Elliptic cohomology theories

Transcription

1 Elliptic cohomology theories Shan Shah September 26, 2012 Master thesis Supervisor: dr. Tilman Bauer Korteweg-de Vries Institute for Mathematics Faculty of Science University of Amsterdam

2 Details Title: Elliptic cohomology theories Author: Shan Shah, student nr Supervisor: dr. Tilman Bauer Second reviewer: dr. André Henriques Date: September 26, 2012 Korteweg-de Vries Institute for Mathematics University of Amsterdam Science Park 904, 1098 xh Amsterdam

3 Abstract This thesis is an exposition of elliptic cohomology theories in the sense of Landweber-Ravenel-Stong. After a summary on generalised cohomology theories, the theory complex cobordism is introduced. Next, a correspondence between certain types of cohomology theories and power series, namely formal group laws, is explained. This is then used to construct cohomology theories from formal group laws that describe the group operation on elliptic curves.

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5 Contents Introduction 1 1 Generalised cohomology theories Generalised cohomology theories and spectra Complex cobordism A bridge to algebra Formal group laws Complex oriented cohomology theories Landweber s exact functor theorem Enter number theory Formal group laws from elliptic curves Jacobi quartics and elliptic cohomology Further comments Summary for a first year student (in Dutch) 39 Bibliography 43 v

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7 Introduction This thesis treats a subject from the field of algebraic topology, namely elliptic cohomology theories. A cohomology theory h is a certain type of algebraic invariant of topological spaces. To every topological space X it associates a possibly infinite sequence of abelian groups..., h 1 (X), h 0 (X), h 1 (X), h 2 (X),..., hopefully extracting geometric data from X and crystallising it in an algebraic form. Cohomology theories are required to satisfy the so called Eilenberg-Steenrod axioms, which provide one with some methods of computing of these groups h n (X). For a long time, cohomology theories mainly came in three flavours: ordinary cohomology, H, defined in terms of singular n-simplices n X, real (KO ) and complex (K ) theory, defined in terms of real, respectively complex vector bundles ξ X, various flavours of cobordism, MU, MSpin, MSO, defined via mappings M X of manifolds with some extra structure into X, under the equivalence relation of two manifolds bounding a third one. Most other interesting theories were variants of these three. In the eighties, however, an entirely new type was constructed, which came from number theory. These are the elliptic cohomology theories, crafted from the formal group laws of elliptic curves. From the viewpoint of chromatic homotopy theory, a subfield of algebraic topology, one sees these theories as lying ordered in a tower: cobordism theories (MU, MSpin, MSO ) elliptic cohomology (Ell ) K-theory (K, KO ) ordinary cohomology (H ) So elliptic cohomology can be seen to lie one chromatic step up from K-theory. This already had applications in detecting elements in the mysterious stable homotopy groups of spheres, which K-theory could not see. Researchers have also found connections to physics and were able to reprove results in number theory, namely Richard Borcherds congruences satisfied by theta series of even unimodular lattices. The subject therefore lies at an emerging intersection between algebraic topology, number theory and physics. 1

8 Applications to the geometry of topological spaces have so far been limited though, because there is no known geometric definition of elliptic cohomology, like we have for the three classical types of cohomology theories. The search for this is thus also a matter of ongoing research. The subject has expanded enormously since its inception, but this thesis focuses on the very first, humble construction of elliptic cohomology theories in the article (Landweber, Ravenel, and Stong 1995). Some background in algebraic topology needed. The reader is assumed to be familiar with singular cohomology, vector bundles and the cohomology theory called complex K-theory. Spectral sequences are used in two places. For Chapter 3, basic knowledge of elliptic curves over fields, and the interpretation of elliptic curves over C as complex tori is also required. Outline of this thesis Chapter 1 starts with a section that introduces generalised cohomology theories. The correspondence between unreduced and reduced theories and between cohomology and homology theories through spectra is explained, and a calculational tool, the Atiyah-Hirzebruch spectral sequence, is given, which will be needed in the next chapter. The next section defines the cohomology theory complex cobordism in both geometric and homotopical terms, which will play a crucial role in the sequel. The first section of Chapter 2 treats formal group laws and is purely algebraic. Then, it is shown how certain cohomology theories, the complex oriented ones, have formal group laws naturally associated to them. Finally, Quillen s theorem on the universality of complex cobordism and Landweber s exact functor theorem are discussed, which sometimes allow one to construct a complex oriented cohomology theory from a formal group law. Chapter 3 takes a number-theoretic turn. There, it is explained how the group operation on an elliptic curve gives rise to a formal group law. Attention is next restricted to a special class of elliptic curves, the Jacobi quartics over C, and an explicit expression for their formal group laws is proven. Using Landweber s theorem from the previous section, cohomology theories are then constructed from them. Acknowledgements First and foremost, I would like to thank my supervisor Tilman Bauer for introducing me to an exciting thesis topic. It is a subject that I found difficult to penetrate and his explanations have been of great help. Furthermore, I am thankful for his understanding and patience with me during the writing of this thesis and I owe my interest in algebraic topology to a course he taught in the first semester of my master. Everything I know about the field, I learned from him. I also thank André Henriques for agreeing to be the second reviewer of my thesis. Next, I am lucky to have studied with fellow students whose enthusiasm for mathematics has been very infectious. For me, there comes great solace from knowing that you are not the only one who does not understand something, 2

9 and us trying to figure things out together and talking about mathematics more freely than as written in the textbooks has helped me tremendously. Finally, I am grateful to my parents, who have supported me throughout my studies. Notations and conventions All rings will be commutative with 1, and ring homomorphisms are required to be unital. For X a space with base-point, its reduced suspension will be denoted by ΣX and its loop space by ΩX. If X and Y are two topological spaces, the set of homotopy classes of maps X Y will written as [X, Y ]. We write X, Y if X and Y also have base-points and we are restricting ourselves to pointed maps. The singular (also called ordinary) cohomology of a space X will be written as H (X). The total space of a vector bundle ξ will be commonly denoted by E(ξ) and γ n for n 1 is the tautological n-dimensional complex vector bundle on the classifying space BU(n). 3

10 Chapter 1 Generalised cohomology theories This chapter contains some prerequisite material for the rest of the thesis. We first discuss our objects of study: generalised cohomology theories and their representing spectra. Next, we quickly introduce a cohomology theory that the reader may not yet be familiar with, namely complex cobordism, which will play a crucial role in the sequel. 1.1 Generalised cohomology theories and spectra When Samuel Eilenberg and Norman Steenrod first wrote down axioms for a homology theory in their paper (Eilenberg and Steenrod 1945), they included the dimension axiom, that is, if h is a homology theory, then h n (pt) = 0 for n 1. This holds for example for singular (also called ordinary) homology. As they explained in their book (Eilenberg and Steenrod 1952), this was simply a normalisation condition to ensure that the index n had a geometric meaning. Moreover, no interesting theories were known which did not satisfy it. This changed in the fifties with René Thom s invention of bordism a theory for which the dimension axiom does not hold at all. Such theories are now called generalised (or extraordinary) homology theories. We first define the notion dual to it. Let T op 2 denote the category of pairs of topological spaces (X, A), that is, A X, and write Ab for the category of abelian groups. Define a functor κ: T op 2 T op 2 by κ(x, A) = (A, ). Definition An (unreduced) generalised cohomology theory h on T op 2 with values in Ab is a sequence {h n, δ n } n Z of contravariant functors h n : T op 2 Ab and natural transformations δ n : h n κ h n+1, that satisfies the following axioms: (i) (Homotopy invariance) If f, g : (X, A) (Y, B) are homotopic maps in T op 2, then the induced homomorphisms h n (f), h n (g): h n (Y, B) h n (X, A) are equal for all n. (ii) (Exactness) For every pair (X, A) T op 2, the sequence h n 1 δ (A, ) n 1 h n (X, A) h n (X, ) h n (A, ), 4

11 where the unmarked arrows are induced by inclusions, is exact. (iii) (Excision) If (X, A) is a pair and U A is open such that U A, then the inclusion (X\U, A\U) (X, A) induces an isomorphism h n (X, A) = h n (X\U, A\U) for all n. (iv) (Additivity) If (X, A) is the disjoint union of a set of pairs (X i, A i ), then the inclusions (X i, A i ) (X, A) together induce for every n an isomorphism h n (X, A) = h n (X i, A i ). i Not all authors include the additivity axiom, but we will. We will abbreviate h n (X) := h n (X, ). Since the dimension axiom is not demanded, the groups h n (pt) can be highly non-trivial. We call these the coefficient groups of the theory. Definition A morphism of generalised cohomology theories T : {h n, δ n } {k n, ε n } is a sequence of natural transformations {T n : h n k n } n Z such that for each n the following diagram commutes: δ n h n κ h n+1 T n T n+1 ε n k n κ k n+1 With these morphisms, generalised cohomology theories form a category. A morphism of generalised cohomology theories T is then an isomorphism if each of the T n s is a natural equivalence of functors. Up until now, for X a space, h (X) := n hn (X) has merely been a graded abelian group. The theory is called multiplicative if this graded group also has the structure of a ring. A triad (X; A, B) is a space X with subspaces A and B. It is called excisive for a cohomology theory h if there exist isomorphisms h n (X, B) = h n (A, A B) and h n (X, A) = h n (B, A B) for all n. Definition Let h be a generalised cohomology theory. A multiplicative structure on h is a family of abelian group homomorphisms h m (X, A) h n (X, B) h m+n (X, A B), x y x y, for all m, n Z, defined for suitable triads (X; A, B), excisive ones in any case. These homomorphisms are called the cup product and are required to satisfy the following axioms: (i) (Naturality) If f : (X; A, B) (X ; A, B ) is a map of triads, then it respects the cup product: f (x y) = f x f y. 5

12 (ii) (Stability) Let (X; A, B) be an excisive triad and i A : h n (X, B) h n (A, A B) and i B : h m (X, A) h m (B, A B) be the maps induced by the inclusions and define maps δ A : h r (A, A B) = h r (A B, B) δ h r+1 (X, A B) and δ B : h r (B, A B) = h r (A B, A) δ h r+1 (X, A B). following diagram should commute: Then the h m (A) h n 1 i A (X, B) h m (A) h n (A, A B) h m+n (A, A B) δ 1 h m+1 (X, A) h n (X, B) h m+n+1 (X, A B) δ A that is, δ(a) x = δ A (a i A x), and similarly for i B and δ B, that is, x δ(b) = ( 1) deg x δ B (i B x b). (iii) (Unit element) There exists an element 1 h 0 (pt) such that 1 x = x = x 1 for all x. (iv) (Associativity) For all x, y and z, (x y) z = x (y z). (v) (Commutativity) For all x and y, (x y) = ( 1) deg(x) deg(y) (y x). Ordinary cohomology is an example of a multiplicative theory and the tensor product of vector bundles induces a multiplication on complex K-theory. For spaces with base-point, the notion of a reduced generalised cohomology theory can sometimes be more appropriate. Let T op denote the category of pointed topological spaces (X, x 0 ), that is, x 0 X, and define Σ: T op T op to be the reduced suspension functor Σ(X, x 0 ) = (ΣX, x 0 ). Definition A reduced generalised cohomology theory h on T op with values in Ab is a sequence { h n, δ n } n Z of contravariant functors h n : T op Ab and natural transformations δ n : hn h n+1 Σ, that satisfies the following axioms: (i) (Homotopy invariance) See the similar axiom in Definition (ii) (Exactness) For every space and subspace (A, x 0 ) (X, x 0 ) T op, the sequence hn (X/A) h n (X) h n (A) is exact for all n, where the first arrow is induced by the projection and the second by the inclusion. (We suppress the base-point in the notation to reduce clutter.) (iii) (Suspension) For every space (X, x 0 ) T op, the suspension morphism is an isomorphism for all n. δ n : hn (X) h n+1 (ΣX) (iv) (Additivity, or the wedge axiom) If (X, x 0 ) is the wedge of a set of spaces (X i, x 0 ), then the inclusions (X i, x 0 ) (X, x 0 ) together induce for every n an isomorphism hn (X) = hn (X i ). i 6

