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VECTOR FIELDS on SPHERES Contents 1. Introduction 1 2. Adams Operations 2 3. K-Theory of Complex and Real Projective Spaces 3 4

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VECTOR FIELDS on SPHERES Contents 1. Introduction 1 2. Adams Operations 2 3. K-Theory of Complex and Real Projective Spaces 3 4 VECTOR FIELDS ON SPHERES JAY SHAH Abstract. This paper presents a solution to the problem of finding the max- imum number of linearly independent vector fields that can be placed on a sphere. To produce the correct upper bound, we make use of K-theory. After briefly recapitulating the basics of K-theory, we introduce Adams operations and compute the K-theory of the complex and real projective spaces. We then define the characteristic class ρk and develop some of its properties. Next, we recast the question of the upper bound into a question about fiber homotopy n equivalent bundles over RP , whose resolution reduces to a calculation in K- theory. Finally, we give the purely algebraic proof that this upper bound is realized. Contents 1. Introduction 1 2. Adams operations 2 3. K-theory of complex and real projective spaces 3 4. The characteristic class ρk 6 5. The upper bound 10 6. Realizing the upper bound 11 Acknowledgments 12 References 12 1. Introduction Let Sn be the n-dimensional sphere. A vector field v on Sn is a continuous assignment x 7! v(x) of a tangent vector v(x) at x for every point x 2 Sn. It is a consequence of the degree of a map between spheres being a homotopy invariant that a non-zero vector field on Sn exists if and only if n is odd (one uses the vector field to construct a homotopy between the identity and the antipodal map). This paper will be concerned with a far-reaching generalization of that result, namely, what is the maximum number of linearly independent vector fields one can put on a sphere? To approach this problem, we will make use of a generalized cohomology the- ory called topological K-theory. Let X be a compact topological space and let Vect Λ(X) be the set of isomorphism classes of real or complex vector bundles over X, Λ = R or C. Vect Λ(X) can be granted the structure of a commutative ring without negatives under direct sum and tensor product. Adjoining formal additive Date: AUGUST 26, 2011. 1 2 JAY SHAH inverses via the Grothendieck group construction forms the ring KΛ(X). KΛ is a contravariant functor from the category of compact spaces to the category of rings, ∗ with maps f : X ! Y giving homomorphisms f : KΛ(Y ) ! KΛ(X) via pull- back of bundles. Since our spaces are compact, homotopic maps induce isomorphic 1 pullbacks of bundles and KΛ descends to a functor on the homotopy category of compact spaces. Now suppose that our spaces are based and let K~Λ(X) = ker(KΛ(X) ! KΛ(∗)) ~ −n ~ n n and KΛ = KΛ(Σ X) for n > 0, where Σ denotes n-fold reduced suspension. ~ n ~ n The Bott periodicity theorem allows us to extend KΛ to all n by defining KΛ(X) = ~ n0 0 0 KΛ (X), n ≡ n mod d and −d < n ≤ 0, where d = 2 if Λ = C and d = 8 if Λ = R. ~ n One can check that the KΛ so defined satisfy the Eilenberg-Steenrod axioms and so give a reduced cohomology theory. ~ n Remark 1.1. In fact, the KΛ constitute a sequence of represented functors. Suppose Λ = C. Complex n-plane bundles are classified by the space BU (n), in the sense that every complex n-plane bundle E over X may be realized as the pullback of a map f : X ! BU (n). We have inclusions in : BU (n) ! BU (n+1) for all n, and we may define BU = colimn!1BU (n). If X is compact and nondegenerately based, ~ then KC(X) = [X; BU × Z], where the brackets denote based homotopy classes of maps. By definition, K~ −n(X) = [X; Ωn(BU × )] for n > 0, and Bott periodicity C Z is the statement that there is a homotopy equivalence BU × Z ' Ω2(BU × Z). We may then use the represented definition to extend the functors K~ n(X) to all spaces C of the homotopy type of CW-complexes. Details may be found in May [9, Ch. 24]. The story is similar for Λ = R, with the classifying space BO in place of BU . Following Karoubi [8], we proceed with the derivation of the upper bound on the number of linearly independent vector fields that can be placed on a sphere. All base spaces will be assumed to be compact and connected. 