Unstable Cohomology Operations: Computational Aspects of Plethories William Mycroft School of Mathematics and Statistics University of Sheffield A thesis submitted for the degree of Doctor of Philosophy supervised by Prof. Sarah Whitehouse December 2017 2 Acknowledgements I would like to thank the following people, without whom this thesis would not have been possible. To my supervisor, Sarah Whitehouse, for your endless patience and support. Our meetings kept me motivated and inspired throughout the course of my studies. To Tilman Bauer and Andrew Baker, for many useful mathematical conver- sations. Without your input, this thesis would be considerably thinner. To the other PhD students at the University of Sheffield, for broadening my mathematical horizons and for your company through some of the drudgery that goes into producing a thesis. To my family, especially my parents, for instilling in me a passion for learning and always being supportive of my interests and activities. To all my friends in Sheffield and elsewhere, for providing entertaining dis- tractions from mathematics. Particular mentions must go to the Broomhill Tavern quiz teams and the DMC contingent of Sheffield University athletics club. To the Vashisht family, for always being extremely welcoming and keeping me well fed with delicious food. Most importantly, to Sabrina. Thank you for everything. Abstract Generalised cohomology theories are a broad class of powerful invariants in algebraic topology. Unstable cohomology operations are a useful piece of structure associated to a such a theory and as a result the collection of these operations is of interest. Traditionally, these operations have been studied through the medium of Hopf rings. However, a Hopf ring does not readily admit algebraic structure corresponding to composition of operations. Stacey and Whitehouse showed that the unstable cohomology operations nat- urally admit the structure of an esoteric algebraic gadget termed a plethory. This plethory contains all the information of the Hopf ring together with additional structure corresponding to the composition of operations. In this thesis, I shall introduce the algebraic theory of plethories and extend with results which will aid computations. I will then illustrate, in a direct fashion, how the unstable cohomology operations admit the structure of a plethory and discuss the implications in this context. Finally, I shall perform some computations of the plethory of unstable cohomology operations for some familiar cohomology theories. Contents Introduction 1 1 Abstract plethories 9 1.1 Algebraic objects in categories . 10 1.1.1 Group objects . 11 1.1.2 Ring and algebra objects . 11 1.1.3 General algebraic objects . 13 1.1.4 Coalgebraic objects . 14 1.2 Birings, plethories and P -algebras . 15 1.2.1 Birings . 17 1.2.2 The composition product . 20 1.2.3 Plethories . 24 1.2.4 P -algebras . 26 1.2.5 A detailed example: λ-rings . 29 1.3 Plethystic theory . 33 1.3.1 Sub-birings and sub-plethories . 34 1.3.2 Augmentations . 36 1.3.3 Ideals . 37 1.3.4 Duality . 40 1.3.5 Primitives . 46 1.3.6 k-Primitives . 51 1.3.7 Super primitives . 53 1.3.8 Linear plethories . 55 1.3.9 Indecomposables . 57 1.3.10 The Frobenius and Verschiebung maps . 59 1.4 Graded plethories . 62 1.4.1 Graded algebraic objects . 62 1.4.2 Graded plethories . 63 v 2 Plethories in topology 67 2.1 The plethory of unstable cohomology operations . 68 2.1.1 Filtrations and topologies . 69 2.1.2 Filtered modules . 71 2.1.3 The filtered tensor product . 74 2.1.4 Filtered algebras . 75 2.1.5 Topological filtrations . 76 2.1.6 Filtered birings . 79 2.1.7 The filtered composition product . 82 2.1.8 Filtered plethories . 85 2.1.9 Filtered P -algebras . 87 2.1.10 Formal plethories . 88 2.1.11 Based and primitive operations . 91 2.2 The suspension isomorphism . 93 2.2.1 Plethories with looping . 97 2.2.2 Stable operations . 100 2.3 Complex orientation . 101 2.3.1 Complex orientation and Hopf rings . 104 2.3.2 The impact on cohomology operations . 109 3 Computations 113 3.1 Singular cohomology . 114 3.1.1 Singular cohomology with rational coefficients . 115 A direct approach . 115 Via the enriched Hopf Ring . 117 3.1.2 Singular cohomology with mod 2 coefficients . 118 A direct approach . 121 Via the enriched Hopf Ring . 124 3.1.3 Singular cohomology with coefficients in Fp ............. 132 A direct approach . 135 Via the enriched Hopf ring . 137 3.2 Complex K-theory . 142 3.2.1 Ungraded operations . 144 The λ-operations . 144 The Adams operations . 151 3.2.2 Graded operations . 153 vi 3.3 The Morava K-theories . 159 3.3.1 Via the enriched Hopf ring . 160 3.3.2 The bialgebra of primitives . 167 3.3.3 A useful filtration . 