13 There exists a bijective correspondence between unreduced and reduced cohomology theories. We namely have a functor π : T op 2 T op that sends a pair (X, A) to (X/A, pt), where pt := A/A. We claim that if h is a reduced cohomology theory, then h := h π defines an unreduced theory. Conversely, if h is an unreduced theory, then h n (X, x 0 ) := h n (X, {x 0 }) is a reduced theory. Details on how to define the boundary maps δ n can be found in (Rudyak 2008, pg. 56). If h is an unreduced cohomology theory and (X, x 0 ) is a pointed space, then the sequence 0 h n( X, {x 0 } ) j h n i (X) h n( {x 0 } ) 0, where i: {x 0 } X and j : (X, ) (X, {x 0 }) are the inclusions, is split exact. So we see that we may consider h n (X) as a subgroup of h n (X): h n (X) = hn (X) h n (pt). Dually to cohomology theories, we have the notions of unreduced and reduced generalised homology theories. These are defined by replacing the contravariant functors in the above definitions by covariant ones. In the additivity axiom, the direct product is replaced with a direct sum. Homology and cohomology theories turn out to be closely related though: they come in pairs and may be seen as two sides of the same coin. The connection is made via certain topological objects, called spectra. Definition A spectrum E is a sequence of spaces E n with base-point, together with base-point preserving maps σ n : ΣE n E n+1. (1.1) Since the reduced suspension functor Σ is left adjoint to the loop space functor Ω in the category of based spaces, (1.1) is equivalent to giving a basepoint preserving map σ n : E n ΩE n+1. Some authors call the above definition a prespectrum and use the term spectrum if the adjoints σ n are homeomorphisms. Let us look at a few examples. Examples (i) For any cw-complex X with base-point the sequence of spaces { Σ n X for n 0, E(X) n = {pt} for n < 0, together with the obvious maps, forms the suspension spectrum Σ X of X. The suspension spectrum Σ S 0 of S 0 is called the sphere spectrum. Its n-th term is S n for n 0. (ii) For any abelian group G, Eilenberg-Maclane spaces K(G, n) are characterised up to homotopy equivalence by the property of connectedness and having just one non-trivial homotopy group π n (K(G, n)) = G. Because the loop space functor shifts homotopy groups back by one, we have ΩK(G, n + 1) = K(G, n), which makes the K(G, n) s into a spectrum. 7

14 (iii) The two-periodic sequence K 2n = Z BU, K 2n+1 = U, where BU is the classifying space of the infinite unitary group U, is a spectrum, the complex K-theory spectrum. There is namely a natural homotopy equivalence Ω(Z BU) U and the Bott periodicity theorem, which claims an equivalence Ω(U) Z BU. Just like for a topological space, one can define homotopy groups of a spectrum E. Applying the reduced suspension to both sides of a pointed map S n+k E k gives a map ΣS n+k σ ΣE k k Ek+1. Since ΣS n+k = S n+k+1, we see that we have a map π n+k (E k ) π n+k+1 (E k+1 ). Definition The n-th homotopy group of a spectrum E is defined as the direct limit π n (E) := lim k π n+k (E k ) We write π (E) := n π n(e). If E is a spectrum and X a pointed cw-complex, one can form a new spectrum X E by defining (X E) n := X E n and the structure maps Σ(X E n ) = X ΣE n X E n+1. It can then be shown (Hatcher 2001, Proposition 4F.2) that the functor h n (X) := π n (X E) defines a reduced generalised homology theory on the category of pointed cw-complexes. The sphere spectrum, for example, gives rise to a theory which is denoted by π S and called stable homotopy, because the groups πn(x) S are nothing but the stable homotopy groups of X. The Eilenberg-Maclane spectrum K(G, n) gives reduced ordinary homology with coefficients in G. Similarly, we have natural maps Σ k X, E n+k Σ Σ k+1 X, ΣE n+k σn Σ k+1 X, E n+k+1, and it can be shown that h n (X) := lim k Σ k X, E n+k defines a reduced generalised cohomology theory. The complex K-theory spectrum is named like that for good reason. Note that h n (pt) = h n (pt) = π n (E). This associating of a (co)homology theory to a spectrum is due to George W. Whitehead in Conversely, also in 1962, Edgar H. Brown and John F. Adams in 1971 showed that under certain conditions, all generalised (co)homology theories can be obtained this way. These theories may therefore be seen as topological objects and can be studied via homotopy theory. We first restrict our notion of a spectrum a bit. Definition A spectrum E is called an Ω-spectrum if all the spaces E n are cw-complexes and the structure maps σ n : E n ΩE n+1 are weak homotopy equivalences. Theorem (Brown, Adams). Let h be a reduced generalised cohomology theory. Then there exists an Ω-spectrum E, unique up to homotopy, such that hn (X) = X, E n for all n and all pointed cw-complexes X. Let h be a reduced generalised homology theory. Then there exists an Ω- spectrum E, unique up to homotopy, such that the reduced homology theory associated to E is equivalent to h. 8

15 The first statement is Brown s and the second is Adams. Proofs can be found in (Hatcher 2001, Theorem 4E.1) and (Switzer 1975, Corollary 14.36). So if one starts with an Ω-spectrum E, the associated reduced cohomology theory h also has the nicer form h n (X) = X, E n. We admit that with unique up to homotopy, something is shoved under the rug. Related to this issue is that one might expect that if a generalised cohomology theory is multiplicative, this extra structure is reflected in the spectrum that represents it. This is indeed the case, but the details turns out to be quite tricky and we will not need this. For calculating the generalised cohomology of a cw-complex, the following tool can be of help. Theorem (Atiyah-Hirzebruch spectral sequence). Let h be a generalised cohomology theory and X a space of the homotopy type of a cwcomplex. Then there exists a multiplicative, natural, half-plane spectral sequence E p,q 2 = H p (X; h q (pt)) converging conditionally to h p+q (X). So the h -cohomology of a cw-complex can be approximated with knowledge of its ordinary cohomology and the coefficient ring of h. One way to construct this spectral sequence is via William Massey s method of exact couples. See for example (Kono and Tamaki 2006). We filter X through its p-skeleta X (p) X and glue together the long exact sequences in h -cohomology for each pair (X (p), X (p 1) ) in the exact couple p h ( X (p 1) ) δ i p h ( X (p), X (p 1)) j p h ( X (p)) We have E p,q 1 = h p+q (X (p), X (p 1) ) = h p+q (X (p) /X (p 1) ) = h p+q ( α Sα) p = hp+q (Sα) p α = hq (S 0 ) α = h q (pt) α = C p (X) h q (pt), where C p (X) is the p-th cellular chain group of X. Also the first differential d 1 of the spectral sequence is nothing but the coboundary δ : h p+q 1 (X (p 1), X (p 2) ) h p+q (X (p), X (p 1) ). Therefore, E p,q 2 = H p (X; h q (pt)). As a final remark of this section, we note that the above expression for h p+q (X (p), X (p 1) ) can also be used to prove the following uniqueness theorem. 9

16 Proposition Let T : h k be a morphism of generalised cohomology theories defined on the category of cw-complexes, such that h and k are bounded below. If T (pt) is an isomorphism, then T is an isomorphism for all pairs of cw-complexes. Just like ordinary cohomology theory is uniquely determined by the dimension axiom, so are generalised cohomology theories, kind of, uniquely determined by their coefficient groups (although one does need a morphism between them beforehand). The Atiyah-Hirzebruch spectral sequence allows you to for example calculate the complex K-theory of complex projective space. We will not do this here, because we will show it in the next chapter for the class of so called complex oriented cohomology theories, to which complex K-theory belongs. In the rest of this thesis, the adjective generalised will be implied when we speak of a (co)homology theory. 1.2 Complex cobordism As mentioned in the Introduction, it is assumed that the reader is familiar with ordinary cohomology and complex K-theory. This section is a quick introduction to a third cohomology theory, called complex cobordism. Not only will this serve as an additional example to guide the general theory later on, but complex cobordism will also turn out to play an important, distinguished role among other cohomology theories. The notion of (co)bordism was already present in the works of for example Poincaré and Pontryagin, but was not yet brought to full fruition until a 1954 paper of René Thom. His work revolved around unoriented and oriented manifolds, but this has been generalised since then to other types of structure. The type that we will treat is the following. By manifolds, we will mean closed, that is, compact without boundary, smooth manifolds. Definition A manifold M is called stably almost complex if there exists a stably complex structure on its tangent bundle T (M), that is, there exists an isomorphism T (M) ε n = ξ for some n and some complex vector bundle ξ on M, where ε n is the trivial real n-dimensional vector bundle on M. In contrast to the stricter demand of a complex structure, odd dimensional manifolds can have such a structure too, allowing you to define the bordism equivalence relation for them. Recall that elements in the n-th singular homology group H n (X) of a space X are generated by singular n-simplices σ : n X under a certain equivalence relation. The idea behind bordism is to replace these standard simplices by manifolds of a fixed type. Definition Let X be a topological space. A singular stably almost complex manifold in X is a pair (M, f), where M is a stably almost complex manifold and f : M X is a map. Two singular stably almost complex manifolds (M.f) and (N, g) in X are called bordant if there exists a stably almost complex manifold W with boundary such that W = M N, together with a map F : W X such that F M = f and F N = g. 10

17 Every pair (M, f) is bordant to itself via W := M I, so this relation is reflexive. It is also transitive, because if we have W 1 = M1 M 2 and W 2 = M2 M 3, then a manifold W 3 can be defined by gluing W 1 to W 2 along M 2. This requires the so called Collar theorem (Hatcher 2001, Proposition 3.42) from differential geometry. Then W 3 = M1 M 3 and so we conclude that bordism is an equivalence relation. Note that if we would not demand that W is compact also, then every manifold M would be bordant to the empty set by taking W := M [0, ), and the equivalence relation would collapse. Let us write MU n (X) for the set of bordism classes of singular n-dimensional stably almost complex manifolds in X for n 0 and define the empty set to be a manifold of every dimension. Then this forms an abelian group under the disjoint union with unit the empty set. This is well-defined, because if W 1 = M M and W 2 = N N, then (W 1 W 2 ) = (M N) (M N ). Definition The n-th complex bordism group MU n (X) of a space X is defined to be the group of bordism classes of singular n-dimensional stably almost complex manifolds in X. Moreover, the graded abelian group MU (X) forms a ring under the cartesian product of manifolds. This is also well-defined. Without giving the boundary operators and proving it, we claim that this defines a multiplicative generalised homology theory, called complex bordism. Contrary to K-theory, for example, it turns out to more convenient to define this than the corresponding cohomology theory, cobordism. Because not every stably almost complex manifold is bordant to the empty set, that is, is the boundary of such a manifold, this is not an ordinary homology theory. We will instead go into more detail describing the spectrum that represents complex (co)bordism. That the resulting theories coincide will also not be shown (this is the so called Pontryagin-Thom construction), but giving this homotopytheoretic definition was in fact Thom s major insight. The bridge between geometric and homotopic topology that he thus afforded is built through Thom spaces. Recall that every complex vector bundle over a paracompact base space admits a Hermitian inner product. This can be achieved by choosing a system of local trivialisations for the bundle, selecting an inner product on each of them and then gluing them together using a partition of unity. Each of the fibers therefore possesses a metric. For the rest of this section we will assume that all vector bundles have a metric, secure in the knowledge that this holds for nice base spaces. Definition Let ξ : E(ξ) B be a constant rank vector bundle over a base space B. The (unit) disc bundle D(ξ) B of ξ is the fibre bundle over B of which the total space consists of those vectors in the fibers of E(ξ) which have length at most 1. The (unit) sphere bundle S(ξ) B of ξ consists of the unit vectors. Definition The Thom space (or Thom complex ) T (ξ) of a constant rank vector bundle ξ is the quotient of the total space of its associated disc bundle by that of its sphere bundle: T (ξ) := D(ξ)/S(ξ). 11

