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Quasifibrations and Bott Periodicity

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Quasifibrations and Bott Periodicity Topology and its Applications 98 (1999) 3–17 Quasifibrations and Bott periodicity M.A. Aguilar 1, Carlos Prieto 2 Instituto de Matemáticas, UNAM, 04510 México, D.F., Mexico Received 10 July 1997; received in revised form 16 July 1998 Abstract We give a proof of the Bott periodicity theorem, along the lines proposed by McDuff, based on the construction of a quasifibration over U with contractible total space and Z × BU as fiber. 1999 Elsevier Science B.V. All rights reserved. Keywords: Bott periodicity; Homotopy groups; Spaces and groups of matrices AMS classification: Primary 55R45, Secondary 55R65 1. Introduction The periodicity theorem of Raoul Bott is one of the most important results in algebraic topology. This theorem is used to define K-theory, which is a generalized cohomology theory that has enormous impact in topology, geometry and analysis. Bott’s original proof [5] used Morse theory (see also [11]). The proofs of Toda [21], Cartan and Moore [8] and Dyer and Lashof [10] were based on homological calculations with spectral sequences. Atiyah and Bott [3] (see also [14,22]) obtained the result from a study of bundles over the product space X × S2, in terms of bundles over X. In [2] Atiyah gave a proof using the index of a family of linear elliptic differential operators (cf. also [4]). More recently, other proofs have appeared. Kono and Tokunaga [15] use cohomology and Chern classes; Latour [16] works with the space of Lagrangians; Giffen [12] and Harris [13] use classifying spaces of categories defined via simplicial spaces; and Bryan and Sanders [7] and Tian [20] use moduli spaces of instantons. In this paper, we give a proof which uses only quasifibrations and linear algebra (and some basic differential topology). It is based on a very beautiful idea of McDuff [17], whose program we develop. This proof is both simpler and more elementary than previous proofs. 1 Email: [email protected]. 2 Email: [email protected]. Author partially supported by CONACYT grant 400333-5-25406E. 0166-8641/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII:S0166-8641(99)00041-3 4 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 We shall construct a quasifibration p :E ! U over the infinite-dimensional unitary group, such that its total space E is contractible and has Z × BU as fiber. This way, we obtain a long exact sequence ···!πi.Z × BU/ ! πi.E/ ! πi .U/ (1.1) ! πi−1.Z × BU/ ! πi−1.E/ !···; where πi.E/ D 0 D πi−1.E/,sothat,fori>1, ∼ ∼ πi.U/ D πi−1.Z × BU/ D πi−1.BU/; (1.2) and, for i D 1, ∼ π1.U/ D Z: (1.3) 1 For the time being, one has (locally trivial) fibrations Ek.C / ! BUk with fiber Uk, where the base spaces are the classifying spaces of the unitary groups given by the colimits of Grassmann manifolds, and the total spaces are the corresponding colimits of Stiefel manifolds, such that, by passing again to the colimit, they determine a (locally trivial) fibration EU ! BU with EU a contractible space and U as fiber (see [19]). On the other hand, consider the path space PBU D ! : I ! BU j !.0/ D x0 of BU, where x0 2 BU is the base point. One knows that PBU is contractible and the map q :PBU! BU, such that q.!/D !.1/, is a Hurewicz fibration with fiber ΩBU. Clearly, if p : E ! B is a quasifibration with fiber F ,andp0 : E0 ! B is a Hurewicz fibration with fiber F 0, such that their total spaces are contractible, then there is a weak 0 ∼ homotopy equivalence F ! F . Moreover, their homotopy groups satisfy πi−1.F / D ∼ 0 πi.B/ D πi−1.F /, i > 1. Therefore, by [18] and the Whitehead theorem, one obtains homotopy equivalences ΩBU ' UandZ × BU ' ΩU. So we have isomorphisms ∼ ∼ πi−2.U/ D πi−2(ΩBU/ D πi−1.BU/; i > 2: Whence, we obtain the desired theorem, as stated below. Theorem 1.4 (Bott periodicity). There is a homotopy equivalence