Geometric Aspects of Quantum Condensed Matter

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1 Geometric Aspects of Quantum Condensed Matter January 15, 2014 Lecture XI y Classification of Vector Bundles over Spheres Giuseppe De Nittis Department Mathematik room denittis.giuseppe@gmail.com W gdenittis.wordpress.com/courses/

2 Recommended Bibliography Classical monographies: [At] Atiyah, M. F.: K -Theory. Benjamin Inc., 1967 [Ha] Hatcher, A.: Vector Bundles and K -Theory [Hu] Husemöller, D.: Fibre Bundles. Springer, 1994 [Ka] Karoubi, M.: K -Theory: An Introduction. Springer, 1978 [LM] Luke, G. & Mishchenko, A. S.: Vector Bundles and Their Applications. Kluwer Ac. Pub., 1998 [MS] Milnor, J. W. & Stasheff, J. D.: Characteristic Classes. Princeton U. Press, and many others...

3 1 Classification of Vector Bundles over Spheres The Rôle of Homotopy Groups Homotopy of the Unitary Group Homotopy of the Orthogonal Group Homotopy Equivalence First Classification Results Outline

4 We proved that: I If X is paracompact then Vec m K X [X,G mk ] ; II If X is compact, there is a finite covering and for some N N. Vec m K X [X,G mk N ] We want to use II in order to classify vector bundles over X = S d for d N when K = C complex case and K = R real case. DEFINITION Hmotopy groups Let X be a topological space. Then π d X,x := { f : S d, X,x modulo homotopy equivalences }. With respect to the composition of maps π 1 X,x is a group, called fundamental group; π d X,x is a commutative group for all d 2; π 0 X is not a group merely a pointed set but provides the set of path-connected components of X. π 0 X,x = 0 for path-connected X.

5 Remark: Let x and y be two base-points of a path-connected spacex. Consider the two homotopy groups π d X,x and π d X,y and a path γ : [0,1] X from x to y. For each [α] π d X,x let us consider the map F : S d [0,1] X given by F γ v,0 := αv v S d F γ,t := γt t [0,1] and extended to the whole space S d [0,1] box principle from obstruction theory. The map γ := F γ,1 is an isomorphism γ : π d X,x π d X,y which depends e only on the homotopy class of γ amongst the paths from x to y. If we take x = y, the assignment γ γ may be thought of as an action of π 1 X,x on π d X,x. Only if this action is trivial, can one speak unambiguously of π d X without reference to any base point x.

6 PROPOSITION Let X be a path-connected space. The inclusion of base point preserving maps into the set of all maps induces a bijection [S d,x] π d X,x/π 1 X,x where on the right we have the relation [α] γ [α] for [γ] π 1 X,x. Proof sketch of. Let j : π d X,x [S d,x] be induced by the inclusion of base point preserving maps into the set of all maps. If [α] π d X,x and [γ] π 1 X,x it is always possible to write down an explicit free homotopy between α and γ α. Hence j factors through the action of π 1 X,x on π d X,x and defines a map j : πd X,x/π 1 X,x [S d,x]. Since X is path connected, any map in [S d,x] can be deformed to a base point preserving map. So j is surjective. Now suppose [α] π d X,x is null-homotopic [S d,x]. Then there is a F : S d [0,1] X such that F,0 = α, F,1 = x. Let γt := F,t. Then α = γ v x which means that j is injective.

7 As a consequence we have Vec m K Sd π d Gm K N,x /π 1 Gm K N,x. In order to classify vector bundles we need to compute π d Gm K N,x for d N and N sufficiently large i.e. N =. The Grassmann manifold are path path-connected, enche π 0 Gm K N,x = 0. Key formulas for the computation of the homotopy groups for K = C,R are the following exact sequences of fibrations and Um UN/UN m UN/Um UN m G m C N Om ON/ON m ON/Om ON m G m R N THEOREM homotopy exact sequence for fibrations Let Z X Y be a base point preserving fibration. Suppose that Y is path-connected. Then there is a long exact sequence of homotopy groups... π d Z,z πd X,x πd Y,y πd 1 Z,z... π0 X,x 0

8 Forgetting in the notation any reference to base points we get π d UN/UN m πd G m C N π d 1 Um πd 1 UN/UN m π d ON/ON m πd G m R N π d 1 Om πd 1 ON/ON m As a consequence we have: PROPOSITION For N big enough and for all d 1 π d G m C N π d 1 Um, πd G m R N π d 1 Om. Proof sketch of. We need to show that π d UN/UN m 0 πd ON/ON m for all d N when N is big enough....