18 So the Thom space is obtained from the total space of the bundle by applying one-point compactification to each fiber, and then identifying all the points at to a single point. This will typically be chosen as its base-point. The Thom space does not depend on the chosen metric. In fact, it will be homeomorphic to the one-point compactification of the total space of the bundle if the base space is compact. If ϕ: ξ η is a map of vector bundles, then ϕ(d(ξ)) D(η) and ϕ(s(ξ)) S(η), so ϕ induces a map of Thom spaces T (ϕ): T (ξ) T (η). It is easy to see that this makes T a covariant functor. Examples We give some examples for the real case. The Thom space of the trivial n-dimensional real vector bundle over a point is obviously S n (S 2n in the complex case). The trivial 1-dimensional real vector bundle on S 1 has as Thom space a torus with a longitude curve contracted to a point. The Thom space of the Möbius bundle on S 1, however, is RP 2. Proposition If ξ and η are vector bundles over base spaces B and B respectively, then there exists a natural homeomorphism T (ξ) T (η) T (ξ η). See (Switzer 1975, Proposition 12.28) for a proof. In particular, if θ n is the trivial n-dimensional vector bundle over a point b 0, we have T (ξ θ n ) = T (ξ) T (θ n ) = T (ξ) S 2n = Σ 2n T (ξ). The product bundle ξ θ n on B {b 0 } can be regarded as the Whitney sum ξ ε n on B, where ε n is the trivial n-dimensional vector bundle over B. So T (ξ ε n ) = Σ 2n T (ξ) also. We will now focus on Thom spaces of certain specific bundles. Recall that for each n 1, the classifying space BU(n) of the group of unitary complex n n- matrices U(n) can be identified with the infinite Grassmannian G n (C ), the n-dimensional subspaces of C. So BU(n) has a tautological bundle E n (C ) on it, of which the fibers are simply the points of BU(n) themselves, considered as vector spaces. Definition Let γ n : E n (C ) BU(n) be the tautological n-dimensional vector bundle on the classifying space BU(n). Define MU(n) as its Thom space: MU(n) := T (γ n ). We can now define the spectrum that represents complex cobordism. Proposition The sequence of spaces MU 2n := MU(n), MU 2n+1 := ΣMU(n), for all n 0, where MU(0) := {pt}, forms a spectrum, denoted by MU. Proof. We need to furnish a base-point preserving map Σ 2 MU(n) MU(n+1). The inclusion of groups U(n) U(n + 1) induces a map of classifying spaces i n : BU(n) BU(n + 1). Let i n(γ n+1 ) be the pullback bundle on BU(n) along i n of the tautological bundle γ n+1 on BU(n + 1). Then we have 12

19 i n(γ n+1 ) = γ n ε 1, where γ n is the tautological bundle, and ε 1 is the trivial line bundle on BU(n). By functoriality of the Thom construction this yields a map T (γ n ε 1 ) T (γ n+1 ), and we saw that Σ 2 MU(n) = MU(n) S 2 = MU(n) T (θ 1 ) = T (γ n ε 1 ), where θ 1 is the trivial line bundle over a point. Alternatively, we could have first considered the (n + 1)-dimensional vector bundle γ n ε 1 on BU(n), and then defined i n : BU(n) BU(n + 1) to be its classifying map. Definition The cohomology theory associated to the spectrum M U is called complex cobordism and denoted by MU. The multiplicative structure of this cohomology theory may be alternatively given as coming from the map MU n MU m MU n+m that is induced from the map BU(n) BU(m) BU(n + m), which in turn comes from the block sum of matrices U(n) U(m) U(n + m). The following small remark will be needed in the next chapter. Proposition There exists a homotopy equivalence BU(1) MU(1). Proof. Since U(1) = S 1, the sphere bundle of the tautological line bundle γ 1 : E 1 (C ) BU(1) is nothing but the universal principal U(1)-bundle. Therefore, its total space S(γ 1 ) is contractible and D(γ 1 ) is homotopy equivalent to MU(1) = D(γ 1 )/S(γ 1 ). In turn, the zero-section BU(1) D(γ 1 ) is also a homotopy equivalence, with homotopy inverse the projection D(γ 1 ) BU(1), since the fibers of D(γ 1 ) are contractible. In other words, taking the Thom space of the tautological line bundle over BU(1) gives you, up to homotopy, BU(1) back. The structure of the coefficient ring of complex cobordism was elucidated independently by John Milnor and Sergei Novikov in It turned out to be a polynomial algebra, similar to an earlier calculation by Thom for unoriented cobordism. Theorem (Milnor, Novikov). There exists a ring isomorphism where deg x 2i = 2i for all i 1. MU (pt) = Z[x 2, x 4, x 6,...], There is no canonical set of generators, but x 2i may be taken to be the class of CP i if i is of the form p 1 for some prime number p. 13

20 Chapter 2 A bridge to algebra After the previous chapter, the reader should be familiar with ordinary cohomology, complex K-theory and (the definition of) complex cobordism. The next chapter will start by introducing elliptic cohomology theories which come from number theory. In order to construct these theories, it is necessary to first forge a bridge between cohomology theories and algebra. This is the goal of the current chapter. We will show that the smaller class of so called complex oriented cohomology theories have certain naturally associated power series, namely formal group laws, to them. Since we saw in Section 1.1 that generalised cohomology theories are essentially topological objects by considering the spectra that represent them, this procedure can be seen to lie in the general vein of algebraic topology: extracting algebraic invariants from topological spaces. In Section 2.3 we will see how this sometimes can be reversed and cohomology theories can be cooked up from a formal group law. This will be the key to constructing elliptic cohomology theories in the next chapter. 2.1 Formal group laws In this section we take a brief excursion from the world of topology to that of algebra. We introduce formal group laws, which are certain power series that were first defined by Salomon Bochner in 1946 in the context of Lie theory. The rough idea behind them is that a formal group law describes a group addition law, while lacking (a priori) a set of elements to be made into a group. Definition A formal group law over a ring R is a two-variable power series F (x, y) R[[x, y]] satisfying (i) F (x, 0) = x R[[x]] (identity), (ii) F (x, y) = F (y, x) R[[x, y]] (commutativity), (iii) F (x, F (y, z)) = F (F (x, y), z) R[[x, y, z]] (associativity). So by (i) and (ii), F (x, y) is of the form F (x, y) = x + y + i,j 1 a ij x i y j for some coefficients a ij R and, furthermore, a ij = a ji for all i and j. The above definition is actually that of a 1-dimensional commutative formal group 14

21 law and n-dimensional ones can arise from, for example, Lie groups. We will not discuss the latter though. One might ask if perhaps an axiom for an inverse is missing, but this turns out to be automatic. Proposition If F is a formal group law over a ring R, then there exists a unique power series i(x) R[[x]] such that F (x, i(x)) = 0 (so also F (i(x), x) = 0). Proof. The proof of the existence is very similar to that of Proposition 2.1.5, so we will skip this. Using the notation x + F y := F (x, y), it becomes obvious that the uniqueness is proven exactly the same as it is done for group inverses. Let us look at some examples of formal group laws. Examples (i) The simplest example is the additive formal group law given by F (x, y) = x + y. It is defined over any ring. (ii) For any element u in a ring R one can define a multiplicative formal group law F (x, y) = x + y + uxy. It is so named because when u = 1, we can write F (x, y) = (x + 1)(y + 1) 1. This can be seen as wanting to multiply x and y, but first needing to shift to the multiplicative identity 1, and afterwards translating back again. x+y 1+xy/c 2 (iii) If c is an invertible element of a ring R, then F (x, y) = defines a formal group law over R. What we mean by such notation is the formal expansion as a Taylor series. So actually, F (x, y) = (x + y) ( 1 xy/c 2 + (xy/c 2 ) 2 (xy/c 2 ) 3 + ). It is sometimes called the Lorentz formal group law since it describes the velocity addition formula in special relativity. If c = 1, it is also the addition formula for the hyperbolic tangent: tanh(x+y) = F (tanh(x), tanh(y)). (iv) Over any ring in which 2 is invertible there exists Euler s formal group law F (x, y) = x 1 y 4 + y 1 x x 2 y 2. The inverse of 2 is needed because the denominators of the coefficients of the expansion of 1 + t are powers of 2. One can check associativity by a calculation, but we will see the true reason that it satisfies the axioms in Chapter 3: it is the formal group law describing the addition on an elliptic curve. Definition A morphism θ : F G of formal group laws defined over a ring R is a one-variable power series θ(x) R[[x]] without a constant term such that θ(f (x, y)) = G(θ(x), θ(y)). We call θ an isomorphism if there exists a reverse morphism τ : G F such that θ(τ(x)) = τ(θ(x)) = x. That this definition is correct is suggested by the notation x + F y := F (x, y). It is clear that formal group laws over a fixed ring R form together with their morphisms a category which we denote by FGL(R). 15

22 If f : R S is a ring homomorphism and F (x, y) = x + y + i,j 1 a ijx i y j is a formal group law over R, then the power series (f F )(x, y) := x + y + i,j 1 f(a ij)x i y j is a formal group law over S. So, f induces a functor f from FGL(R) to FGL(S). For an inverse τ of a morphism θ, it is actually only needed to demand that θ(τ(x)) = τ(θ(x)) = x, because having no constant term and being a morphism is automatic: we have θ(0) = 0, so τ(0) = τ(θ(0)) = 0 and secondly, τ ( G(x, y) ) = τ ( G(θτ(x), θτ(y)) ) = τ ( θ(f (τx, τy)) ) = F ( τ(x), τ(y) ). The standard proof that an inverse of a morphism is unique, holds. We give a condition for its existence. Proposition A morphism θ : F G of formal group laws defined over a ring R is an isomorphism if and only if the coefficient a 1 of x in θ(x) is a unit in R. Proof. The only if statement is clear. For the if statement, we will construct inductively a sequence of polynomials τ n (x) R[x] such that θ(τ n (x)) x (mod x n+1 ). Start by defining b 1 := a 1 and τ 1 (x) := b 1 x. Suppose next that τ n (x) has been constructed with the desired property. Then θ(τ n (x)) = x + cx n+1 (mod x n+2 ) for some c R. Define b n+1 := a 1 1 c and τ n+1(x) := τ n (x) + b n+1 x n+1. Then θ ( τ n+1 (x) ) θ ( τ n (x) ) + a 1 b n+1 x n+1 x + cx n+1 cx n+1 x (mod x n+2 ). So for τ(x) := n=1 b nx n R[[x]] we have θ(τ(x)) = x, because τ(x) = lim τ n (x). Repeating the same procedure for τ(x), instead of for θ(x), gives a power series ψ(x) R[[x]] such that τ(ψ(x)) = x. We then see that also τ ( θ(x) ) = τ ( θ ( τ ( ψ(x) ))) = τ ( ψ(x) ) = x. The inverse i(x) from Proposition of a formal group law F is an example of an endomorphism. We namely have i(0) = F (i(0), 0) = 0 and the equality i(f (x, y)) = F (i(x), i(y)) can be proven by repeated use of commutativity and associativity and the uniqueness of i(x). Other examples are provided by Definition Let F be a formal group law over a ring R. Define for every n 1 the n-series of F inductively as [n] F (x) := F (x, [n 1] F (x)) and [0] F (x) := 0, or in alternative notation, [n] F (x) = x + F + F x. }{{} n times That these are endomorphisms of F can be checked via induction and, again, repeated use of commutativity and associativity. We have for example [n] F (x) = nx for the additive formal group law, and [n] F (x) = (x + 1) n 1 for the multiplicative formal group law if u = 1. The n-series of F is an automorphism if and only if n is invertible in R, because [n] F (x) always starts as nx +. We now come to the notion of height of formal group laws over certain types of rings. 16

