On the Structure of Mackey Functors and Tambara Functors Kristen Luise Mazur DeWitt, NY Master of Science, University of Virginia, 2011 Bachelor of Science, Lafayette College, 2008 A Dissertation presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy Department of Mathematics University of Virginia May, 2013 Abstract The stable homotopy groups of a G-spectrum are Mackey functors. Moreover, in 2004, Morten Brun showed that the zeroeth stable homotopy group of a commutative G-ring spectrum is a Mackey functor with added structure. More specifically, it is a Tambara functor. Thus, for G the cyclic group of prime power order, we endow the category of G-Mackey functors with a equivariant symmetric monoidal structure such that G-Tambara functors are the equivariant commutative monoids. This equivariant structure relies on the construction of symmetric monoidal norm functors from the categoryofH-MackeyfunctorstothecategoryofG-Mackeyfunctorsforallsubgroups H of G, and we devote most of Chapter 2 to defining these functors. We focus on the elegant and concrete nature of this new equivariant structure and provide numerous examples. WeendbydiscussingsomeresultsonTambarafunctorsthatfollowdirectly from the computability of these norm functors. Acknowledgements I would like to thank my advisor, Mike Hill, for sharing his knowledge of mathematics and for his support and patience over the past few years. I would also like to thank Nick Kuhn and Greg Arone for teaching me so much about algebraic topology. Finally, many thanks to my friends and family. This dissertation would not be complete without their constant love and encouragement. i Contents 1 Introduction 1 1.1 Mackey Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 The Category of G-Mackey Functors . . . . . . . . . . . . . . . . . . 14 1.2.1 The Box Product . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Green Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Tambara Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.1 The Norm of a Sum in a C -Tambara Functor . . . . . . . . 26 pn 1.4.2 The Norm of a Transfer in a C -Tambara Functor . . . . . . 30 pn 1.4.3 A Definition for C -Tambara Functors . . . . . . . . . . . . . 33 pn 1.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.4.5 The Box Product of Tambara Functors . . . . . . . . . . . . . 41 2 An Equivariant Symmetric Monoidal Structure on the Category of ii Mackey Functors 42 2.1 G-Symmetric Monoidal Structures . . . . . . . . . . . . . . . . . . . . 42 2.2 Constructing the Norm Functors NG . . . . . . . . . . . . . . . . . . 47 H 2.2.1 Part 1: H is Maximal in G . . . . . . . . . . . . . . . . . . . . 47 2.2.2 Part 2: H is Any Subgroup of G . . . . . . . . . . . . . . . . 65 2.3 The New G-Symmetric Monoidal Structure . . . . . . . . . . . . . . . 86 2.3.1 Tambara Functors are the G-Commutative Monoids . . . . . . 88 2.4 Two Interesting Consequences . . . . . . . . . . . . . . . . . . . . . . 96 2.4.1 Building Tambara Functors From Commutative Rings . . . . . 96 2.4.2 Green Functors Can Have Multiple Tambara Functor Structures100 1 Chapter 1 Introduction In algebraic topology we would often like generalized cohomology theories to remem- ber the symmetries of a given topological space X. Thus, we equip X with an action of some finite group G and extend the definition of a generalized cohomology theory to an RO(G)-graded cohomology theory so that it reflects the group action. Just as a generalized cohomology theory can be represented by a spectrum, an RO(G)-graded cohomology theory can be represented by a G-spectrum, which is a spectrum appro- priately endowed with an action of G. Therefore, many topologists have passed from studying topological spaces to studying G-spectra and hence developed the field of equivariant stable homotopy theory. For good references on this subject refer to [8], [10], and [3]. A large part of equivariant stable homotopy theory involves the study of the ho- motopy groups of G-spectra. However, the equivariant stable analogues of homotopy groups are more than just groups. Letting Sk be the kth sphere spectrum and [Sk,X] denote the set of homotopy classes of maps Sk → X, if X is a spectrum (with no 2 group action) then for all k in Z we define the kth homotopy group of X by π (X) = [Sk,X]. k Thus, we might guess that if we endow X with a G-action then we should define the kth stable homotopy group of X by π (X) = [Sk,X]G k where now [Sk,X]G is the set of homotopy classes of G-equivariant maps Sk → X. However, this definition does not suffice because we require that the stable homotopy groups of a G-spectrum not only remember G-action information, but also record in- formation about the action of all subgroups of G. Therefore, as described in Chapters 6 of [14] and 12 of [10], if X is a G-spectrum then to define the kth stable homotopy