Bott Periodicity Theorem Davide Matasci 2018 Bott Periodicity Theorem 2018 Remark 1 Just to recall... Ke(X) can be identified with the ker(K(X) ! K(x0)) for x0 2 X. All functions are assumed to be continuous and a vector bundle p: E ! B is identified with E. Definition 2 For a topological space (X; τX ) and an equivalence relation ∼ on X we define a new topology (called quotient topology) τY on Y = X= ∼ by: −1 τY = fU ⊂ Y j q (U) 2 τX g where q : X ! X= ∼ is the canonical map. The space (Y; τY ) is called quo- tient space. Definition 3 A map f : X ! Y is called quotient map if: • f is surjective; −1 • U 2 τY iff f (U) 2 τX , where τX , τY are the topologies on X resp. Y . Remark 4 The canonical map q : X ! X= ∼ is always a quotient map, provided that X= ∼ is equipped with the quotient topology. Definition 5 The cone CX of a space X is defined by CX = (X×[0; 1])=(X× f0g). Definition 6 The suspension SX of a space X is defined by taking X × [0; 1] and collapsing X × f0g to a point and X × f1g to another point. SX can be seen as double cone on X. Definition 7 The reduced suspension ΣX of a space X is defined by ΣX = SX=(fx0g × [0; 1]) for some x0 2 X. Remark 8 Suspension and reduced suspension are homotopically equivalent. For the proof, see [1]. Definition 9 Given f : X ! Y , we define Sf : SX ! SY by Sf([x; t]) = [f(x); t]. With this definition, S becomes a functor T op ! T op. SnX is the space obtained by applying S to X n times. Davide Matasci 2 Bott Periodicity Theorem 2018 Definition 10 Let 'i : Gi ! G(i + 1) be an homomorphism between groups (for all 0 ≤ i ≤ n). The sequence G1 ! G2 ! ::: ! Gn is called exact if ker 'i+1 = Im 'i for all i. Definition 11 A partition of unity subordinate to an open cover fUαgα of X is a collection f'β : X ! [0; 1]gβ such that: •8 β9α such that supp('β) ⊂ Uα (i.e. each 'β is supported in one of the Uα's); P • β 'β = 1, where the sum is finite in a neighbourhood of each point of X. Definition 12 A space (X; τ) is called paracompact if it is Hausdorff and for each open cover fUαgα of X there is a partition of unity f'βgβ subordinate to the cover. Proposition 13 (1.18 in [1]) Every compact Hausdorff space is paracom- pact. For the proof, see [1]. Proposition 14 (2.9 in [1]) Let X be compact and Hausdorff, A ⊂ X be closed subspace. Let {: A,! X be the inclusion and let q : X ! X=A be the quotient map. Then q A −!{ X −! X=A induces (applying Ke, a contravariant functor) the exact sequence Ke(q) Ke({) Ke(X=A) −−! Ke(X) −−! Ke(A) Proof Let q∗ = Ke(q) and i∗ = Ke({). We only have to show that the sequence is exact, i.e. that Im(q∗) = ker({∗). Since qi = 0 it holds that Ke(qi) = i∗q∗ = 0 and thus the inclusion Im(q∗) ⊂ ker({∗) is proven. To show the other inclusion, we take an element [E] 2 ker(i∗) ⊂ Ke(X) and want to find an element [F ] 2 Ke(X=A) such that q∗[F ] := [q ∗ F ] = [E]. This is equivalent (by definition of []) to the following: Davide Matasci 3 Bott Periodicity Theorem 2018 We take a vector bundle E ! X with i∗E ≈ ξ0 (i.e. i∗[E] = 0 = [ξ0]) Ke(A) and look for a vector bundle F ! X=A such that q∗F ≈ E. i∗E ≈ ξ0 means that E ! X is trivial over A. Let h: p−1(A) ! A × Cn be the trivialization over A. Let E=h = E= ∼ with h−1(x; v) ∼ h−1(y; v) for all x; y 2 A. Since p(h−1(x; v)) = x for all x 2 A and q is a projection, the composition p q E ! X ! X=A satisfies qp(h−1(x; v)) = qp(h−1(y; v)) for all x; y 2 A and thus induces a projection E=h ! X=A. The goal is now to show that E=h ! X=A is a vector bundle. This will be the F we're looking for. It's enough to find a trivialization of E over a neighbourhood U of A, since this induces a trivialization of E=h over U=A. This is in turn enough to prove that E=h ! X=A is a vector bundle since