23 Proposition Let p be a prime number and R be a ring such that p = 0 in R. If θ : F G is a morphism between formal group laws over R such that the coefficient θ (0) of x in θ(x) is zero, then there exists an h 0 such that θ(x) = ϕ(x ph ) for some ϕ(x) R[[x]]. Proof. By the definition of a morphism, we have θ(f (x, y)) = G(θ(x), θ(y)). Now take the partial derivative with respect to y of both sides and evaluate at y = 0: θ (x) F y G( ) (x, 0) = θ(x), θ(0) θ (0) = 0. y Since ( F/ y)(x, 0) has constant term 1, θ (x) must be 0. Therefore, θ(x) = ϕ 1 (x p ) for some ϕ 1 R[[x]]. If the coefficient of x p in θ(x) is also zero, we can proceed iteratively and repeat the same procedure for ϕ 1, but first we must prove that ϕ 1 is a morphism of formal group laws over R too. Write F (x, y) = a ij x i y j and define F (p) (x, y) := a p ij xi y j. Then F (p) is a formal group law over R because a a p is an endomorphism of R. We have ϕ 1 ( F (p) (x p, y p ) ) = ϕ 1 ( F (x, y) p ) = θ ( F (x, y) ) So ϕ 1 is a morphism from F (p) to G. = G ( θ(x), θ(y) ) = G ( ϕ 1 (x p ), ϕ 1 (y p ) ). Definition The largest possible integer h in Proposition is called the height of the morphism θ. If θ = 0, then we say that its height is. The height of F is the height of the endomorphism [p] F : F F. The additive formal group law has infinite height because its p-series is simply px. It can in fact be proven that it is the only one with this height. The multiplicative formal group law F has height 1 because for u = 1, for example, we have [p] F (x) = (1 + x) p x p 1 = x p (mod p). Definition A formal group law F over a ring L is called universal if for any other formal group law G over any ring R there is a unique ring homomorphism θ : L R such that θ (F ) = G. By the usual argument, a universal formal group law is unique up to unique isomorphism, that is, if there were two of them, F over L and F over L, then θ : L L with θ (F ) = F is a ring isomorphism. We now prove that such a thing exists. Theorem There exists a universal formal group law. Proof. We will construct L as the quotient of a ring by an ideal. Let Z[a ij ] = Z[a 11, a 12, a 21,..., a ij,...] be the polynomial ring over Z in formal variables a ij for all integers i, j 1. Define the power series F (x, y) := x + y + a ij x i y j Z[a ij ][[x, y]]. i,j 1 17

24 over Z[a ij ] and set b ijk Z[a ij ] to be the elements determined as follows: F ( F (x, y), z ) F ( x, F (y, z) ) = i,j,k b ijk x i y j z k. Let I be the ideal of Z[a ij ] generated by the differences a ij a ji and the coefficients b ijk. Then the ring L := Z[a ij ]/I satisfies the desired properties by letting the map to any other formal group law be the obvious one. The ring L constructed in this proof is called the Lazard ring, named after Michel Lazard. From topological considerations which we will see in Section 2.3, the ring Z[a ij ] is typically graded by giving a ij degree 2(i + j 1). If we assign to x and y degree 2, then the power series F (x, y) is homogeneous of degree 2. So F (F (x, y), z) and F (x, F (y, z)) are also of degree 2, which means that the coefficients b ijk Z[a ij ] are homogeneous of degree 2(i + j + k 1). Therefore, the ideal I is generated by homogeneous elements and this grading on Z[a ij ] descends to a grading on L. So the construction of the universal formal group law is very simple. Just like the universal enveloping algebra of a Lie algebra, you simply take something big and divide out the relations that make it fit just right into the universal property. Similarly, the explicit structure of our object is not obvious at all. The universal enveloping algebra requires the involved Poincaré-Birkhoff-Witt theorem and in 1955 Lazard proved the following result. Theorem (Lazard). As a graded ring, L is isomorphic to Z[x 1, x 2,...], where deg x i = 2i for all i 1. See (Adams 1974) for a proof. The theorem implies that formal group laws are abundant: for any ring R simply choosing a countable set of elements of R will give a formal group law over R. A further corollary is that given a surjection of rings f : R S, any formal group law F S over S can be lifted to one over R, F R, meaning that f (F R ) = F S. So, there is a surjection FGL(R) FGL(S). 2.2 Complex oriented cohomology theories In this section we tie in the previous one on formal group laws with our story on cohomology theories. We introduce the restricted class of so called complex oriented cohomology theories and show how each of them has a naturally associated formal group law. A complex oriented cohomology theory is a generalised cohomology theory h that has a certain extra set of tools for working with complex vector bundles. There are many equivalent formulations for what these tools are, but one of them is that h has Chern classes, similar to ordinary cohomology. So, roughly, it has a sequence of natural transformations c 1, c 2,... from the vector bundle functor Vect C on the category of topological spaces, to h. Since we know that each of the functors Vect C n is representable by a classifying space BU(n), by Yoneda s lemma, it is the same to specify arbitrary elements in the cohomology ring h (BU(n)). We put some restrictions on these elements though, because our theory of Chern classes should also satisfy properties like a normalisation and additivity. 18

25 We will only define what it means for a cohomology theory to have first Chern classes. It is namely possible to uniquely construct from these higher Chern classes via the splitting principle, but we will not need those. Recall that BU(1) has, up to homotopy, the explicit description CP. Definition A multiplicative cohomology theory h is called complex oriented if there exists a class x h (CP ) such that h (CP 1 ) is freely generated by the element i (x) h (CP 1 ) as an h (pt)-module. Here, i denotes the natural inclusion CP 1 CP. Such an element x is an orientation of the theory. Notice that CP 1 = S 2 = Σ 2 S 0, so h (CP 1 ) = h (Σ 2 S 0 ) = h 2 (S 0 ) = h 2 (pt). Therefore, h (CP 1 ) already is a free h (pt)-module with generator the multiplicative unit 1 h 0 (pt) = h 2 (CP 1 ). We do not require that i (x) = 1 though, or even that i (x) lies in h 2 (CP 1 ). Our definition merely implies that i (x) = u 1 for an invertible element u h (pt). We define for the tautological line bundle γ 1 on CP its first Chern class c 1 (γ 1 ) h (CP ) as c 1 (γ 1 ) := x. For a general line bundle ξ on a space B, classified by a map ϕ: B CP, we define its first Chern class c 1 (ξ) to be c 1 (ξ) := ϕ (x) h (B). This c 1 is indifferent to isomorphisms of vector bundles and is furthermore natural, that is, if f : B B is a map and f (ξ) is the pullback of ξ along f, then c 1 (f (ξ)) = f (c 1 (ξ)). The classifying map of f (ξ) namely is ϕ f. Since the trivial line bundle B C over B is classified by the base-point map B CP, and x is a reduced cohomology class, its first Chern class is zero. Note that a complex oriented cohomology theory might allow for different choices of orientation. Let us look at some examples. Examples (i) Ordinary cohomology is complex oriented by taking as an orientation the usual generator x of H (CP ) = Z[x], where deg x = 2. We namely know that H (CP 1 ) = Z[x]/(x 2 ) and that the morphism H (CP ) H (CP 1 ) induced by the inclusion is the projection. (ii) We know about complex K-theory that K (pt) = Z[u, u 1 ], where deg u = 2, and that K 0 (CP 1 ) = Z[y]/(y 1) 2, where y = [ξ], the class of the tautological line bundle ξ on CP 1 and 1 is the trivial line bundle. So K 0 (CP 1 ) is generated by [ξ] 1. Because the tautological line bundle γ 1 on CP restricts to ξ on CP 1, it follows that the class x = [γ 1 ] 1 K 0 (CP ) provides complex K-theory with an orientation. For reasons related to grading though, it turns out to be more convenient to choose x = u 1 ([γ 1 ] 1), so that x K 2 (CP ). (iii) We saw in Proposition that there exists a homotopy equivalence f : CP = BU(1) MU(1). This map defines an element [f] CP, MU(1) = MU 2 (CP ), which restricts to the inclusion in CP 1, CP = S 2, CP = π 2 (CP ) = Z. Because CP 1, CP = MU 2 (CP 1 ), [f] is an orientation. 19

26 We will now prove that for a complex oriented cohomology theory h with orientation x h (CP ), there is an isomorphism h (CP ) = h (pt)[[x]]. First consider for each n 1 the morphism of graded h (pt)-algebras h (pt)[x] h (CP n ). That x n+1 is sent to zero can be seen as follows. One can cover CP n by the standard n + 1 affine open subsets U i = {[z j ] : z i 0}. Since these are contractible and x is a reduced cohomology class, x restricts on each U i to zero. Therefore, x h (CP n, U i ) for all i. By the multiplicativity of h this means that x n+1 h (CP n, U 1 U n+1 ) = h (CP n, CP n ) = 0. We now claim Proposition The induced homomorphism h (pt)[x]/(x n+1 ) h (CP n ) of graded h (pt)-algebras is an isomorphism. Proof. Consider the Atiyah-Hirzebruch spectral sequence E p,q 2 = H p (CP n ; h q (pt)) h p+q (CP n ) of CP n. The E 2 -page is isomorphic to h (pt)[x]/(x n+1 ). Because we know that x h (CP n ), x must survive to the E -page and so all the differentials vanish on x. They are furthermore h (pt)-algebra morphisms and so also vanish on the entire E 2 -page. Therefore the spectral sequence collapses at the E 2 -page. So this coincides with the known result H (CP n ) = Z[x]/(x n+1 ) for ordinary cohomology. To calculate h (CP ) we would now like to take inverse limits of both sides, hoping that the inverse limit of the right hand side can be pushed through h to become a direct limit. Homology functors do commute with direct limits, but the situation turns out to be more subtle for cohomology. There exists a sort of error term, an obstruction, identified by John Milnor in We will need the following piece of homological algebra. Definition Let A 1 f 1 A2 f 2 A3 f 3 be an inverse system of abelian groups indexed by the positive integers and δ : n A n n A n the map given by δ(a 1, a 2,...) = ( a 1 f 1 (a 2 ), a 2 f 2 (a 3 ),... ). The first derived functor of the inverse limit functor on the system {A n } is defined as the cokernel of δ and denoted by lim 1 A n. Notice that the inverse limit of {A n } can also be expressed in terms of δ: it is nothing but its kernel. Milnor used the above functor as follows. Proposition (Milnor). For h any reduced or unreduced cohomology theory and X 0 X 1 an increasing sequence of cw-complexes with union X, there exists a short exact sequence 0 lim 1 h 1 (X i ) h λ (X) lim h (X i ) 0, where λ is the natural map that sends an element of h (X) to its sequence of images in h (X i ). 20

