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Equivariant and Isovariant Function Spaces UC Riverside UC Riverside Electronic Theses and Dissertations Title Equivariant and Isovariant Function Spaces Permalink https://escholarship.org/uc/item/7fk0p69b Author Safii, Soheil Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA RIVERSIDE Equivariant and Isovariant Function Spaces A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by Soheil Safii March 2015 Dissertation Committee: Professor Reinhard Schultz, Chairperson Professor Julia Bergner Professor Fred Wilhelm Copyright by Soheil Safii 2015 The Dissertation of Soheil Safii is approved: Committee Chairperson University of California, Riverside Acknowledgments I would like to thank Professor Reinhard Schultz for all of the time and effort he put into being a wonderful advisor. Without his expansive knowledge and unending patience, none of our work would be possible. Having the opportunity to work with him has been a privilege. I would like to acknowledge Professor Julia Bergner and Professor Fred Wilhelm for their insight and valuable feedback. I am grateful for the unconditional support of my classmates: Mathew Lunde, Jason Park, and Jacob West. I would also like to thank my students, who inspired me and taught me more than I could have ever hoped to teach them. Most of all, I am appreciative of my friends and family, with whom I have been blessed. iv To my parents. v ABSTRACT OF THE DISSERTATION Equivariant and Isovariant Function Spaces by Soheil Safii Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, March 2015 Professor Reinhard Schultz, Chairperson The Browder{Straus Theorem, obtained independently by S. H. Straus in the 1960s and W. Browder in the 1970's, states that for manifolds with smooth group actions, isovariant and equivariant homotopy equivalences are the same provided the spaces involved satisfy some dimensional restrictions on their fixed point sets. A proof of this theorem was presented in 2013 by R. Schultz using contributions from surgery theory. In particular, the proof requires an equivariant version of C.T.C Wall's π −π Lemma. In this dissertation we demonstrate that the dimensional restrictions of the Browder{ Straus Theorem are sharp by providing examples, just outside of those dimensional restrictions, for which the conclusions of the theorem do not hold. Additionally, we discuss the implications of these examples with respect to the Equivariant π − π Lemma. vi Contents List of Figures ix 1 Introduction1 1.1 Overview..................................1 1.2 Organization...............................3 2 Preliminaries4 2.1 G-Spaces..................................5 2.2 The Browder{Straus Theorem......................6 2.3 Equivariant and Isovariant Function Spaces..............9 3 Replacing the Equivariant Function Space 12 3.1 Fibrations................................. 13 3.2 A Model for the Equivariant Function Space.............. 15 4 Replacing the Isovariant Function Space 23 4.1 Pullbacks................................. 23 4.2 The Dula{Schultz Model......................... 27 4.3 The Unreduced Suspension........................ 32 4.4 A Model for the Isovariant Function Space............... 33 5 Analysis of the Forgetful Map 36 5.1 The Stretched Suspension Map..................... 36 5.2 The Forgetful Map............................ 40 6 Comparing the Spaces In Dimensions Satisfying the Standard Gap Hypothesis 45 6.1 Fiber Bundles............................... 46 6.2 Spectral Sequences............................ 49 6.3 Comparing the Spaces.......................... 50 vii 7 Comparing the Spaces Outside of the Dimensions Satisfying the Standard Gap Hypothesis 55 7.1 Known Results Required for the Spectral Sequence Computations.. 56 7.2 Spectral Sequence Computations for the Equivariant Function Space. 58 7.3 Spectral Sequence Computations for the Isovariant Function Space.. 59 7.4 Examples Demonstrating that the Dimensional Restrictions of Browder{ Straus are Optimal............................ 61 7.5 Equivariant π − π Lemma........................ 68 Bibliography 70 viii List of Figures (1) V 0 (1) V 0 2.3.2.1 EG (S ;S ) and IFG (S ;S ) with dim V = 2.......... 10 3.2.0.1 Polar Zones of SV with dim V = 2.................. 16 3.2.0.2 Arctic and Antarctic Circles of SV with dim V = 2........ 16 ∗ V 0 3.2.0.3 EG(S ;S ) with dim V = 2..................... 17 ∗ V 0 3.2.4.1 Restriction of the domain of EG(S ;S ) with dim V = 2..... 20 ∗ V 3.2.4.2 FG(S(V ) × [−c; c];S ) with dim V = 2............... 21 ∗ V 3.2.4.3 FG(S(V );FG([0; 1];S )) with dim V = 2............... 21 4.2.0.1 Tropical Zone of SV with dim V = 2................. 27 (1) V 0 4.2.0.2 Dula{Schultz Model of IFG (S ;S ) with dim V = 2....... 28 4.3.1.1 Suspension Functor applied to FG(V ) with dim V = 2....... 33 0 5.1.5.1 Functor σ applied to FG(V ) with dim V = 2............ 40 5.2.2.1 σ0(f) applied to T with dim V = 2................. 42 5.2.2.2 σ0(f) applied to S(V ) × [−c; c] with dim V = 2........... 