over the other points of X=A a local trivialization is already given (induced by the local trivialization of E over X). By the discussion before Lemma 1.1 in [1], there are sections s1; : : : ; sn : A ! −1 E such that for all a 2 A : s1(a); : : : ; sn(a) are linearly independent in p (a). Let fUjgj2J be an open cover of A such that E is trivial over all of the Uj's. The existence of these sets follows from the fact that E ! X is a vector bundle, and J is finite since A is compact. −1 si restricts to A \ Uj ! p (A). Using the trivialization we can regard si −1 as a map : A \ Uj ! p (aj) for some aj 2 A \ Uj. More precisely we have: si −1 hj n πj n A \ Uj −! p (A \ Uj) −! (A \ Uj) × C −! C −1 n where hj : p (Uj) ! Uj ×C is a local trivialization and πj is the projection. 0 −1 0 Taking si(x) := hj (aj; πj ◦ hj ◦ si(x)) for some aj 2 A \ Uj we see that si in 0 −1 fact restricts to a single fiber: si(x) 2 p (fajg) for all x. We can "recover" 0 si from si with −1 0 si(x) = hj (x; πj ◦ hj ◦ si(x)) Thus, as claimed, we can w.l.o.g. assume that si restricts to a single fiber −1 p (aj). Since X is a normal topological space, by the Tietze extension theo- rem we can extend these maps to si;j : Uj ! E. Formally, I use the fact that A \ Uj is closed in Uj with respect to the relative topology. Davide Matasci 4 Bott Periodicity Theorem 2018 Let f'j j j 2 Jg [ f'g be a partition of unity subordinate to the (open) cover fUj j j 2 Jg [ fX − AgofX. This exists because X, being Hausdorff compact, is paracompact (see Proposition 1.8). Using this partition we get P that ti := j2J 'jsi;j is a section extending sj to a map X ! E. −1 Over a fiber p (a) (for an element a 2 A) these extended section ti form a basis, i.e det(tj : j 2 J) 6= 0. Since the determinant is continuous (or alternatively, the linear independence is an open condition) this property holds true even in a small neighbourhood of each fiber. This means that −1 n (tj : j 2 J) is a linear-space isomorphism between p (a) and a × C for all a 2 Uj. Only remaining to be shown is that q∗(E=h) ≈ E. E k E=h p q X X=A This follows from the Universal property of the pullback (the uniqueness- up-to-isomorphism part) because the quotient map E ! E=h, denoted by k in the diagram, takes each fiber over an element x 2 X isomorphically to the fiber over q(x) 2 X=A (shown above...). Remark 15 ξn denotes the n-dimensional trivial vector bundle. • V ectn(X) = f"n − dim vector bundles over X"g= ≈ with E ≈ F iff E and F are isomorphic; • Ke(X) = f"vector bundles over X"g= ∼ with E ∼ F iff 9n; m ≥ 0 : E ⊕ ξn ≈ F ⊕ ξm; 0 0 0 0 • K(X) = fE − F g= ≈K with E − F ≈K E − F iff E ⊕ F ≈s E ⊕ F , n n where E ≈s F iff 9n ≥ 0 : E ⊕ ξ ≈ F ⊕ ξ . The next Lemma shows that for contractible subspaces A in X, we may take X=A instead of X as a base for vector bundles. Lemma 16 (2.10 in [1]) Let X and A as in the proposition above. If A is contractible, q : X ! X=A induces a bijection q∗ : V ectn(X=A) ! V ectn(X) given by Ke(q) for every n 2 N. Proof We have to find an inverse of the map q∗. Let p: E ! X be an n-dimensional vector bundle. The image of p under inverse is going to be E=h, with h as in the proposition above (since A is compact and contractible, Davide Matasci 5 Bott Periodicity Theorem 2018 p is trivial over A by Corollary 1.8 in [1]). As the fact that E=h is a vector bundle is already been proven, only to show is that E 7! E=h is well defined, i.e. that E=h does not depend on the choice of the map h. Suppose we have two trivializations h0 and h1. These are homeomor- −1 n −1 phisms p (A) ! A × C . Because h1 = (h1h0 )h0, we see that h1 and h0 n −1 differ by a matrix ga 2 GLn(C ) over each a 2 A (i.e. in the fiber p 8(a)). Let g : A ! GLn(C) be the map a 7! ga.