27 For a proof, see (Hatcher 2001, Theorem 3F.8). So, because lim 1 is the obstruction towards lim commuting with a cohomology functor to become lim, it is important to know when it vanishes. There exists a more general so called Mittag-Leffler condition for this to happen, but for our needs it suffices to notice that lim 1 vanishes when all the maps f n in an inverse system of abelian groups are surjective. To prove the surjectivity of δ in this case we need to find for every element (b 1, b 2,...) in n A n an element (a 1, a 2,...) such that a 1 f 1 (a 2 ) = b 1, a 2 f 2 (a 3 ) = b 2 and so forth. Set a 1 = 0 and define inductively a 2 such that f 1 (a 2 ) = b 1, a 3 such that f 2 (a 3 ) = a 2 b 2 and so forth. This helps to finally prove Theorem Let h be a complex oriented cohomology theory with orientation x h (CP ). Then there exists an isomorphism h (CP ) = h (pt)[[x]]. Let π 1, π 2 : CP CP CP be the projection maps and π i (γ 1) be the pullback bundles on CP CP of the tautological line bundle γ 1 on CP. Define x i := c 1 (π i γ 1) = π i (x) h (CP CP ) to be their first Chern classes. Then there also exists an isomorphism h (CP CP ) = h (pt)[[x 1, x 2 ]]. Proof. The maps i : h (CP n+1 ) h (CP n ) are surjective because they are precisely the natural maps h (pt)[x]/(x n+2 ) h (pt)[x]/(x n+1 1 ). So lim vanishes and h (CP ) = lim h (CP n ) = lim h (pt)[x]/(x n+1 ) = h (pt)[[x]]. The second claim can be proven similarly, going through the Atiyah-Hirzebruch spectral sequence and Milnor s exact sequence again, or via the trick of considering h (CP ) as a cohomology theory. This result agrees with what we already know about ordinary cohomology, H (CP ) = Z[x], because in a cohomology ring one usually only considers homogeneous elements. The coefficient ring Z of ordinary cohomology is concentrated in degree zero, so homogeneous elements are finite sums, and the difference between Z[[x]] and Z[x] is ignored. The coefficient ring of a general complex oriented cohomology theory though may have elements of negative degree, so homogeneous elements can be infinite sums. We are now ready to associate a formal group law to a complex oriented cohomology theory. Recall that the set of isomorphism classes of complex line bundles Vect C 1 (B) over a base space B forms an abelian group under the tensor product. The unit element is given by the trivial line bundle over B and dual bundles are inverses. In ordinary cohomology, the reflection of this group operation in the cohomology of B via the first Chern class is very simple: we have c 1 (ξ 1 ξ 2 ) = c 1 (ξ 1 ) + c 1 (ξ 2 ) (2.1) for any two line bundles ξ 1 and ξ 2 on B. This might already be known to the reader, but we will prove it later anyway. One might ask if, for a general complex oriented cohomology theory, there always exists a formula that expresses 21

28 c 1 (ξ 1 ξ 2 ) in terms of c 1 (ξ 1 ) and c 1 (ξ 2 ). We show that this is indeed the case: it is a formal group law, and the right hand side of (2.1) should be read as the additive formal group law. Assume the data of Theorem Then define a power series F (x 1, x 2 ) h (pt)[[x 1, x 2 ]] as the first Chern class of the line bundle π 1(γ 1 ) π 2(γ 1 ) on CP CP : F (x 1, x 2 ) := c 1 (π 1γ 1 π 2γ 1 ) h (CP CP ) = h (pt)[[x 1, x 2 ]]. Since x i := c 1 (πi γ 1), we see that we have expressed the first Chern class of this tensor product as a power series in the first Chern classes of the factors. This holds in greater generality, since π1(γ 1 ) π2(γ 1 ) is the universal line bundle for tensor products. That is, if ξ 1 and ξ 2 are line bundles on a base space B, classified by maps ϕ 1, ϕ 2 : B CP respectively, then these ϕ i s can be written as composites B Φ CP CP πi CP, where Φ = (ϕ 1, ϕ 2 ). So ξ i = ϕ i (γ 1) = Φ (πi γ 1). Because the tensor product of pullbacks along a fixed map is the pullback along that map of the tensor product, we also have ξ 1 ξ 2 = Φ (π1γ 1 π2γ 1 ): E(ξ 1 ξ 2 ) E(π 1γ 1 π 2γ 1 ) E 1 (C ) ξ 1 ξ 2 π γ 1 γ 1 π2 γ 1 1 Φ B CP CP CP We can therefore express the first Chern class of this tensor product as c 1 (ξ 1 ξ 2 ) = c 1 Φ (π 1γ 1 π 2γ 1 ) = Φ c 1 (π 1γ 1 π 2γ 1 ) = Φ F (x 1, x 2 ) = F (Φ x 1, Φ x 2 ). Here, we used the naturality of c 1 in the second step, and the fact that Φ is an h (pt)-algebra homomorphism in the last one. Next, we have Φ x i = Φ c 1 (π i γ 1 ) = c 1 (Φ π i γ 1 ) = c 1 (ξ i ). So c 1 (ξ 1 ξ 2 ) = F (c 1 ξ 1, c 1 ξ 2 ). (A little caution needs to be taken here a power series in h (B) might not make sense. If B is a finite cw-complex, however, then the c 1 (ξ i ) are nilpotent. The classifying maps ϕ i of the ξ i namely are homotopic to maps B CP n for n the dimension of B by cellular approximation, and we learned that h (CP n ) = h (pt)[x]/(x n+1 ). So F (c 1 ξ 1, c 1 ξ 2 ) is an innocent polynomial.) Applying this formula to specific tensor products of line bundles over CP CP and (CP ) 3 shows that F (x 1, x 2 ) is a formal group law. If ε 1 is the trivial line bundle over CP CP, then we know that c 1 (ε 1 ) = 0. So F (x 1, 0) = F (c 1 π 1γ 1, c 1 ε 1 ) = c 1 (π 1γ 1 ε 1 ) = c 1 (π 1γ 1 ) =: x 1. Because π 1(γ 1 ) π 2(γ 1 ) = π 2(γ 1 ) π 1(γ 1 ), commutativity of F (x 1, x 2 ) also follows. It is for checking of the associativity that we need to consider (CP ) 3. 22

29 A similar calculation as for Theorem results in h ( (CP ) 3) = h (pt)[[x 1, x 2, x 3 ]], where x i := c 1 (π i γ 1). By the associativity up to isomorphism of the tensor product of vector bundles, we then have F ( x 1, F (x 2, x 3 ) ) = F ( c 1 π 1γ 1, c 1 (π 2γ 1 π 3γ 1 ) ) = c 1 ( π 1 γ 1 (π 2γ 1 π 3γ 1 ) ) = c 1 ( (π 1 γ 1 π 2γ 1 ) π 3γ 1 ) = F ( c 1 (π 1γ 1 π 2γ 1 ), c 1 π 3γ 1 ) ) = F ( F (x 1, x 2 ), x 3 ). This link between cohomology theories and formal group laws was perhaps first observed in section 5 of (Novikov 1967), where Sergei Novikov and Aleksandr Miščenko discuss the formal group law associated to complex cobordism. We remark that the map µ: CP CP CP that classifies π 1(γ 1 ) π 2(γ 1 ) happens to be the map ( [x0 : x 1 : ], [y 0 : y 1 : ] ) [x 0 y 0 : x 0 y 1 + x 1 y 0 : : i x iy k i : ] that makes CP an H-space. It is homotopy associative and strictly commutative, and this can also be used to prove that F (x 1, x 2 ) is a formal group law. Therefore, to define a complex oriented cohomology theory and extract a formal group law from it requires no references to vector bundles and their Chern classes whatsoever. We could simply define F (x 1, x 2 ) as µ (x). This is the approach taken in, for example, (Adams 1974), but it is the author s opinion that it makes the story fairly mysterious. As mentioned before, a theory might have different choices of orientation. This does not really matter for the formal group law though. Proposition If h is a complex oriented cohomology theory and x and x are two orientations for it, then the associated formal group laws F and F over h (pt) are isomorphic. Proof. We know that h (pt)[[x]] = h (CP ) = h (pt)[[x ]], so there exists a series θ(x ) h (pt)[[x ]] such that x = θ(x ). Moreover, θ(x ) can not have a constant term and its coefficient of x is a unit in h (pt). Since µ (x ) =: F (x 1, x 2) and µ is a ring homomorphism, we have µ (θ(x )) = θ(f (x 1, x 2)). In turn, µ (θ(x )) = µ (x) =: F (x 1, x 2 ). We also know that h (pt)[[x 1, x 2 ]] = h (CP CP ) = h (pt)[[x 1, x 2]], where x i = πi (x) and x i = π i (x ) for i = 1, 2. Again, because the πi s are ring homomorphisms, we have πi (θ(x )) = θ(x i ). In turn, x i = πi (x) = π i (θ(x )). This proves that θ(x ) is an isomorphism from F to F. It is also easily checked that if G is a formal group law over h (pt) that is isomorphic to F, then there exists an orientation y of h such that its associated formal group law is G. The formal group law of a complex oriented cohomology theory h will be denoted by F h if the choice of orientation is not important. 23

30 Example Let us prove that the formal group law associated to ordinary cohomology is indeed additive, as stated earlier. Consider the inclusion CP CP CP CP. Because the difference (CP CP )\(CP CP ) consists of cells of dimension higher than, or equal to 4 only, we have H 2 (CP CP ) = H 2 (CP CP ). In turn, we have the wedge isomorphism j 1 j 2 : H2 (CP CP ) = H 2 (CP CP ) = H 2 (CP ) H 2 (CP ), where the maps j 1, j 2 : CP CP CP CP CP are the inclusions. Because the bundle π 2(γ 1 ) on CP CP is trivial on the first factor, the tensor product π 1(γ 1 ) π 2(γ 1 ) restricts to j 1(π i γ 1) = (id CP ) (γ 1 ) = γ 1 on the first factor. So the effect on the first Chern class of π 1(γ 1 ) π 2(γ 1 ) is that j 1( c1 (π 1γ 1 π 2γ 1 ) ) = c 1 (γ 1 ) H 2 (CP ). The same holds for j 2. Together, they give (j 1 j 2) ( c 1 (π 1γ 1 π 2γ 1 ) ) = c 1 (γ 1 ) + c 1 (γ 1 ). Because the maps π i j i : CP CP are the identity maps, we have j i (c 1π i γ 1) = c 1 (γ 1 ). Since j 1 j 2 is an isomorphism, we conclude that c 1 (π 1γ 1 π 2γ 1 ) = c 1 (π 1γ 1 ) + c 1 (π i γ 1 ). Example Let us use the element x = u 1 ([γ 1 ] 1) K 2 (CP ) from Examples 2.2.2(ii) as an orientation of complex K-theory. For a line bundle ξ on a base-space B, classified by a map ϕ: B CP, we then have c 1 (ξ) = ϕ (x) = ϕ ( u 1 ([γ 1 ] 1) ) = u 1( ϕ ([γ 1 ]) 1 ), because ϕ is a ring homomorphism. Next, we have ϕ ([γ 1 ]) = [ϕ γ 1 ] and ϕ γ 1 = ξ. So c 1 (ξ) = u 1 ([ξ] 1). If we apply this formula to the line bundle π 1(γ 1 ) π 2(γ 1 ) on CP CP, we can rewrite it as c 1 (π 1γ 1 π 2γ 1 ) = u 1( [π 1γ 1 π 2γ 1 ] 1 ) = u 1( [π 1γ 1 ] [π 2γ 1 ] 1 ) = u 1( [π 1γ 1 ] 1 ) + u 1( [π 2γ 1 ] 1 ) + u 1( [π 1γ 1 ] [π 2γ 1 ] [π 1γ 1 ] [π 2γ 1 ] + 1 ) = c 1 (π 1γ 1 ) + c 1 (π 2γ 1 ) + u c 1 (π 1γ 1 ) c 1 (π 2γ 1 ). This shows that the formal group law associated to complex K-theory is multiplicative: F K (x 1, x 2 ) = x 1 + x 2 + ux 1 x 2. Not every formal group law can be obtained from a complex oriented cohomology theory. An example is given on (Rudyak 2008, pg. 443), where it is shown that the formal group law over MU (pt) Z/2Z induced by the modulo 2 reduction MU (pt) MU (pt) Z/2Z can not come from such a theory. We close this section by mentioning the following theorem. Theorem For any complex oriented cohomology theory h, there exists a unique morphism of multiplicative cohomology theories such that T (F MU ) = F h. T : MU h 24

