages of H-spaces . 26 3.5 Localization . 29 4 Higher Cohomology Operations . 33 4.1 Secondary operations . 33 4.2 Higher operations . 38 vii 5 Degree Sets . 43 5.1 Multi-degrees . 43 5.2 A criterion for degrees . 45 5.3 Self-maps of SU(3) .......................... 48 5.4 Maps between principal SU(2)-bundles over S5 ........... 50 5.5 Self-maps of Sp(2) ........................... 52 5.6 Maps between principal Sp(1)-bundles over S7 ........... 54 6 Self-maps of SU(4) ............................. 60 6.1 Low dimensions . 60 6.2 Eliminating higher obstructions . 64 7 Future Directions . 67 7.1 General questions about H-spaces . 67 7.2 Problems about specic H-spaces . 69 APPENDICES A Index of Notation . 72 B The Leray-Serre Spectral Sequence . 74 B.1 Basic information . 74 B.2 The bration ............... 75 K(Z2; 4) ! X4 ! K(Z; 3) Bibliography . 78 viii CHAPTER 1 INTRODUCTION Given two closed oriented n-manifolds M and N, one often wishes to know whether there is a continuous map f : M ! N of nonzero degree; that is, such that f∗[M] 6= 0 2 Hn(N). An ideal answer would not only produce such a map, but also determine all possible such that there is a continuous with . One of k 2 Z f f∗[M] = k[N] the foundational results of algebraic topology, that up to homotopy there is exactly one degree-d self-map of the n-sphere, f : Sn ! Sn, is of this form. Other well- known results along these lines include the fact that there is a map f :Σg ! Σg0 of positive degree between oriented surfaces if and only if g ≥ g0, and that an oriented n-manifold M always has maps of positive degree into Sn. Our goal is to construct maps of given degree between compact simple Lie groups of the same dimension, and to do the same for some related spaces that arise naturally, most of which are H-spaces (a sort of Lie group up to homotopy). The main tool we use is the Postnikov tower, rst introduced by Postnikov [16] in 1951. This object provides a way to reconstruct the homotopy type of a space X from its homotopy groups and a set of Postnikov invariants kn, though there are some caveats (as detailed in, for example, [18]). By carefully matching up the 1 Postnikov invariants of two spaces X and X0, one can construct maps φ : X ! X0 with desired characteristics. Using this technique, we prove the following: Theorem 5.5.1. Let be the generators of the cohomology groups 3 x3; x7 H (Sp(2); Z) and 7 , respectively. There is a map such that ∗ H (Sp(2); Z) φ : Sp(2) ! Sp(2) φ (x3) = ∗ t3x3 and φ (x7) = t7x7 if and only if t3 ≡ t7 (mod 12). This is a result originally due to Maruyama-Oka (see [6]), but our technique, and in particular elucidation of the secondary cohomology operation relating x3 and x7, does not seem to have appeared in print to date. This can be extended to show: 7 Theorem 5.6.2. Let X; Y be principal Sp(1)-bundles over S , and let t3; t7 be dened analogously to above. There is a map φ : X ! Y realizing t3 and t7 if and only if certain congruences involving t3 and/or t7 are satised. We defer the specics to 5.6. Finally, we use Postnikov towers and some tech- niques from the theory of H-spaces to prove: Theorem 6.1.1. Let be the generator of i for . Then there xi H (SU(4); Z) i = 3; 5; 7 ∗ is a map φ : SU(4) ! SU(4) such that φ (xi) = tixi for i = 3; 5; 7 if and only if t3 ≡ t5 (mod 2) and t3 ≡ t7 (mod 6). The structure of this document is as follows. In Chapter 2, we review the notion of a Postnikov tower, and prove a series of lemmas that will be useful in constructing maps of given degree later on. Chapter 3 reviews H-spaces and H-maps, nishing with a criterion for an H-map of Postnikov stages 0 to be ex- φn−1 : Xn−1 ! Xn−1 tended to an H-map of the succeeding stages, 0 . In Chapter 4, we φn : Xn ! Xn 2 explore secondary and higher cohomology operations, which are necessary for ex- plicitly representing the Postnikov invariants of Sp(2), SU(4), and related spaces. Chapter 5 explains the multi-degree of a map, and after applying our techniques to some results of Lafont-Neofytidis [9] on SU(2)-bundles over S5, we prove Theorem 5.5.1 and Theorem 5.6.2. In Chapter 6, we go on to prove Theorem 6.1.1 using H- space machinery, and Chapter 7 concludes by listing some conjectures and questions that constitute a natural continuation of our program. In what follows, all spaces are assumed to be 1-connected CW-complexes, and (co)homology with unspecied coecients has coecients in Z unless otherwise noted. We will write ρm for the coecient homomorphism on (co)homology induced by reduction , and for the generator of n (when is clear Z ! Zm ιn H (K(G; n); G) G from context). In general, we will notate the Bockstein of the coecient sequence ·n by to distinguish it from the Bockstein of the mod- Steenrod Z −! Z ! Zn βe β p algebra. 3 CHAPTER 2 POSTNIKOV TOWERS In what follows, we will cast almost everything in the language of Postnikov towers, which provide a canonical homotopy decomposition for appropriately nice spaces. To that end, in this section we introduce Postnikov towers and establish some key facts that will be useful in our study. Comprehensive references about Postnikov towers are dicult to nd, so we will cite a number of dierent ones; the main ones are, in increasing order of technicality, [4], [1], and [2]. 2.1 The Postnikov tower of a space As noted above, a Postnikov tower is a way to decompose a 1-connected CW-complex X into homotopy-invariant pieces (we will assume from now on that X is such a space). It is dened as follows: Denition 2.1.1. If X is a 1-connected CW-complex, a Postnikov tower for X is a sequence of 1-connected CW-complexes q3 q4 qn X2 o X3 o ··· o Xn−1 o Xn o ··· together with a set of Postnikov sections pn : X ! Xn such that 4 1.
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