Division Algebras and Parallelizable Spheres III Seminar on Vectorbundles in Algebraic Topology ETH Zürich Ramon Braunwarth May 8, 2018 These are the notes to the talk given on April 23rd 2018 in the Vector Bundles in Algebraic Topology Seminar. In the two talks before, we discussed Adam s Theorem and we used the Splitting Principle to prove it. What remains is proving the Splitting Principle. For this, we first state and prove the Leray-Hirsch theorem for K-theory, which will be useful for proving the Splitting Principle. Theorem 1 (Leray-Hirsch). Let p : E B be a fiber bundle with E and B compact Hausdorff spaces and with fiber F such that K (F ) is free. Suppose that there exist classes c 1,..., c k K (E) that restrict to a basis for K (F ) in each fiber F. If either (a) B is a finite cell complex, or (b) F is a finite cell complex with all cells having even dimension, then K (E) as a module over K (B) is free with basis {c 1,..., c k }. Remark 2. Before we prove this theorem, let us make some preliminary remarks: (i) The module multiplication of two elements β K (B) and γ K (E) is defined by β γ = p (β)γ. (ii) We notice that the conclusion of the theorem is equivalent to saying that for the inclusion map i : F E, the following map is an isomorphism: Φ : K (B) K (F ) K (E), b j i (c j ) p (b j )c j j j (iii) In the case of a trivial bundle E = F B, we can choose the classes c i as the pullbacks of a basis of K (F ) under the projection E F. The proof of the Leray-Hirsch theorem now consists of four parts: First, we define a commutative diagram from two long exact sequences. Second, we use this diagram and the five-lemma to prove that the map Φ defined above is an isomorphism if (a) 1
holds. Third, we prove the same statement in case (b) for product bundles. Finally, in the fourth part, we prove (b) in the general case. Proof of Leray-Hirsch Theorem. Part 1: The Commutative Diagramm Consider a subspace B B. Let E := p 1 (B ) be the pullback bundle of B under p. Consider the following diagramm, where Φ is the map from Remark 2: K (B, B ) K (F ) K (B) K (F ) K (B ) K (F ) Diagram ( ) The map Φ rel is defined by the same formula as Φ with the small change that p (b i )c i is now the relative product K (E, E ) K (E) K (E, E ). The right hand side map Φ is defined using the restrictions of the c i to E. Claim: The above diagram commutes and the rows are exact. The bottom row is of course exact and the top row is exact since tensoring with the free module K (F ) replaces an exact sequence by the direct sum of several copies of itself. We show that the diagram commutes by factoring Φ as the composition Φ = Θ Ψ with b i i (c i ) i Ψ i which allows us to add a middle row to ( ): p (b i ) i (c i ) Θ i p (b i )c i, K (B, B ) K (F ) K (B) K (F ) K (B ) K (F ) Ψ rel Ψ Ψ K (E, E ) K (F ) K (E) K (F ) K (E ) K (F ) Θ rel Θ Θ The upper squares commute, because inclusion and pullback commute and the lower squares commute by Proposition 2.15 in [1]. We have thus proven the claim. Part 2: Proof of case (a) We will prove that the map Φ is an isomorphism by showing that both Φ rel and Φ in ( ) are isomorphisms and invoking the five-lemma. The general strategy will be very similar in the following two parts of the proof. The reader may want to look out for the following steps every time: (i) We define a commutative diagram with the appropriate spaces, where we want to use the five-lemma (ii) We define a second diagram sharing a map with the first that can be shown to be an isomorphism this 2