31 Proofs can be found in (Kono and Tamaki 2006, Theorem 3.30) or (Adams 1974, pg. 52). Complex cobordism can therefore be seen as the universal complex oriented cohomology theory. The theorem also implies that for a cohomology theory, being complex oriented means the same thing as having a multiplicative morphism from MU to the theory. 2.3 Landweber s exact functor theorem In this section we state the Landweber exact functor theorem, which allows you to reverse the procedure of the previous section under certain conditions. That is, given a formal group law, construct a complex oriented cohomology theory such that its associated formal group law is the one that you started with. We will refer to the literature for a proof of this theorem, and instead give a simple application by constructing complex K-theory from its formal group law. The starting point is the following theorem by Quillen. We saw in Examples that complex cobordism has a complex orientation. Therefore, there exists a unique ring homomorphism θ MU : L MU (pt) from Lazard s universal ring to the coefficient ring of the theory such that θ MU (F ) = F MU, where F is the universal formal group law and F MU is the formal group law associated to complex cobordism. Notice that L and MU (pt) happen to be isomorphic as rings by Lazard s and Milnor s and Novikov s structure theorems and In 1969 Daniel Quillen proved the following, surprising, stronger statement, inspired by ideas about universal cohomology theories in algebraic geometry. Theorem (Quillen). The classifying map θ MU : L MU (pt) is an isomorphism of graded rings. So, while Theorem says that complex cobordism s formal group law is universal among all that come from complex oriented cohomology theories, Quillen claims that it is universal among all formal group laws, period! It is nothing but Lazard s universal one. This is a stronger statement by the example from the previous section of a formal group law that can not be obtained from a complex oriented cohomology theory. A proof can be found in (Adams 1974, p. 75). To the author s knowledge, it is as of 2012 still not known whether complex cobordism itself can be constructed through algebraic data, without referencing manifolds or vector bundles. This theorem is the reason for the strange grading that we put in Section 2.1 on L. In Examples we namely gave x and y in the formal group law F MU degree 2. So if we want F MU (x, y) MU 2 (CP CP ) to have degree 2 also, the coefficients a ij of the terms x i y j in F MU (x, y) should have degree 2(i + j 1). Moving on, this theorem makes it reasonable to ask whether perhaps every formal group law F over any ring R comes from a complex oriented cohomology theory and whether we then could also explicitly construct such a theory. One candidate would be the functor X MU (X) MU (pt) R (2.2) 25

32 on some category of topological spaces, where R has been given some grading and is a module over MU (pt) = L via Quillen s theorem. Here, we give the tensor product of two graded modules M and N over a ring S the canonical grading (M S N) n := i+j=n M i S N j. This is indeed a multiplicative cohomology theory because MU is, except that perhaps the exactness axiom is not satisfied; tensoring an exact sequence with an arbitrary ring might not preserve exactness. Furthermore, Proposition If (2.2) satisfies the exactness axiom of a cohomology theory, then this theory has coefficient ring R, is complex oriented and its formal group law is isomorphic to F. Proof. The first statement is clear. If x is a complex orientation for MU, then x 1 is an orientation for our theory. For the third statement, name our theory h F. Then we have for every n h F (CP n ) = ( MU (pt)[x]/(x n+1 ) ) MU (pt) R = ( MU (pt) MU (pt) R ) [x]/(x n+1 ) = R [x]/(x n+1 ). Here, in the first line x is still an orientation for MU, but in the second and third line it denotes the orientation x 1 for h F. Taking inverse limits using Milnor s theorem as in the previous section shows that h F (CP ) = R [[x]]. Similarly, h F (CP CP ) = R [[x 1, x 2 ]]. It is convenient to use the definition of a formal group law via the H-space map µ: CP CP CP, as explained in the previous section. We then have µ (x) =: F MU (x 1, x 2 ) for the induced map on MU. Now consider the induced map on h F : µ : MU (CP ) MU (pt) R MU (CP CP ) MU (pt) R. Then µ (x 1) = F MU (x 1, x 2 ) 1. Because the map MU (pt) R that is present in this tensor product sends the coefficients of F MU to those of F, the ring isomorphism between MU (pt) MU (pt) R and R gives an isomorphism between F MU (x 1, x 2 ) 1 and F. Another motivation to define your cohomology theory like this is the Conner- Floyd isomorphism. In 1966, Pierre Conner and Ed Floyd constructed a natural transformation MU K of homology theories, making K (pt) into a MU (pt)-module. They then proved that this induces an isomorphism MU (X) MU (pt) K (pt) = K (X). This showed that complex K-theory is determined by complex cobordism, although they did use the knowledge beforehand that K is a homology theory. We would like to imitate this isomorphism. An obvious condition for exactness is that R is a flat module over the ring MU (pt). This is quite strong though and it seems plausible that a weaker condition than flatness could allow (2.2) to become a cohomology theory, since the functor MU (pt) R does not need to be exact for all MU (pt)-modules, but only for those of the form MU (X, A) and the morphisms between them. And indeed, in the 1970 s Peter Landweber found sufficient criteria on the map MU (pt) R to satisfy our purposes. His theorem initially constructs a homology theory instead. 26

33 We define for every prime number p a sequence of elements (u 1, u 2, u 3,...) of MU (pt), as the coefficients of x p, x p2,... in the p-series [p] FMU (x) = px + + u 1 x p + + u 2 x p2 + of the (universal) formal group law of MU over MU (pt). These elements thus depend on p, but we will supress this in the notation. We can now formulate Landweber s condition. Theorem (Landweber s exact functor theorem). Let M be a graded MU (pt)-module such that for each prime number p the sequence of elements (p, u 1, u 2,...) of MU (pt) is M-regular, that is, multiplication by p on M and by u n on M/(p, u 1,..., u n 1 )M for all n 1 is injective. Then the functor X MU (X) MU (pt) M on the category of cw-complexes is a homology theory. For finite cw-complexes, the associated cohomology theory is given by X MU (X) MU (pt) M, where M is M with the opposite grading, that is, M n = M n. This proven in (Landweber 1976). We will apply the theorem in the special case when M is a graded ring R with a formal group law F over it. The elements p, u 1, u 2,... will often be interpreted as the corresponding coefficients of the p-series of F. We give two simple applications. Example Let F H (x, y) = x + y be the additive formal group law of ordinary cohomology with coefficients in Q. Let us grade Q by Q 0 = Q and Q n = 0 for n 0. We have [p] FH (x) = px, so the u n s associated to p are all zero. Since multiplication by p on Q is obviously injective and the quotient ring Q/(p) is zero, Landweber s conditions are satisfied and X MU (X) MU (pt) Q defines a homology theory. In fact, by the uniqueness of ordinary homology, this is singular homology with coefficients in Q. The additive formal group law over any Q-algebra would work too, but, for example, Z would not meet the conditions. So not all formal group laws give rise to cohomology theories in this way. Example Let F K (x, y) = x+y +uxy be the multiplicative formal group law of complex K-theory over its coefficient ring K (pt) = Z[u, u 1 ], where deg u = 2. Writing F K (x, y) = u(x + u 1 )(y + u 1 ) u 1, one sees that for every prime number p, [p] FK (x) = u p 1 (x + u 1 ) p u 1. The coefficients of x pn in this series are u p 1 for n = 1 and 0 for n > 1. Let us now fix p. We are asking whether the sequence (p, u p 1, 0, 0,...) is K (pt)- regular. The multiplications by p on Z[u, u 1 ] and by u p 1 on (Z/pZ)[u, u 1 ] 27

34 are certainly injective and the quotient ring (Z/pZ)[u, u 1 ]/(u p 1 ) is zero. So our checking is done and X MU (X) MU (pt) K (pt) defines a homology theory. This is complex K-theory homology. In the next section we will study a more dramatic application of Landweber s theorem, namely to formal group laws of elliptic curves. 28

35 Chapter 3 Enter number theory In this chapter number theory makes a sudden appearance. We will show how elliptic curves have naturally associated formal group laws to them, derived from the group structure on their rational points. This is classic and is used for example in (Silverman 2009) in a proof of the weak Mordell-Weil theorem. Next, we demonstrate how for certain families of elliptic curves, these formal group laws satisfy the conditions of Landweber s theorem from the previous chapter. We thus obtain many so called elliptic cohomology theories. 3.1 Formal group laws from elliptic curves Let us first agree on what an elliptic curve is. Definition An elliptic curve over a field K is a non-singular projective algebraic curve C of genus 1 over K, together with a specified point O C. Some subtility is actually needed here; we should define the curve to consist of points with coordinates in an algebraic closure of K and then require that O is a K-rational point, that is, it has coordinates in K. It is well-known that using the Riemann-Roch theorem, one can construct for every elliptic curve an isomorphism to a cubic curve cut out of KP 2 by a Weierstrass equation Y 2 Z + a 1 XY Z + a 3 Y Z 2 = X 3 + a 2 X 2 Z + a 4 XZ 2 + a 6 Z 3, where a 1,..., a 6 K. Furthermore, the point O gets sent to [0 : 1 : 0] by this isomorphism. Conversely, if for a Weierstrass equation a quantity called its discriminant K is non-zero, then this equation defines an elliptic curve and [0 : 1 : 0] is often chosen as O. The set of rational points on an elliptic curve C form an abelian group by defining the sum of two points P and Q to be the third intersection point of the line through O and P Q with C, where P Q is the third intersection of the line through P and Q. Under this operation, O is the identity element and the inverse of P is the third intersection of the line through P and O, where O is the third intersection of tangent line to O. If an elliptic curve is given by a Weierstrass equation, one can write down explicit equations for sum and inverse. See for example (Silverman 2009, p. 53). 29

36 We next show how for an elliptic curve over a field K given by an affine Weierstrass equation y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6, (3.1) its coordinate functions x and y can be expanded into formal power series in one variable. Using these, the addition on the elliptic curve can then be expressed as a formal group law. We will examine the group structure near its identity, so it will be convenient to make the change of variables z := x y and w := 1 y, (3.2) that is, [X : Y : Z] [ X : Z : Y ] in projective coordinates. The identity element O = [0 : 1 : 0] now namely becomes the point (z, w) = (0, 0) in the affine (z, w)-plane. The equation (3.1) turns into w = z 3 + a 1 zw + a 2 z 2 w + a 3 w 2 + a 4 zw 2 + a 6 w 3 := f(z, w). (3.3) We now wish to repeatedly substitute this expression for w into itself, so that w becomes a power series in z: w = z 3 + (a 1 z + a 2 z 2 )w + (a 3 + a 4 z)w 2 + a 6 w 3 (3.4) = z 3 + (a 1 z + a 2 z 2 )f(z, w) + (a 3 + a 4 z)f(z, w) 2 + a 6 f(z, w) 3 = z 3 + (a 1 z + a 2 z 2 ) ( z 3 + (a 1 z + a 2 z 2 )w + (a 3 + a 4 z)w 2 + a 6 w 3) =. + (a 3 + a 4 z) ( z 3 + (a 1 z + a 2 z 2 )w + (a 3 + a 4 z)w 2 + a 6 w 3) 2 + a 6 ( z 3 + (a 1 z + a 2 z 2 )w + (a 3 + a 4 z)w 2 + a 6 w 3) 3 So the series can be written as w = z 3 + a 1 z 4 + (a a 2 )z 5 + (a a 1 a 2 + a 3 )z 6 + = z 3 (1 + A 1 z + A 2 z 2 + ), (3.5) for certain polynomials A n Z[a 1,..., a 6 ]. It is not obvious whether this procedure actually converges to a power series w(z) Z[a 1,..., a 6 ][[z]], where this ring is complete with respect to the ideal (z), such that w(z) = f(z, w(z)). To be more precise in what we have been doing, define for every n 1 inductively the sequence of polynomials Then we set f 1 (z, w) := f(z, w) and f n+1 (z, w) := f n ( z, f(z, w) ). w(z) := lim n f n(z, 0). (3.6) Proposition The limit (3.6) exists in the complete ring Z[a 1,..., a 6 ][[z]] and it satisfies w(z) = f(z, w(z)). 30