group of X we must first consider πH(X) = [Sk ∧(G/H) ,X]G k + for every subgroup H of G. Then as H varies these abelian groups fit together to form a G-Mackey functor, denoted π (X), and we define the kth stable homotopy group k of X to be this Mackey functor. Moreover, acommutativeG-ringspectrumisaG-spectrumwithextrastructurein the form of a commutative multiplication, and the zeroeth stable homotopy group of a commutative G-ring spectrum has extra structure as well. In fact, it has even more structurethanonemightinitiallyexpect. ThecollectionofG-Mackeyfunctorsformsa 3 category, and this category has a symmetric monoidal product called the box product and denoted . A unital commutative monoid under the box product is called a G- (cid:64) Green functor, and so we might surmise that if X is a commutative G-ring spectrum then π (X) is a commutative G-Green functor. While this statement is not altogether 0 false, it is incomplete since it does not describe the entirety of the structure of π (X). 0 In fact, in 2004 Morten Brun proved that the zeroeth stable homotopy group of a commutative G-ring spectrum is a G-Tambara functor [1]. Thus, Brun proved that π (X) is a commutative G-Green functor with the extra structure of multiplicative 0 transfer maps called norm maps. It would therefore be beneficial to develop an equivariant symmetric monoidal structure on the category of Mackey functors that is reminiscent of the box product but such that Tambara functors become the equivariant commutative ring objects. Such a structure is called a G-symmetric monoidal structure, and the commutative ring objects are called G-commutative monoids [4]. We define this structure in terms of tensoring over finite G-sets. In particular, if we let SetFin be the category of G finite G-sets and Mack be the category of G-Mackey functors then, in essence, a G G-symmetric monoidal structure is a map (−)⊗(−) : SetFin ×Mack → Mack . G G G Then a given Mackey functor M is a G-commutative monoid if the map (−)⊗M : SetFin → Mack G G 4 extends to a functor. More specifically, the Mackey functor M is a G-commutative monoid if a map X → Y of finite G-sets induces a map X ⊗ M → Y ⊗ M of Mackey functors. We refer to Section 2.1 of this thesis for complete definitions of a G-symmetric monoidal structure and a G-commutative monoid. Let G be C , the cyclic group of prime power order. The goal of this thesis is pn to endow the category of G-Mackey functors with a G-symmetric monoidal structure such that a given Mackey functor is a G-commutative monoid if and only if it has the structure of a Tambara functor. The G-symmetric monoidal structure that we create relies heavily on the construction of symmetric monoidal norm functors that build a G-Mackey functor from an H-Mackey functor for all subgroups H in G. Thus, most of this thesis consists of proving the following theorems. Theorem. There exist strong symmetric monoidal functors NG : Mack → Mack H H G for all subgroups H of G. Theorem. For all subgroups H < K < G the functor NG is isomorphic to the H composition of functors NGNK. K H We then define the map (−)⊗(−) by G/H ⊗M = NGi∗ M for all subgroups H of G and H H (X (cid:113)Y)⊗M = (X ⊗M) (Y ⊗M) for all finite G-sets X and Y (cid:64) 5 where the functor i∗ : Mack → Mack is the restriction functor. H G H The category of G-Mackey functors already supports a G-symmetric monoidal structure [4]. However, this structure is difficult to unravel. Indeed, its definition requires the passage to G-spectra! Moreover, we have been unable to identify all Tambara functors as the G-commutative monoids under this structure. Therefore, there are two advantages of the new construction presented in this thesis. First, it is concrete and computable. In particular, we will define the norm functors (and thus a new G-symmetric monoidal structure) without passing to G-spectra. Instead, we will explicitly describe the functors using the algebraic and categorical properties of Mackey functors. Finally, in Section 2.3.1 we prove the result given below. Theorem. Under this new G-symmetric monoidal structure a Mackey functor is a G-commutative monoid if and only if it has the extra structure of a Tambara functor. In the remaining sections of Chapter 1 we provide definitions and examples of G-Mackey functors, G-Green functors and G-Tambara functors for any finite abelian group G. We also describe the box product for C -Mackey functors and Tambara pn functors. We build the new C -symmetric monoidal structure on the category of pn C -Mackey functors in Chapter 2, and we devote Section 2.2 to manufacturing the pn norm functors. We end by discussing two interesting consequences of this equivariant symmetricmonoidalstructure. First, thenormfunctorswillprovidefunconstructions of Tambara functors, and we also describe how to use the norm functors to endow a