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* * * Stable Homotopy and the J-Homomorphism

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* * * Stable Homotopy and the J-Homomorphism *** Stable Homotopy and the J-Homomorphism Eva Belmont Advisor: Prof. Michael Hopkins Submitted: March 19, 2012 Submitted to the Harvard University Department of Mathematics in partial fulfillment of the requirements for the degree of AB in Mathematics 2 Acknowledgements I would like to thank my advisor, Prof. Michael Hopkins, for introducing me to the beautiful world of stable homotopy theory. I am grateful for the wisdom he shared, and for the time and energy he took out of a busy life to start one student down the road of curiosity and amazement. I would also like to thank my parents for their unflagging moral support, and my friends for being there for me during this process. Contents Contents 3 1 Introduction to K-theory and the J-homomorphism4 1.1 The Hopf fibration................................. 4 1.2 Vector bundles ................................... 7 1.3 Definition of K-theory............................... 9 2 Some useful tools 11 2.1 The stable category ................................ 11 2.2 Adams operations ................................. 13 2.3 Localization..................................... 15 2.4 Thom complexes .................................. 16 3 The e-invariant 21 3.1 Bernoulli numbers ................................. 21 3.2 Defining the e-invariant .............................. 22 3.3 The image of e ................................... 23 s 4 A splitting for π2k−1 28 4.1 (Almost) surjectivity................................ 28 4.2 The groups Je(X).................................. 29 4.3 A special case of the Adams conjecture...................... 31 A Cannibalistic class computation 34 Bibliography 38 3 Chapter 1 Introduction to K-theory and the J-homomorphism One of topology's earliest goals was to study the degree of maps Sn ! Sn: the number of times that the first copy of Sn winds around the second. The dawn of algebraic topology brought the insight that this col- lection of maps could be given the structure of a group which is n n isomorphic to Z: each k 2 Z corresponds to a map S ! S of de- gree k. This is not hard to see, once the idea has been made precise, so now you may want to ask the natural follow-up question: how do we classify maps Sn ! Sm for any n; m? Half of this is again easy: if n < m, there are no such maps, other than (up to equivalence) the trivial map that collapses the entire domain to a point. Unfor- tunately, if n > m, this question is so hard that topologists are still stumped. n Our knowledge is frustratingly spotty. The groups πn+1S and n 2 πn+2S have been known since the 1940's (see [23]), but πnS remains unclassified. We know n 2n that πn+k(S ) are finite, except for π4m−1S , which are finitely generated. The groups up n through π19+nS are known in entirety; more have been calculated, and for still others, only certain prime components are known. n However, there is a range of groups that are more tractable. If n ≥ i + 1, then πi+nS is s independent of n; these are called the stable groups, and are denoted πi . We still do not know the full story of the stable groups, but we know a piece of it: this thesis tells part of that story. The exposition here is based primarily on Adams' papers [2]-[5], filtered through the insight of my advisor Prof. Michael Hopkins, who taught me this material. Using K-theory over R, it is possible to compute explicitly a cyclic group that is a direct s summand of every πi . Here we will work over C instead, which dodges some technicalities; the price is that we land a factor of 2 away in one computation from proving that there is a splitting. x1.1 The Hopf fibration m As a starting point for studying πn(S ), let us take a nontrivial example in low dimension: 4 Chapter 1. Introduction to K-theory and the J-homomorphism 5 3 2 3 4 ∼ 2 2 ∼ 1 the Hopf fibration S ! S . View S as a subset of R = C , and use the fact that S = CP . 2 1 3 2 The natural quotient map C ! CP induces a natural map S ! S . The preimage of 1 [x; y] 2 CP is the set f(λx, λy): j(λx, λy)j = 1g =∼ λ : jλj2 = (jxj2 + jyj2)−1 ∼ ∗ ∼ 1 = C = S : We say that S1 is the fiber, and S1 ! S3 ! S2 is a fibration; we will return to fiber maps later. 