U.U.D.M. Project Report 2020:17 K-Theory and An-Spaces William Hedlund Examensarbete i matematik, 30 hp Handledare: Thomas Kragh Examinator: Denis Gaidashev Juni 2020 Department of Mathematics Uppsala University Uppsala university Institution of Mathematics K-Theory and An-Spaces Author: Supervisor: William Hedlund Thomas Kragh June 18, 2020 Abstract We define the reduced and unreduced K-theory rings and prove the Bott periodicity theorem. With this we construct the exact six-term loop in K-theory induced by a pair (X; A) of spaces. This loop is applied along with Adams operations, which we construct, to prove Adams' theorem on the Hopf invariant, implying that the only 0 1 3 7 spheres with H-space structure are S , S , S , and S . We define An-spaces and the Stasheff polytopes, and use singular chains to construct an An-algebra structure on the singular chain complex of such a space. Sammanfattning Vi definierar de reducerade och oreducerade K-teoriringarna och bevisar Botts pe- riodicitetssats. D¨armed konstruerar vi den exakta sextermiga kretsen i K-teori som induceras av ett par (X; A) av rum. Denna krets anv¨ands tillsammans med Adamso- perationer, som vi konstruerar, f¨or att bevisa Adams sats om Hopfinvarianten, som medf¨or att de enda sf¨arer som b¨ar H-rumsstruktur ¨ar S0, S1, S3, och S7. Vi definierar An-rum och Stasheffpolytoperna, och anv¨ander singul¨ara kedjor f¨or att konstruera en An-algebrastruktur p˚adet singul¨ara kedjekomplexet av ett s˚adant rum. 1 Introduction The notion of an H-space generalizes that of a topological group, by dropping the require- ments of inverses and associativity, according to this definition: Definition 1.1. An H-space is a topological space X with a map µ: X × X ! X, which we call the multiplication, such that µ(x; e) = µ(e; x) = x for every x 2 X, for some element e 2 X which we call the identity of the multiplication. We can examine how invariants in algebraic topology can tell us which spaces can be equipped with such a multiplication. It is a famous result that the only spheres which admit H-space structure are S0, S1, S3, and S7. We reproduce the proof of this statement using K-theory, in proposition 2.13 and theorem 2.5. The main part of this thesis is dedicated to the construction of enough machinery of K-theory for the proof of this. In the last section we introduce An-spaces, which interpolate, for n from 2 to 1, between H-spaces and spaces with strictly associative multiplication. For the section on K-theory our reference [1] is Hatcher's Vector Bundles and K-Theory, whose exposition we generally follow throughout. For the section on An-spaces our references are the two parts of Stasheff's Homotopy Associativity of H-spaces, from which we take our notation and quote some facts about the Stasheff polytopes. 2 K-Theory K-theory is an algebraic object associated to a topological space, which is constructed from vector bundles over that space. A map f : X ! Y pulls back vector bundles over Y to ones over X in a way which makes K-theory a contravariant functor. We will be able to understand the K-theory of spheres quite explicitly, and this will enable us to answer which spheres can be H-spaces. Since K-theory is constructed from vector bundles, we define those and state some useful properties. 2.1 Some Properties of Vector Bundles Definition 2.1. A vector bundle over a space X is a space E and a surjection p: E ! X, along with a complex vector space structure on p−1(x) for each x 2 X, such that there is an −1 n open cover fUαg of X with a homeomorphism hα : p (Uα) ! Uα × C for each α, with n possible depending on α, restricting to a vector space isomorphism p−1(x) ! fxg × Cn for each x 2 Uα. X is called the base space of the bundle, and E the total space. The space p−1(x) is called the fibre over x, and the maps hα are called local trivializations. All the fibres over each connected component must have the same dimension, but if the base space is disconnected our definition allows the fibre dimensions to vary over the components. Often the vector bundle is denoted just by its total space, i.e. we talk about the vector bundle E in the notation above. 