37 This turns out to be a consequence of Hensel s lemma (Silverman 2009, Lemma IV.1.2). Now that w(z) has been constructed, we can also express our coordinates x and y from our Weierstrass equation as a power series in z. Filling in w(z) in (3.2), we get x(z) = z w(z) = and y(z) = z2 w(z) = 1 z 3 + and these also have coefficients in Z[a 1,..., a 6 ]. We have thus obtained formal solutions (x(z), y(z)) of (3.1). The next step is then to express the addition on our elliptic curve in terms of these power series. Let z 1 and z 2 be formal variables, define w i := w(z i ) and let P i = (z i, w i ) for i = 1, 2 be two formal points on our curve. Then the line w = λz + ν, where ν = w 1 λz 1, through the P i s has slope λ = w 2 w 1 z 2 z 1 = n=3 A n 3 z n 2 z n 1 z 2 z 1, where we used the expansion (3.5). Factoring z n 2 z n 1, we see that λ lies in Z[a 1,..., a 6 ][[z 1, z 2 ]]. To find the third point at which the line intersects our curve, we substitute w = λz + ν in (3.4): λz + ν = z 3 + (a 1 z + a 2 z 2 )(λz + ν) + (a 3 + a 4 z)(λz + ν) 2 + a 6 (λz + ν) 3. We know by definition two roots of this cubic in z: z 1 and z 2. Therefore, we can express the third one, which we call z 3, in z 1, z 2 and the a i s. By looking at the quadratic term, we see that so (z 1 + z 2 + z 3 ) = a 1λ + a 2 ν + a 3 λ 2 + 2a 4 λν + 3a 6 λ 2 ν 1 + a 1 λ + a 4 λ 2 + a 6 λ 3, z 3 = z 1 z 2 a 1λ + a 2 ν + a 3 λ 2 + 2a 4 λν + 3a 6 λ 2 ν 1 + a 1 λ + a 4 λ 2 + a 6 λ 3. Since λ has no constant term, the denominator appearing here can be expanded into a power series, and we see that z 3 also lies in Z[a 1,..., a 6 ][[z 1, z 2 ]]. This is the z-coordinate for the third point of intersection. Its w-coordinate is obtained by filling in z 3 (z 1, z 2 ) in the equation of our line: w 3 := λ(z 1, z 2 )z 3 (z 1, z 2 ) + ν(z 1, z 2 ). Since the points P 1, P 2 and (z 3, w 3 ) are collinear, they add up to the identity O of the elliptic curve, which means that z 3 is the z-coordinate of the inverse of the sum P 1 + P 2. We know that the inverse of a point (x, y) on the curve in the affine (x, y)-plane is given by (x, y a 1 x a 3 ). By our change of coordinates z = x/y, the z-coordinate of the inverse of a point (z, w) is therefore given by i(z) = x(z) y(z) + a 1 x(z) + a 3 Z[a 1,..., a 6 ][[z]]. So the z-coordinate of the sum P 1 + P 2 is F (z 1, z 2 ) := i(z 3 (z 1, z 2 )). This can be calculated to begin as F (z 1, z 2 ) = z 1 + z 2 a 1 z 1 z 2 a 2 (z 2 1z 2 + z 1 z 2 2) +. 31

38 It is a two-variable power series lying in Z[a 1,..., a 6 ][[z 1, z 2 ]] and from the axioms of the group structure on our elliptic curve and knowing that the identity O has (z, w)-coordinates (0, 0), it follows that F (z 1, z 2 ) is a formal group law. We state the following theorem, which will be needed later. Theorem The formal group law of an elliptic curve over a field of positive characteristic has either height 1 or 2. A proof can be found in (Silverman 2009, Corollary IV.7.5). 3.2 Jacobi quartics and elliptic cohomology After this discussion on general elliptic curves, we will now restrict our attention to the family of Jacobi quartics over C. It is to this family that we will associate cohomology theories. We demonstrate how these quartics arise, by using the interpretation of elliptic curves over C as complex tori via the Weierstrass -function. See for example (Knapp 1992, Chapter VI). Recall that every elliptic curve over C is of the form y 2 = 4x 3 g 2 x g 3 for certain g 2, g 3 C. Proposition Let C/C be an elliptic curve given by an equation y 2 = 4x 3 g 2 x g 3 = 4(x e 1 )(x e 2 )(x e 3 ) (3.7) for certain e i C. Then there exists a change of variables such that its equation becomes η 2 = 1 2δξ 2 + εξ 4, where δ = 3 2 e 1 and ε = (e 1 e 2 )(e 1 e 3 ). Proof. By the Uniformisation theorem (Knapp 1992, Corollary 6.32) there exists a lattice Λ C with basis ω 1, ω 2 C and associated Weierstrass -function (z) such that x = (z) and y = (z). Define the meromorphic function f(z) := 1/ (z) e 1 on C, where we pick the branch (z) e 1 = 1 z +. Now consider the curve in CP 2 that is parametrised by ξ := f(z) and η := f (z). Expressing ξ in terms of x: ξ = 1 1 =, (z) e1 x e1 gives us x = ξ 2 + e 1. Similarly, it follows from ( ( (z) ) ) 1/2 (z)( ) 3/2 η = e1 = (z) e1 2 = y 2 (x e 1) 3/2 = y 2 ξ3 that y = 2ηξ 3. Substituting these into (3.7) gives which can be rewritten as 4η 2 ξ 6 = 4ξ 2 (ξ 2 + e 1 e 2 )(ξ 2 + e 1 e 3 ), η 2 = ( 1 + ξ 2 (e 1 e 2 ) )( 1 + ξ 2 (e 1 e 3 ) ) = 1 + (3e 1 e 1 e 2 e 3 )ξ 2 + (e 1 e 2 )(e 1 e 3 )ξ 4 = 1 + 3e 1 ξ 2 + (e 1 e 2 )(e 1 e 3 )ξ 4, 32

39 where we used that it follows from comparing coefficients in (3.7) that e 1 + e 2 + e 3 = 0. This is the desired equation. The discriminant of the polynomial 1 2δξ 2 + εξ 4 in the variable ξ can be calculated to be 2 8 ε(δ 2 ε) 2. Definition The Jacobi quartic family of elliptic curves over C is given by y 2 = 1 2δx 2 + εx 4, (3.8) where δ, ε C are such that the discriminant := ε(δ 2 ε) 2 0. By looking at the partial derivatives and using the facts that ε 0 and δ ± ε, it is easily calculated that a Jacobi quartic actually has a single singularity at [0 : 1 : 0]. However, after a resolution of this singularity CP 2 [x:x2 :y:1] CP 3, one does obtain an elliptic curve. We prove the following addition formula about the function f(z) appearing in the above proof. Lemma The function f(z) = 1/ (z) e 1 in the proof of Proposition satisfies f(z + w) = f(z)f (w) + f (z)f(w) 1 εf(z) 2 f(w) 2. (3.9) Proof. Let R(z, w) be the function on C 2 at the right hand side of (3.9). Then consider its total differential dr(z, w) = R z We first calculate using the quotient rule dz + R w dw. R z (1 εf(z) 2 f(w) 2) 2 = ( f (z)f (w) + f (z)f(w) ) (1 εf(z) 2 f(w) 2) ( f(z)f (w) + f (z)f(w) ) ( 2εf(z)f (z)f(w) 2). Since f(z) and f (z) satisfy the Jacobi quartic equation f (z) 2 = 1 2δf(z) 2 + εf(z) 4, we can use this to calculate f (z) by differentiating both sides. The result is f (z) = 2δf(z) + 2εf(z) 3. Filling this in, and considering our equation modulo the abelian group S of symmetric functions in the variables z and w gives R z (1 εf(z) 2 f(w) 2) 2 2εf(z) 3 f(w) (1 εf(z) 2 f(w) 2) 2 + 2εf(z)f(w) 3 f (z) 2 (mod S), 33

40 where we used that f (z)f (w) S, f(z)f (w) 2εf(z)f (z)f(w) 2 S and 2δf(z)f(w) S. With this, we see that R z (1 εf(z) 2 f(w) 2) 2 2εf(z) 3 f(w) 2ε 2 f(z) 5 f(w) 3 + 2εf(z)f(w) 3 f (z) 2 2εf(z) 3 f(w) 2ε 2 f(z) 5 f(w) εf(z)f(w) 3( 1 2δf(z) 2 + εf(z) 4) (mod S). Therefore, R/ z = s(z, w) for some symmetric function s(z, w) S. Because R(z, w) is symmetric in z and w, we must also have R/ w = s(w, z) = s(z, w). So dr(z, w) = s(z, w) (dz + dw) = s(z, w) d(z + w). If z and w add up to a constant c, we have dr(z, w) = 0, so R(z, w) = c for some other constant c. But then also c = R(0, c). Because f has a zero at the lattice point 0 and it follows from f (z) 2 = 1 2δf(z) 2 + εf(z) 4 that f (0) = 1, we have R(0, c) = f(c) = f(z + w). Let Λ C be a lattice corresponding to a Jacobi quartic C : y 2 = 1 2δx 2 + εx 4. Because the map C/Λ C given by z [f(z) : f (z) : 1] is a complex analytic isomorphism of complex Lie groups, the above lemma tells us something about formulas for the group structure on C. Let namely P 1 = (f(z), f (z)) and P 1 = (f(w), f (w)) for z, w C be two points on C. Then their sum P 1 + P 2 is given by (f(z + w), f (z + w)), and we have just obtained the expression (3.9) for its x-coordinate in terms of the coordinates of P 1 and P 2. On the other hand, this x-coordinate is also equal to the formal group law F C (f(z), f(w)) of C. Writing x = f(z) and y = f(w), we conclude Theorem The formal group law of a Jacobi quartic (3.8) C is the following generalisation of Euler s formal group law: where F C (x, y) = x R(y) + y R(x) 1 εx 2 y 2, (3.10) R(x) = 1 2δx 2 + εx 4. Next, we observe the following fact. Lemma The formal group law (3.10) is defined over the ring Z[ 1 2 ][δ, ε]. Proof. Expand 1 + t = (1 + t) 1/2 as a binomial series ( 1/2 ) n=0 n t n. We have ( ) ( ) 1/2 1/2 := 0, = 1 ( ) ( 1/ , 2 2 = 1) 2 2! and for n 3, ( ) 1/2 = n 1 2 ( 1 2 1) ( 1 2 n + 1) n! = ( 1)n (2(n 1) 1 ) 2 n. n! 34

41 Comparing the prime factorisations of the numerator and denominator shows that the denominator is a power of 2. Therefore, the coefficients of ( 1/2 ) n=0 n t n lie in Z[ 1 2 ]. We are now ready to associate cohomology theories to this formal group law via Landweber s exact functor theorem. Let us write for convenience M := Z[ 1 2 ][δ, ε]. We give this ring a grading through deg δ = 2 and deg ε = 4. Theorem (Landweber-Ravenel-Stong). For each of the choices of polynomial S(δ, ε) = 1, ε, δ 2 ε,, the functor Ell S(δ,ε) (X) := MU (X) MU (pt) M [S(δ, ε) 1 ] on the category of cw-complexes is a homology theory. Here, M [S(δ, ε) 1 ] is a module over MU (pt) via the map MU (pt) M induced by the formal group law (3.10). Proof. We first demonstrate the result for S(δ, ε) = 1 or. The Landweber condition is satisfied for p = 2, since 2 is invertible in M [S(δ, ε) 1 ], so no further checking is needed. Let now p > 2. Multiplication by p on M [S(δ, ε) 1 ] is indeed injective. Next, Landweber has shown in (Landweber 1988) that if we define the homogeneous Legendre polynomials P n (δ, ε) M as the coefficients of the following Taylor expansion: (1 2δx 2 + εx 4 ) 1 2 =: P n (δ, ε)x 2n, then we have the congruence u 1 P (p 1)/2 (δ, ε) (mod p). Since P n (1, 1) = 1 for all n, we have u 1 0 (mod p), and therefore its multiplication on M [S(δ, ε) 1 ]/(p) is also injective. We next claim that u 2 is a unit in M [S(δ, ε) 1 ]/(p, u 1 ). This will not only imply injectivity of multiplication by u 2 on this ring, but will also finish the proof since the quotient of M [S(δ, ε) 1 ]/(p, u 1 ) by u 2 is then 0. Suppose that u 2 is not invertible in the ring. It would then lie in a maximal ideal m. Our Jacobi quartic y 2 = 1 2δx 2 + εx 4 is now defined over the field (M [S(δ, ε) 1 ]/(p, u 1 ))/m of characteristic p and we have u 1 = u 2 = 0 for the coefficients u 1 and u 2 of its formal group law. This means, however, that its height is strictly larger than 2, which contradicts Theorem Proving the result for S(δ, ε) = ε, δ 2 ε requires a Chudnovsky-Landweber congruence, which claims that n=0 (δ 2 ε) (p2 1)/4 ε (p2 1)/4 (mod (p, u 1 )). A proof can be found in (Franke 1992). The corollary is that, modulo (p, u 1 ), inverting δ 2 ε is equivalent to inverting ε and thus also to inverting := ε(δ 2 ε) 2. Therefore, the proof goes through for S(δ, ε) = ε, δ 2 ε as well. So even each elliptic curve can be used to produce different flavours of an elliptic cohomology theory, depending on which element you invert. Note that 1 has degree 12, so the ring M [ 1 ] and also the theory Ell are 12-periodic. Similarly, the theories Ell ε and Ell δ2 ε are 4-periodic. 35