2 Alternatively, recall that we can decompose CP as the open 4-cell f[1; a; b]: a; b 2 Cg, the open 2-cell f[0; 1; c]: c 2 Cg, and the \point at infinity" [0; 0; 1]. The boundary of the 4-cell is the limit of circles f[1; Ra; Rb]: jaj2 + jbj2 = 1g =∼ f(Ra; Rb)g 2 R · S3 as R ! 1. So the attaching map A : @(4-cell) = @f (a; b): a; b 2 C g ! f[0; a; b]g is homotopic to the limit of the maps AR :(Ra; Rb) ! [0; Ra; Rb]. Each AR is homotopic to the Hopf fibration defined above, and so A = limR!1 AR is also just the Hopf fibration. n Other than being an easy example of a nontrivial element of πm(S ) for m > n, the Hopf fibration is also worthy of consideration because it is the n = 2 case of the Hopf construction, 2n−1 n n−1 a method for obtaining maps Hf : S ! S , given the initial data of a map f : S × Sn−1 ! Sn−1. The reason why anyone would care about maps Sn−1 × Sn−1 ! Sn−1 goes back to an old problem that asks which spheres Sn−1 are Hopf spaces { topological spaces with a monoid structure. Of course, the map Sn−1 × Sn−1 ! Sn−1 in question is just the monoid multiplication. (In addition to being of independent interest, this problem is also tied to the determination of the number of linearly independent vector fields on spheres, which Adams solved definitively in 1961 and laid out in [1].) Now we describe the Hopf construction itself. Note that a map Sn−1 × Sn−1 ! Sn−1 is equivalent to a map Sn−1 ! Maps(Sn−1;Sn−1); since rotations of Sn−1 can be naturally extended to rotations of Dn that leave the origin fixed, our H-space multiplication µ can be written as an assignment Sn−1 ! Maps(Dn;Dn). Equivalently, there are extensions n−1 n n n n−1 n 2n−1 µ+ : S × D ! D and µ− : D × S ! D . To describe a map out of S , treat S2n−1 as the boundary of a 2n-cell, and decompose as follows: S2n−1 = @D2n = @(Dn × Dn) = (@Dn × Dn) [ (Dn × @Dn) = (Sn−1 × Dn) [ (Dn × Sn−1) : 2n−1 If we use µ+ on the first piece and µ− on the second piece, then we have a map S ! n n n−1 n D [ D . But since µ+ and µ− are extensions of µ, they agree on S ⊂ D , and the hemispheres Dn [ Dn can be glued along Sn−1 to form Sn. n Having found a way to generate elements of the elusive group π2n−1S , we are now motivated n to aim higher and look for elements of any πmS . Happily, only a small modification of the Chapter 1. Introduction to K-theory and the J-homomorphism 6 above construction is necessary to produce useful results. Instead of the starting H-space n−1 n n multiplication S ! Maps(D ;D ), let us begin with any element f 2 πiO(n), which can i n n 2n−1 be written as a map S × R ! R . Then the decomposition of S can be imitated to write Sn+i = @Dn+i+1 = @(Di+1 × Dn) = (@Di+1 × Dn) [ (Di+1 × @Dn) = (Si × Dn) [ (Di+1 × Sn−1) Send the second piece to a point, and the first piece to Dn using the chosen map f. Gluing these together produces a map Sn+i ! Dn [ {∗} = Sn. One can show that this is a n homomorphism when regarded as a map of homotopy groups πiO(n) ! πn+iS . This is the J-homomorphism.1 In the rest of this thesis, we will be working over C instead of R, because it simplifies the arguments in several places. Going forward, the J-homomorphism will refer to the complex version n π2k−1U(n) ! π2k−1+nS which is constructed in an analogous manner. The beauty of this construction is that most of the groups πkU(n) are known: ( 0 if k is even πk(U(n)) = Z if k is odd for large enough n (in a way that is described more precisely in Section 2.1). Thus, we n know how to compute a subgroup Im(J) of π2k+n−1S . Understanding the image of the J-homomorphism will eventually allow us to prove that Im(J) is a direct summand of one n of the homotopy groups π2k+n−1S . This, in turn, will be accomplished by constructing a sequence J n π2k−1U(n) ! π2k+n−1S ! Q=Z ; (1.1.1) whose study will form the heart of this thesis. In the remainder of this chapter, we will introduce K-theory. In the second chapter, we lay out more background { constructions that will be invaluable to the rest of the discussion. In chapter 3, we will define the map n e : π2k+n−1S ! Q=Z and determine its image. In chapter 4, we will study the composition (1.1.1), and will discuss the ideas necessary to obtain a splitting of the (stable) homotopy groups of spheres.
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