1 The notions of direct sum and tensor product of vector spaces carry over to bundles. For vector bundles E1 and E2 the direct sum E1 ⊕ E2 and tensor product E1 ⊗ E2 have fibres which are the direct sums/tensor products of the fibres of E1 and E2. The same distributivity holds: E1 ⊗ (E2 ⊕ E3) = E1 ⊗ E2 ⊕ E1 ⊗ E3. Given a bundle E ! Y and a map f : X ! Y , there is a pullback bundle f ∗E, whose fibre over x 2 X is the fibre of E over f(x). This pullback respects composition of maps, ∗ ∗ ∗ ∗ ∗ ∗ and the operations on vector bundles: (f ◦ g) E = g (f E), f (E1 ⊕ E2) = f E1 ⊕ f E2, ∗ ∗ ∗ f (E1 ⊗ E2) = f E1 ⊗ f E2. An isomorphism between vector bundles E1 and E2 over the same base space is a home- −1 −1 omorphism E1 ! E2 which maps p1 (x) to p2 (x) by a vector space isomorphism for each ∼ x in the base space. E1 = E2 means that the bundles are isomorphic. We will extend this notion to bundles over homeomorphic spaces: if f : X ! Y is a homeomorphism, then an isomorphism between E1 ! X and E2 ! Y (over the map f) is a homeomorphism between total spaces which maps the fibre over x 2 X to the fibre over f(x) by an isomorphism. ∼ ∗ Equivalently E1 ! X and E2 ! Y are considered isomorphic if E1 = f E2. If two maps f and g are homotopic, then they pull back bundles in the same way: f ∗E ∼= f ∗E for every bundle E. A trivial bundle over X is a bundle isomorphic to the product X × Cn. If the base space X is compact and Hausdorff we have the following fact: for every bundle E ! X there is a bundle E0 ! X such that E ⊕ E0 is a trivial bundle. In our work we shall always work with compact Hausdorff spaces, in large part in order to apply this fact. The pullback of a trivial bundle is trivial. If the base space X is contractible, every bundle over it is trivial. This is because the identity of X can be factored up to homotopy through the one-point space, over which every bundle is trivial (inspecting the definition reveals that a bundle over a point is just a vector space collapsed to that point). ∗ For a subspace A ⊂ X, we can consider the restriction of a bundle E to A, EjA = ιAE. n n n n−1 For the sphere S , the two hemispheres D+ and D− which intersect in the equator S are contractible spaces, so the restrictions of bundles to either one are trivial. On the h−1 n n n−1 n−1 k + −1 n−1 h− n−1 k intersection D+ [ D− = S we then have a map S × C −−! p (S ) −! S × C . At each x 2 Sn−1 this map is an isomorphism of Ck, and we can consider it as giving a map n−1 S ! GLk(C). This map is called the clutching function, or transition function, of the bundle. Conversely, one can construct a bundle over Sn given a clutching function on the equator, by gluing together two copies of Dn × Ck along the two @Dn × Ck in the way given by the clutching function. We have the following fact on clutching functions: two isomorphic bundles give homo- topic clutching functions, and conversely, two homotopic clutching functions give isomor- phic bundles. So we can classify bundles over spheres by homotopy classes of functions n−1 S ! GLk(C). 2 2.2 The K-Rings K-theory is a functor which maps a topological space to a set of equivalence classes of complex vector bundles over it, given a group structure via direct sum of vector bundles. Note that we are considering complex K-theory, so \vector bundle" always mean \complex vector bundle". Also it is advantageous to assume only that the dimension of a bundle is locally constant, so the fibre dimensions can vary if the base space is disconnected. We assume the base spaces to be compact Hausdorff, to have all technical properties available. We let "n ! X denote the trivial vector bundle of dimension n. Definition 2.2. Two vector bundles E1 and E2 are stably isomorphic, denoted E1 ≈s E2, if n ∼ n E1 ⊕ " = E2 ⊕ " for some number n. This is an equivalence relation; the only nontrivial thing is to check transitivity: assuming n ∼ n m ∼ m n+m ∼ n+m ∼ m+n E1 ⊕" = E2 ⊕" and E2 ⊕" = E3 ⊕" , we have E1 ⊕" = E2 ⊕" = E3 ⊕" . Our aim is to make a group from these equivalence classes, with direct sum as group operation. Since stable isomorphism requires dimensions to agree, the identity would clearly have to be the trivial bundle of dimension zero, but then we cannot have inverses since adding a bundle cannot decrease the dimension. Though there are no inverses, there is a cancellation property which we can use to 0 construct a group of formal differences of vector bundles: define E1 − E1 to be equiv- 0 0 0 alent to E2 − E2 if E1 ⊕ E2 ≈s E2 ⊕ E1.