Z × BU ' ΩU; hence, for i > 2 there is an isomorphism ∼ πi.U/ D πi−2.U/; or, equivalently, ∼ πi.BU/ D πi−2.BU/: Or, in other terms, again by (1.2) one has that ∼ ∼ ∼ 2 πi.Z × BU/ D πiC1.U/ D πiC1(ΩBU/ D πi(Ω BU/I i.e., we obtain an isomorphism ∼ 2 πi.Z × BU/ D πi(Ω BU/; M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 5 which is another usual version of Bott periodicity. In particular, from (1.3) and Theo- rem 1.4, we obtain that Z if i is even, πi.BU/ D 0ifi is odd. 2. Preliminaries Let −∞ 6 p 6 q 6 1 (not three of them equal) and define q Cp D z : Z ! C j zi D 0 for almost all i and if i 6 p or i>q ; with the usual topology in the finite-dimensional case and the topology of the union Cq D Cq C1 D C1 C1 D C C0 Df g in the infinite-dimensional case. Therefore, 0 , 0 , 0 , 0 0 , q 1 etc. All spaces Cp are then subspaces of C−∞. With these definitions, we have that if −∞ 6 1 Cq D − 6 6 Cq ⊕ Cr D Cr <p q< ,thendim p q p; moreover, if p q r,then p q p. Cq Df j Cq We have the Grassmann manifold Gn. 0 / W W is a subspace of 0 of dimension g D C1 D Cq n and BUn Gn. 0 / colimq Gn. 0 /, where the colimit is taken with respect to the maps Cq ! CqC1 Gn. 0 / Gn. 0 / ⊂ Cq D ⊕ ⊂ Cq ⊕ CqC1 D CqC1 given by sending W 0 to W W 0 0 q 0 . Hence, BUn can be seen f j C1 g as the set W W is a subspace of 0 of dimension n . If L is any linear operator, then we shall denote by E1.L/ D ker.L − I/ the space of eigenvectors of L with eigenvalue 1. Definition 2.1. Given k 2 Z we define the shift operator by k coordinates 1 1 tk : C−∞ ! C−∞ by tk.z/i D zi−k . These shift operators are continuous isomorphisms such that t0 D Iand tk ◦ tl D tl ◦ tk D tkCl . nC1 ! ⊂ C1 Definition 2.2. Given n,thereisamapjn :BUn BUnC1 sending W to 1 C ⊕ t1.W/ ⊂ C . So, we define BU as BU D colimBUk: k In order to compare this definition with a different way of stabilizing, we have to prove a lemma. But before that, we give a definition. ⊂ Cl Cm ⊂ Cm Definition 2.3. Take W k and let m be such that k W.ThenW= k will denote Cm f g the orthogonal complement of k in W,i.e.,if ekC1;:::;em is the canonical basis for Cm f g Cm k , we complete it to an orthonormal basis ekC1;:::;em;w1;:::;wq of W;thenW= k f g Cm ⊕ Cm D is spanned by w1;:::;wq andwehavethat k .W= k / W. 6 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 Lemma 2.4. There is a homeomorphism ! 0 D ⊂ C1 j 1 Ck ⊂ , D Φ :BU BU W 0 dim W< and 0 W k 0 : 2 Ck ⊂ D Proof. Take W BUn and let k be the largest integer such that 0 W.DefineΦn.W/ Ck 2 0 ! 0 t−k.W= 0/ BU . Obviously, the map Φn :BUn BU determines in the colimit the desired map Φ. 0 Φ is surjective, since, if W 2 BU and dim W D n,thenW 2 BUn and Φn.W/ D W, because in this case k D 0. (In fact, the map Ψ : BU 0 ! BU such that W 7! W is the inverse.) It is also injective, since, if V 2 BUm and W 2 BUn are such that Φm.V / D Φn.W/, Cp ⊂ Cq ⊂ then, if p and q are the largest integers such that 0 V and 0 W, one has Cp D Cq t−p.V = 0 / t−q .W= 0 /: (2.5) Therefore, the dimensions m − p and n − q coincide. Without losing generality, we may 6 − D − > Cq assume that p q, so that, in particular, q p n m 0. Applying tq and adding 0 on the left to both sides of (2.5), yields, on the left, Cq ⊕ Cp D Cq−p ⊕ Cq ⊕ Cp 0 tq−p.V = 0 / 0 q−p tq−p.V = 0 / D Cq−p ⊕ Cp ⊕ Cp D Cq−p ⊕ 0 tq−p. 0 V= 0 / 0 tq−p.V /; which is the image of V in BUmCq−p D BUn; and, on the right, Cq ⊕ Cq D I 0 t0.W= 0 / W n D n D n ◦···◦ mC1 hence jm.V / W,wherejm jn−1 jm , and, thus, V and W represent the same element in BU. 2 f p q Definition 2.6. Define BU DfW j C−∞ ⊂ W ⊂ C−∞; −∞ <p6 q<1g,whichis f p f p covered by the subspaces BU DfW 2 BU j C−∞ ⊂ W and p is maximalg, p 2 Z. 0 0 f 0 Clearly, the map W 7! C−∞ ⊕ W determines a homeomorphism BU ! BU ; f k f 0 similarly, W 7! t−k.W/ determines a homeomorphism BU ! BU , so that one has a canonical homeomorphism Z × BUf 0 ! BUf f k f given by the composite .k; W/ 7! tk.W/ 2 BU ,! BU.
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