42 3.3 Further comments With some modifications in the verification for u 1, the above proof goes through for elliptic curves in other disguises as well, such as the Legendre form or even the general Weierstrass equation (3.1). See (Landweber, Ravenel, and Stong 1995, Section 4.12). Since the Landweber-Ravenel-Stong result, people have begun to study even more general theories: Definition An elliptic cohomology theory consists of (i) an elliptic curve C over a ring R, (ii) a multiplicative cohomology theory h which is even, that is, h n (pt) = 0 whenever n is odd, and weakly periodic, that is, the natural map h 2 (pt) h (pt) h n (pt) h n+2 (pt) is an isomorphism for all n, (such theories are complex orientable) (iii) a ring isomorphism h (pt) = R and an isomorphism Ĉ = Spf h (CP ) of formal groups over h (pt) = R. Here, Ĉ is the formal completion of C along its identity section and Spf h (CP ) denotes the formal scheme associated to h (CP ). So this uses the theory of elliptic curves over more general rings than fields, as described in for example the book Algebraic Geometry by Robin Hartshorne. One reason why cohomology theories associated to Jacobi quartics are particulary interesting is that not only do their formal group laws have an attractive, explicit form, but they induce elliptic genera. Genera are bordism invariants of manifolds, which have already been studied since before the invention of elliptic cohomology in the eighties. Examples are the L-, also called the signature genus and the Â-genus, and these can be seen as degenerate elliptic genera. When we constructed a cohomology theory from an elliptic formal group law, we used the intermediate step that there exists a ring homomorphism θ Ell : MU (pt) Z[ 1 2 ][δ, ε, S(δ, ε) 1 ] by Quillen s theorem. Recall that in Section 1.2, we defined complex (co)bordism in terms of stably almost complex manifolds. A similar construction can be made for oriented manifolds. Here, one demands in the definition of being bordant that W = M N, where N is N with the orientation reversed. The resulting theory is called oriented bordism and denoted by MSO. Definition Let Λ be a commutative Q-algebra. An (oriented) genus ϕ with values in Λ is a ring homomorphism ϕ: MSO (pt) Λ. In other words, a genus assigns Q-algebra elements to oriented manifolds, only depends on the bordism class and, additionally, ϕ(m N) = ϕ(m)+ϕ(n), ϕ(m N) = ϕ(m)ϕ(n) and ϕ(m) = 0 if M is a boundary. It is known that (Stong 1968, pg. 177) that there exists an isomorphism MSO (pt) Q = Q[x 4, x 8, x 12,...], where deg x 4i = 4i and x 4i may be taken to be the bordism class of CP 2i. Because a genus takes values in a Q-algebra, torsion elements in MSO (pt) are sent to zero and so a genus is uniquely determined by its values on the spaces CP 2i. We form the following generating series. 36

43 Definition The logarithm g(x) := log ϕ (x) of a Λ-valued genus ϕ is defined as the power series g(x) := n=0 ϕ(cp 2n ) 2n + 1 x2n+1. Note that g(x) is odd, that is, g( x) = g(x), and that it has linear term x. Conversely, any such power series determines a genus. Definition A Λ-valued genus ϕ is called an elliptic genus if its logarithm g(x) has the form for elements δ, ε Λ. g(x) = x 0 dt 1 2δt2 + εt 4 This notion was first formulated by Serge Ochanine in 1985 to answer a question of Peter Landweber and Robert Stong on when a genus has a stronger multiplicativity property for certain bundles on spin manifolds. It turned out that genera have this property if and only if they are elliptic. This formed the original starting point of elliptic cohomology. Stably almost complex manifolds are oriented, so there exists a forgetful ring homomorphism MU (pt) MSO (pt). It can be shown that the morphism θ Ell : MU (pt) Z[ 1 2 ][δ, ε] induced from the formal group law of a Jacobi quartic factors through MSO (pt) and then is an elliptic genus. Setting δ = ε = 1, one obtains the L-genus and δ = 1/8 and ε = 0 gives the Â-genus. Note that these are elliptic genera, but can not be obtained from Jacobi quartics, because there we demanded that = ε(δ 2 ε) 2 0. So these genera should be seen as degenerate cases. Another interest in elliptic cohomology theories coming from Jacobi quartics comes from looking at their coefficient ring, or equivalently, the codomain of the elliptic genera associated to them. It is well known that the coefficients g 2 and g 3 in a Weierstrass equation y 2 = 4x 3 g 2 x g 3 may be interpreted as modular forms with respect to the modular group SL 2 (Z), namely as Eisenstein series of weight 4 and 6 respectively. Similarly, one can show (Landweber, Ravenel, and Stong 1995, Section 5) that the coefficients δ and ε in a Jacobi quartic equation y 2 = 1 2δx 2 + εx 4 can be interpreted as modular forms of weight 2 and 4 respectively, for the congruence subgroup {( ) } a b Γ 0 (2) := SL c d 2 (Z) : c 0 (mod 2). The ring of modular forms for Γ 0 (2) is isomorphic to C[δ, ε], just like the ring for SL 2 (Z) is isomorphic to C[g 2, g 3 ]. So if we take S(δ, ε) = 1, then the coefficient ring Z[ 1 2 ][δ, ε] of an elliptic cohomology theory is nothing but the ring of modular forms for Γ 0 (2) whose q-expansions at the cusp z = i have coefficients in Z[ 1 2 ], and its elliptic genus takes values in this ring. Inverting the other values for S(δ, ε) corresponds to holomorphicity of the modular forms at other cusps of the modular curve belonging to Γ 0 (2). In Example and we constructed (co)homology theories using Landweber s theorem and these turned out to be already known to us, namely 37

44 singular homology and complex K-theory. Their other, geometric definitions, in terms of singular n-simplices and vector bundles respectively, are well known. Similarly, we saw that complex cobordism has a homotopic description through Thom spaces, but that it can also be defined via mappings of manifolds. The usefulness of these three theories has largely been thanks to these interpretations. Since the conception of elliptic cohomology in the eighties, people have sought after such a geometric description of this theory also, but none has been found so far. This is a subject of much current research. 38

45 Summary for a first year student (in Dutch) Deze scriptie gaat over een bepaald type cohomologie-theorie in het vakgebied genaamd algebraïsche topologie. In de topologie bestudeert men ruimtelijke vormen op continue vervormingen na. Knijpen, uitrekken en kneden van zo een vorm is toegestaan, omdat men alleen geïnteresseerd is in die eigenschappen die hetzelfde blijven onder zulke vervormingen. Knippen en snijden in vormen mag niet dit verandert de vorm op een fundamentele manier. Hierdoor wordt de topologie ook wel informeel de rubbermeetkunde genoemd en misschien is het beste voorbeeld nog steeds de oude grap dat een topologist het verschil niet ziet tussen een koffiemok en een donut. Deze kunnen met een beetje fantasie namelijk zonder scheuren in elkaar overgebracht worden: Figuur 3.1. Een koffiemok en een donut hebben in de topologie dezelfde vorm. (Bron: html) Een donut en een bol zijn echter wél wezenlijk anders van vorm: de één heeft een gat en een bol zal er nooit een krijgen zonder te moeten knippen en plakken. Je kunt je afvragen hoe je een vorm kunt herkennen als deze op oneindig veel verschillende manieren tot andere gedaantes te kneden is, en of je er dan nog wel iets zinnigs over kan zeggen. De algebraïsche topologie weet hier een oplossing op. In dit vakgebied kent men algebraïsche invarianten toe aan ruimtelijke vormen. Dat wil zeggen, ken aan iedere vorm een bepaald algebraïsch object toe, zodanig dat dit object niet verandert als je de vorm zou kneden. Als je dan twee vormen hebt waarvan je kunt laten zien dat de toegekende algebraïsche objecten verschillend zijn, betekent dit dat je vormen niet in elkaar overgebracht kunnen worden. Op deze manier vertaalt de algebraïsche topologie meetkundige, 39

46 ruimtelijke problemen in algebraïsche, in de hoop dat door zuivere formules te bestuderen het probleem makkelijker wordt. Figuur 3.2. De bol heeft Euler-karakteristiek = 2 en voor de torus is dit = 0. Dit klinkt misschien wat abstract, dus laten we naar een voorbeeld van een algebraïsche invariant kijken. Neem een 2-dimensionaal oppervlak, zoals een bol, een torus of een dubbele torus, en deel het oppervlak op in vlakken door er punten en lijnen er tussen op te tekenen. Dit moet wel op een nette manier gebeuren: ieder punt moet via een lijn verbonden zijn met een ander punt en lijnen mogen elkaar niet buiten de getekende punten snijden. Bekijk dan het getal V E + F, waar V het aantal punten op het oppervlak is, E het aantal lijnen en F het aantal vlakken waarin het is opgedeeld. Voor bovenstaande opdeling van de bol krijgen we bijvoorbeeld het getal 2, en 0 voor de torus. Dit getal verandert duidelijk niet als je je oppervlak op een continue manier vervormt. Wat echter bewezen kan worden, en zeer bijzonder is, is dat het ook niet van de gekozen opdeling afhangt! Dit getal wordt het Euler-karakteristiek genoemd, naar de 18de-eeuwse wiskundige Leonhard Euler. Het is een algebraïsche invariant met waarden in de gehele getallen. Omdat we zien dat de bol en de torus niet dezelfde Euler-karakteristiek hebben, bewijst dit onze eerdere opmerking dat ze niet in elkaar overgebracht kunnen worden. Wat nu als twee oppervlakken dezelfde Euler-karakteristiek hebben? Zo heeft bijvoorbeeld ook de Möbius-strip waarde 0, terwijl deze niet vervormd kan worden tot een torus. In zo een geval kan de Euler-karakteristiek je dus niets nuttigs vertellen. Daarom probeert men in de algebraïsche topologie ook geavanceerdere, meer verfijnde algebraïsche objecten, dan slechts getallen, aan ruimtelijke vormen toe te kennen. Op die manier kan men een groter arsenaal aan methoden ontwikkelen om vormen van elkaar te onderscheiden en te bestuderen. Een cohomologie-theorie is een voorbeeld van zo een geavanceerde invariant. Aan een ruimtelijke vorm X kent deze een rijtje abelse groepen h 0 (X), h 1 (X), h 2 (X),... toe, onafhankelijk van hoe X gekneed is, die elk een stukje informatie over X bevatten. In de jaren tachtig is er een nieuw soort cohomologie theorie uitgevonden, de elliptische cohomologie-theorieën. In deze scriptie wordt uitgelegd hoe deze geconstrueerd worden aan de hand van data van een elliptische kromme. Dit is een speciaal soort kromme die over bijvoorbeeld de complexe getallen altijd beschreven kan worden als de punten (x, y) C 2 die voldoen aan een vergelijking y 2 = x 3 + Ax + B voor bepaalde complexe getallen A en B. Het bijzondere aan dit soort krommen is dat hun verzameling van punten altijd een abelse groepsstructuur blijkt te hebben. Deze extra structuur maakt elliptische 40