42 5.2.2.3 e ◦ f applied to S(V ) with dim V = 2................ 43 ix Chapter 1 Introduction 1.1 Overview The main result of this dissertation demonstrates that the dimensional restrictions of the Browder{Straus Theorem are sharp. The motivation for the Browder{Straus Theorem essentially goes back to the fun- damental question of what it means for two mathematical objects to be the same. To address this question and be able to classify mathematical objects that share certain properties requires some notion of equivalence. For topological spaces, it is natural to consider two spaces to be the same if they are homeomorphic. However, it is often useful to consider the weaker notion of homotopy equivalence. When working with topological spaces with the additional structure of a topological group action, there are two different notions of homotopy equivalence, namely equivariant and isovariant 1 homotopy equivalence. In the 1960's, substantial progress was made in further classifying spaces within the same homotopy type using geometric topology. This translated to analogous results for the classification of spaces up to isovariant homotopy equivalence. These advances also led to results for classification up to equivariant homotopy equivalence, provided the spaces under consideration satisfy (among other conditions) some version of the Gap Hypothesis, which is a restriction on the dimension of the fixed point set relative to the dimension of the space. This connection motivates the Browder{Straus Theorem, independently obtained by S. H. Straus in the 1970s [Str72] and W. Browder in the 1980s [Bro87], which gives conditions for which equivariant and isovariant homotopy equivalences are the same, the most prominent condition being that the spaces involved satisfy a version of the Gap Hypothesis. The proof of this result was not published until 2013 when R. Schultz [Sch13] presented a proof using contributions from surgery theory. In particular, the proof requires an equivariant version of C.T.C. Wall's π − π Lemma. In this dissertation we will construct examples, just outside of the dimensions satisfying the Standard Gap Hypothesis, for which the conclusions of the Browder{ Straus Theorem do not hold. 2 1.2 Organization The sections of this dissertation will break down as follows: Chapter 2 introduces the necessary basic definitions for G-spaces, leading up to the presentation of the Browder{Straus Theorem. The chapter concludes with the construction of the equivariant and isovariant function spaces whose homotopy groups will be used to construct our examples. Chapters 3 and 4 focus on finding appropriate models for these spaces. These models will allow us to compare their homotopy groups. Chapter 5 compares the spaces by analyzing the forgetful map from the isovariant function space to the equivariant function space, and the corresponding map between the models for these spaces. Chapter 6 introduces a spectral sequence converging to the homotopy groups of G-bundle maps developed by R. Schultz [Sch73] and applies this spectral sequence to our function spaces. Chapter 7 includes the necessary computations of the spectral sequences from Chapter 6 to demonstrate the difference in the homotopy groups of the function spaces. This result will allow us to present examples, in dimensions outside of those satisfying the Standard Gap Hypothesis, for which the conclusions of the Browder{ Straus Theorem do not hold. The chapter concludes with a brief discussion of how these examples further imply that the Equivariant π − π Lemma [Sch13] cannot be extended to higher dimensions. 3 Chapter 2 Preliminaries We begin with some basic definitions and notation for topological spaces with the added structure of group actions. This section is followed by a discussion of equivariance and isovariance, motivating and leading up to the presentation of the Browder{Straus Theorem. The final section of this chapter will define the function spaces that will be the central focus of this dissertation and whose homotopy groups will lead to our desired examples demonstrating that the dimensional restrictions of the Browder{Straus Theorem are optimal. We will also include the statement of the adjoint property for function spaces, which will be used repeatedly throughout this dissertation. 4 2.1 G-Spaces The definitions in this section are all standard and can be found in [Bre72]. Definition 2.1.1. A topological group G is a Hausdorff space with a continuous map m: G × G ! G, denoted by (g; h) 7! gh, that forms a group such that the inverse map i: G ! G, defined by i(g) = g−1, is also continuous. Definition 2.1.2. For a topological group G, a G-space is a Hausdorff topological space X along with a continuous G-action ·: G × X ! X, denoted by (g; x) 7! g · x, satisfying: i) g · (h · x) = (gh) · x, for any g; h 2 G and x 2 X; ii) e · x = x, for any x 2 X, where e 2 G is the identity.
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