way, (iii) We follow our construction backwards to obtain the desired statement. Recall that we assume that B is a finite cell complex. We prove (a) by a double induction on the dimension of B and, with given dimension, on the number of cells in B. For n = 0, B is a finite discrete set, so every vector bundle is trivial and hence the K-rings consist of only one equivalence class. For the induction step, we assume that B is n-dimensional and obtained from a complex B by attaching a n-cell e n. As above, let E := p 1 (B ). By induction on the number of cells in B we may assume that Φ is an isomorphism. In order to show that also Φ rel is an isomorphism, we consider a characteristic map ϕ : (D n, S n 1 ) (B, B ) for the attached cell e n. By Corollary 1.8 in [1] we know that since D n is contractible, the pullback bundle ϕ (E) is trivial, so we obtain the following commutative diagram: K (B, B ) K (F ) K (D n, S n 1 ) K (F ) Φ rel Φ 1 Φ 2 K (E, E ) K (ϕ (E), ϕ (E )) K (D n F, S n 1 F ) The horizontal maps to the left are isomorphisms, since ϕ restricts to a homeomorphism in the interior of D n, hence it induces homeomorphisms B/B D n /S n 1 and E/E ϕ (E)/ϕ (E ). We consider the diagram ( ) with (B, B ) replaced by (D n, S n 1 ): K (D n, S n 1 ) K (F ) K (D n ) K (F ) K (S n 1 ) K (F ) Commutative diagram for (B, B ) = (D n, S n 1 ) ( ) Here, we may assume by induction that Φ is an isomorphism, since S n 1 is of dimension n 1. Since D n deformation retracts to a point, the map Φ is an isomorphism by the zero-dimensional case and Corollary 1.8 in [1]. Therefore we can apply the five lemma to this diagram and we obtain that Φ rel is an isomorphism. Now notice that Φ rel in ( ) is exactly the right hand side Φ 2 in the diagram before. It follows that Φ rel in ( ) is an isomorphism and thus by the five-lemma, also Φ is an isomorphism, which is what we wanted to prove. Part 3: Proof of part (b) for product bundles We first show the statement for the case of a product bundle E = F B. This will then be the basis for the argument in the general case using the local triviality condition. By comparing the formulas from the definition of the external product and Remark 2(ii), we observe that for product bundles Φ is the external product µ : K (B) K (F ) K (E), so we can exchange the roles of B and F in the above diagrams and use a similar rationale as above: We consider the diagram ( ) for an arbitrary compact Hausdorff space F, a finite cell complex B having all cells of even dimension and B being constructed by attaching an n-cell e n to a subcomplex 3
B. We again have exact rows in the diagram. Now to the five-lemma part of the proof: We want to show that Φ is an isomorphism. If we can show that Φ rel is an isomorphism in this situation, then by induction we find that Φ is an isomorphism as well. With the five-lemma we obtain that Φ is an isomorphism. Φ rel iso. Ind. = Φ iso. 5 lem. = Φ iso. = Success! We prove that Φ rel is an isomorphism: We note that B/B = S n. Hence, we can replace the pair (B, B ) by the pair (D n, S n 1 ) and use the diagram ( ). The map Φ is an isomorphism since D n deformation retracts to a point. Next, we explain why Φ in the same diagram is an isomorphism: Recall the consequences of Bott Periodicity on page 60 in [1], where we proved that the external product K(S n ) K(X) K(S n X) is an isomorphism for even n, thus also Φ is an isomorphism for even n. For odd n, we replace K 0 (S n ) by K 1 (S n 1 ) and vice versa, so we have an isomorphism by the same argument. It follows that Φ rel is an isomorphism by the five-lemma, which finishes the proof of the product bundle case. Part 4: Proof of part (b) for nonproducts The proof will essentially follow along the lines of the previous two parts with the important difference that B is just a compact Hausdorff space and not a cell complex. We therefore need a more subtle statement for the induction. Let us for this purpose define the following condition (in [1], its name is good, which is really boring): Let U B. If for all compact V U the map Φ : K (V ) K (F ) K (p 1 (V )) is an isomorphism, then U is called sparkly. Since by the local triviality condition each point has a trivial neighbourhood, Part 3 shows that we obtain a covering of B with sparkly sets. As B is compact, finitely many sparkly sets suffice. We thus want to show that for two sparkly sets U 1 and U 2, their union U 1 U 2 is still sparkly. Consider a set V U 1 U 2. We have V = V 1 V 2, where V i = V U i for i {1, 2}. We prove that such a set V is sparkly by using again the commutative diagram from above and the five-lemma. The diagram ( ) for (B, B ) = (V, V 2 ) looks like this: K (V, V 2 ) K (F ) K (V ) K (F ) K (V 2 ) K (F ) Commutative diagram for (V, V 2 ) ( ) Since V 2 U 2 and we assumed U 2 to be sparkly 1, we have that Φ in ( ) is an isomorphism. In order to invoke the five-lemma, we now show that Φ rel in ( ) is an isomorphism as well: The quotient space V/V 2 is homeomorphic to V 1 /(V 1 V 2 ), so we can equvalently show that the map Φ rel for V 1 /(V 1 V 2 ) is an isomorphism. One last time, we need the commutative diagram: 1 Note: We are not claiming that Bono is a vampire! 4