171 A Biring relations 177 B Functor cartography 179 C A primer on algebraic geometry 181 D Generalised cohomology theories 183 E Hopf algebras and Hopf rings 185 F Extensions of Hopf algebras 187 References 190 vii viii Nomenclature Algebra T tensor algebra w.r.t. V variety of algebras. F Frobenius. V Verschiebung. V∗ graded variety of algebras. # crossed product. D linear dual. Hker Hopf kernel. h−; −i functional evaluation. Hcoker Hopf cokernel. −b completion. Plethories ⊗b completed tensor product. ∆+ co-addition. Sweedler product. ~ "+ co-zero. I augmentation ideal. σ co-additive inverse. J coaugmentation quotient. ∆× co-multiplication. × P primitives. " co-unit. γ co-linear structure. Pk k-primitives. β alternative co-linear structure. A super primitives. ◦ composition. h−i free module on a set. u composition unit map. Pn[x] truncated polynomial algebra. ι, ιn composition unit element. T tensor algebra. I initial plethory. S symmetric algebra. composition product. S free functor on k-primitives. k b completed composition product. Ψ free functor on super primitives. !S 1-action. ix Ω looping. Hn(−; R) singular cohomology. Hopf algebras and Hopf rings K(G; n) Eilenberg-MacLane space. ∗∗ -multiplication. Ap mod p Steenrod algebra. A∗ mod p dual Steenrod algebra. ◦◦ -multiplication. p P n Steenrod power. comultiplication. Sqn Steenrod square. " counit. β Bockstein. [λ] image of the right unit. Vect(−) isomorphism classes of complex ι, ιn augmentation. vector bundles. r∗ induced map. ξn canonical n-dimensional bundle. FHR(−) free Hopf ring. K(−) ungraded complex K-theory. K∗(−) graded complex K-theory. λ-rings U infinite unitary group. λn n-th λ-operation. BU classifying space of U. n n-th Adams operation. K(n)∗(−) n-th Morava K-theory. Pn;Pn;m universal polynomials. T one point space. Λ universal λ-ring. Ω loop space functor. Ω representing plethory for Λ. Categories Topology Ab abelian groups. h(−) ungraded cohomology theory. Algk commutative unital k-algebras. :com H representing space of h(−). Algk associative unital k-algebras. ∗ E (−) graded cohomology theory. FAlgk filtered k-algebras. E Ω-spectrum for E∗. CAlgk complete Hausdorff k-algebras. ∗ Bialgk bicommutative biunital En n-th representing space of E (−). k-bialgebras. E∗(−) completion of E∗(−). b :com Bialgk associative, cocommutative, E∗(−) pro-finite E∗(−). biunital k-bialgebras. x FBialgk filtered k-bialgebras. CModk complete Hausdorff k-modules. 0 Biringk;k0 k-k -birings. Monoid associative unital monoids. 0 FBiringk;k0 filtered k-k -birings. Plethoryk k-plethories. CPlethory complete Hausdorff CBiringk;k0 complete Hausdorff k k-k0-birings. k-plethories. ΩPlethory k-plethories with looping. Coalgk cocommutative counital k coalgebras. Ring commutative unital rings. Ho Quillen's homotopy category. CRing complete Hausdorff rings. Hopf k Hopf algebras over k. SemiRing commutative unital semi-rings. 0 HopfRingk;k0 k[k ]-Hopf rings. Set sets. Modk k-modules. Top topological spaces. 0 kModk0 k-k -bimodules. TwBialgk twisted k-bialgebras. FModk filtered k-modules. Pro-C pro-objects in C. xi xii Introduction The raison d'^etreof algebraic topology is to find computable algebraic invariants that classify topological spaces up to homeomorphism: deformations under which we are al- lowed to stretch and bend our object, but crucially not pinch or tear. A common `joke' states that a topologist is a mathematician who cannot tell a donut from a coffee mug. The problem of finding such invariants has proved very difficult and we can only com- pute reasonable algebraic invariants in very special cases. However, great progress has been made towards the easier problem of classifying spaces up to homotopy equivalence: deformations under which we are additionally allowed to pinch solid regions down to a point. Rather powerful homotopy invariants of a based space X are the homotopy groups de- noted πn(X) and defined for all integers n ≥ 0. These have the property that if f : X ! Y is a continuous map of spaces then we have an induced map f∗ : πn(X) ! πn(Y ) in a way which respects composition. Thus if X is homotopy equivalent to Y then we have ∼ induced isomorphisms πn(X) = πn(Y ) for all n. If we exclude somewhat pathological ex- amples the converse statement is true by the Whitehead theorem and thus the homotopy groups provide a method to prove a continuous map is a homotopy equivalence. Unfor- tunately, the homotopy groups are extremely difficult to compute and even for simple spaces such as n-dimensional spheres some of the homotopy groups are still unknown. The computation of the homotopy groups of spheres is in some sense the holy grail of algebraic topology. The homology groups of a space X are much more computable homotopy invariants denoted by Hn(X) and defined for all integers n ≥ 0. Just as for the homotopy groups maps of spaces induce maps of homology groups in a manner which is compatible with the composition.