2. Adams operations A cohomology operation in K-theory is a natural transformation from KΛ to k itself. The following theorem asserts the existence of certain operations Λ, termed Adams operations. k Theorem 2.1. There exist natural ring homomorphisms Λ : KΛ(X) −! KΛ(X), defined for all integers k, which satisfy the following properties: • 1 and −1 are the identity. −1 is complex conjugation. 0 assigns to a Λ R C Λ bundle over X the trivial bundle with fibers of the same dimension. k k • If ξ is a line bundle, then Λ(ξ) = ξ . k l kl • Λ Λ = Λ . p p • Λ(x) = x mod p for any prime p. ~ 2n k n • If x 2 KΛ(S ), then Λ(x) = k x. 1To capture this property, it suffices to consider paracompact spaces. VECTOR FIELDS ON SPHERES 3 • Let β denote the periodicity isomorphism in K-theory. The following two diagrams commute. β β ~ ~ 2 ~ ~ 8 KC(X) / KC(Σ X) KR(X) / KR(Σ X) k k k k C C R R K~ (X) / K~ (Σ2X) K~ (X) / K~ (Σ8X) C kβ C R k4β R even k • Let ch : KC(X) ! H (X; Q) be the Chern character and define H (x) = krx for x 2 H2r(X; Z). The following diagram commutes. ch even KC(X) / H (X; Q) k k C H K (X) / Heven (X; ) C ch Q • Let c : KR(X) ! KC(X) be given by complexification of bundles. The following diagram commutes. c KR(X) / KC(X) k k R C KR(X) c / KC(X) The Adams operations derive from the exterior power operations λk, which we now construct. We must extend the usual exterior power construction on bundles to virtual bundles. For any bundle ξ over X let 2 2 k k × λ(ξ) = 1 + λ(ξ)t + λ (ξ)t + ::: + λ (ξ)t + ::: 2 KΛ(X)[[t]] : k L i j By the formula λ (ξ ⊕ η) = i+j=k λ (ξ) ⊗ λ (η), we have λ(ξ ⊕ η) = λ(ξ)λ(η). × Hence λ extends to a homomorphism λ : KΛ(X) ! KΛ(X)[[t]] . For x 2 KΛ(X) define λk(x) to be the kth coefficient in λ(x). k k Let σi be the ith symmetric polynomial and let πk = x1 + ::: + xn be the kth power sum. By the theory of symmetric polynomials, there exists a polynomial Qk, independent of n for n ≥ k, such that πk = Qk(σ1; :::; σk): k 1 k Now define Λ(x) = Qk(λ (x); :::; λ (x)) for k > 0. Theorem 2.1 mandates the 0 −1 −k −1 k definition of Λ and Λ , and by the relation Λ = Λ Λ we define Adams operations for all integers k. The proof of Theorem 2.1 can be found in numerous sources, such as Adams [1]. 3. K-theory of complex and real projective spaces In this section we compute the complex K-theory of the complex and real pro- jective spaces and the real K-theory of the real projective spaces. In the course of the computation we will make use of the Atiyah-Hirzebruch spectral sequence, whose definition and properties are given below. 4 JAY SHAH Theorem 3.1. Let X be a finite CW-complex and let Xp be its p-skeleton. Let n n n n p−1 KΛ(X) be filtered by the groups KΛ;p(X) = ker(KΛ(X) ! KΛ(X )). There exists a multiplicative spectral sequence arising from this filtration that converges to KΛ(X), such that p;q ∼ p q • E1 (X) = C (X; KΛ(∗)); p;q ∼ p q • E2 (X) = H (X; KΛ(∗)); p;q ∼ p+q p+q p+q • E1 (X) = GpKΛ (X) = KΛ;p (X)=KΛ;p+1(X). p;q p+r;q−r+1 Here ∗ denotes a point. The differential dr : Er ! Er shifts degree by (r; −r+1). The multiplication on the E2 page is given by the cup product in ordinary cohomology. For a proof of most of this theorem, see Atiyah and Hirzebruch [5]. There the identification of the multiplication on the E2 page is only asserted; a proof of this assertion may be found in Dugger [7]. q By Bott periodicity, the groups KΛ(∗) are periodic with period 2 for Λ = C and 8 for Λ = R, and they are given as follows. q 0 1 2 3 4 5 6 7 K−q(∗) 0 0 0 0 C Z Z Z Z K−q(∗) =2 =2 0 0 0 0 R ZZ ZZ Z Z With these preliminaries in hand, we proceed to compute. Let η be the canon- ical complex line bundle over CP n−1 and ξ be the canonical real line bundle over RP 2n−1. n−1 n Theorem 3.2. KC(CP ) = Z[t]=t , where the generator t is given by η −1. The operation k is given by k ((η − 1)s) = (ηk − 1)s. C C Proof. It is a theorem (Atiyah [3], Proposition 2.7.1, p. 102) that for any decom- posable vector bundle E = ΣLi over X (the Li being line bundles), KC(P (E)) is generated as a KC(X)-algebra by the tautological line bundle H subject to the single relation Y (H − Li) = 0: Apply this theorem to the case X a point, E = Cn to obtain the indicated descrip- tion of K ( P n−1).
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