9 ... We use the stable homotopy Theorem. The natural inclusion Ur Ur + 1 induce a morphism ı : π k Ur πk Ur + 1 which is an isomorphism if k 2r 1 and it is surjective if k 2r. Likewise, the natural inclusion Or Or + 1 induce a morphism ı : π k Or πk Or + 1 which is an isomorphism if k r 2 and it is surjective if k r 1. Applying the homotopy sequence to the fibre maps one gets UN m UN UN/UN m ON m ON ON/ON m π d UN m ı π d UN 0 πd UN/UN m 0 πd 1 UN m ı... since ı is an isomorphism for N big enough. The same holds true in the real case.

10 We arrived at the following formulas: Vec m C Sd π d 1 Um, Vec m R S d π d 1 Om /Z2, where we used π 1 Um 0, π1 Om Z2. To classify complex vector bundles over spheres we need to compute the homotopy π k Ur for the unitary groups; To classify real vector bundles over spheres we need to compute the homotopy π k Or for the orthogonal groups and to study the Z 2 -action of π 1 Or over these groups.

11 1 Classification of Vector Bundles over Spheres The Rôle of Homotopy Groups Homotopy of the Unitary Group Homotopy of the Orthogonal Group Homotopy Equivalence First Classification Results Outline

12 The first homotopy groups of Ur are summarized in the following table: π k Ur k = 0 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 r = 1 0 Z r = 2 0 Z 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 r = 3 0 Z 0 Z 0 Z Z 6 0 Z 12 r = 4 0 Z 0 Z 0 Z 0 Z Z 24 r = 5 0 Z 0 Z 0 Z 0 Z 0 In the stable regime black entries the homotopy groups are fixed by the Bott Periodicity: 0 if k even π k Ur = Z if k odd k 2r. Z r! if k = 2r

13 The corresponding classification rank m of complex vector bundle over S d is given by Vec m C Sd d = 1 d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = 8 d = 9 m = 1 0 Z m = 2 0 Z 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 m = 3 0 Z 0 Z 0 Z Z 6 0 Z 12 m = 4 0 Z 0 Z 0 Z 0 Z Z 24 m = 5 0 Z 0 Z 0 Z 0 Z 0 In particular, in the stable regime i.e. for fibers big enough the Bott Periodicity provides Vec C S even Z, Vec C S odd 0.

14 The complex Bott Periodicity In order to show that π k Ur = { 0 if k even Z if k odd k 2r 1 one uses the stable homotopy Theorem which assures the existence of an isomorphism ı : π k Ur πk Ur + 1, if k 2r 1 and the following explicit computation: LEMMA Bott For all k = 0,1,2,... π 2k Uk + 1 Z, π2k+1 Uk

15 1 Classification of Vector Bundles over Spheres The Rôle of Homotopy Groups Homotopy of the Unitary Group Homotopy of the Orthogonal Group Homotopy Equivalence First Classification Results Outline

16 The first homotopy groups of Or are summarized in the following table: π k Or k = 0 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 r = 1 Z r = 2 Z 2 Z r = 3 Z 2 Z 2 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 r = 4 Z 2 Z 2 0 Z 2 Z 2 2 Z 2 2 Z 2 12 Z 2 2 Z 2 2 r = 5 Z 2 Z 2 0 Z Z 2 Z 2 0 Z 0 r = 6 Z 2 Z 2 0 Z 0 Z 0 Z Z 24 r = 7 Z 2 Z 2 0 Z Z Z 2 2 In the stable regime black entries the homotopy groups are fixed by the Bott Periodicity: Z 2 if k = 0,1 mod. 8 π k Or = 0 if k = 2,4,5,6 mod. 8 k r 2. Z if k = 3,7 mod. 8

17 The corresponding classification rank m of real vector bundle over S d is given by Vec m R Sd d = 1 d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = 8 d = 9 m = 1 Z m = 2 Z 2 N m = 3 Z 2 Z 2 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 m = 4 Z 2 Z 2 0 Z N 0????? m = 5 Z 2 Z 2 0 Z Z 2 Z 2 0 Z 0 m = 6 Z 2 Z 2 0 Z 0? 0?? m = 7 Z 2 Z 2 0 Z Z Z 2 2 The action of π 0 Om Z2 on π d 1 Om is trivial if m is odd or if d < m. In the stable regime i.e. for fibers big enough the Bott Periodicity provides Vec R S d+n8 d = 1 d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = 8 Z 2 Z 2 0 Z Z

18 The real Bott Periodicity In order to show that Z 2 if k = 0,1 mod. 8 π k Or = 0 if k = 2,4,5,6 mod. 8 Z if k = 3,7 mod. 8 k r 2. one uses the stable homotopy Theorem which assures the existence of an isomorphism ı : π k Or πk Or + 1, if k r 2 and the following explicit computation: LEMMA Bott For all k = 0,1,2,... Z 2 if k = 0,1 mod. 8 π k Ok if k = 2,4,5,6 mod. 8. Z if k = 3,7 mod. 8