K (V 1, V 1 V 2 ) K (F ) K (V 1 ) K (F ) K (V 1 V 2 ) K (F ) Commutative diagram for (V 1, (V 1 V 2 ) ( ) Since we assume that U 1 is sparkly and both V 1 and V 1 V 2 are compact subsets of U 1, we have that both Φ and Φ in ( ) are isomorphisms. Hence, by the five-lemma so is Φ rel in ( ) and thus in ( ). Applying the five-lemma one last time in ( ) yields that U 1 U 2 is sparkly. Thus, by induction we obtain an isomorphism ( k ) Φ : K (B) K (F ) = K U i K (F ) K (E), i=1 which finishes the proof of the Leray-Hirsch theorem. With the Leray-Hirsch theorem proven, we can now focus on proving the splitting principle. Before we restate the Splitting Principle and start its proof, we show an application of the Leray-Hirsch theorem, which will be very usefull in the proof below. Example 3. Let E X be a vector bundle with fibers C n and compact X. We then have an associated projective bundle p : P (E) X with fibers CP n 1, where P (E) is the space of lines in E, i.e. one-dimensional linear subspaces of fibers of E. If we now consider P (E) as our new base space, there is the canonical line bundle L P (E). This L consists of the vectors in the lines in P (E). By Proposition 2.24 in [1], in each fiber CP n 1 of P (E) the classes 1, L,..., L n 1 restrict to a basis for K (CP n 1 ). It now follows from the Leray-Hirsch theorem that K (P (E)) is a free K (X)-module with basis 1, L,..., L n 1. p (E) E P (E) X Pullback in Example 3 Let us remind ourselves of the Splitting Principle: Theorem 4 (Splitting Principle). Given a vector bundle π : E X with X compact Hausdorff, there is a compact Hausdorff space F (E) and a map p : F (E) X such that the induced map p : K (X) K (F (E)) is injective and p splits as a sum of line bundles. The strategy of the following proof is using the projective bundle and the basis for K (P (E)) from Example 3 to iteratively split off orthogonal line bundles from the pullback bundle p (E). From this process we will obtain a flag bundle that will fulfill the conclusion of the theorem. Recall that for an n-dimensional vector bundle E B, the associated flag bundle F (E) B is the n-fold product of P (E) and has fibers consisting of n-tuples of orthogonal lines through the origin in R n. 5 p
Proof of the Splitting Principle. Consider again the situation from Example 3. The fact that 1 is an element of the basis implies that p : K (X) K (P (E)) is injective. Claim: The pullback bundle p (E) over P (E) contains the line bundle L P (E) as subbundle. We have the formulas L = {(l, v) P (E) E v l}, p (E) = {(l, v) P (E) E p(l) = π(v)}. It is easy to see that (l, v) L : p(l) = π(v). Hence L p (E) and we have the following diagram: L p (E) E π P (E) p X It follows that the pullback bundle splits as p (E) = L E for E P (E) the subbundle orthogonal to L with respect to some inner product. We repeat this process by forming P (E ) (the space of pairs of orthogonal lines in fibers in E. We can again split off a line bundle and continue this process a finite number of times until we obtain the flag bundle F (E) X, whose points are n-tuples of orthogonal lines in each fiber. If fibers of E have different dimensions over different connected components of X, we do this separately on each component. By construction, the pullback of E over F (E) splits as a sum of line bundles and the map F (E) X induces an injective map on K, since the latter is a composition of injective maps by the very first statement in this proof. References [1] A. Hatcher. Vector Bundles and K-Theory. Version 2.2, November 2017. 6