488 Characteristic Classes Gilmore R and Letellier C (2006) The Symmetry of Chaos Alice Ott E (1993) Chaos in Dynamical Systems. Cambridge: Cambridge in the Land of Mirrors. Oxford: Oxford University Press. University Press. Gilmore R and Pei X (2001) The topology and organization of Solari HG, Natiello MA, and Mindlin GB (1996) Nonlinear unstable periodic orbits in Hodgkin–Huxley models of receptors Physics and Its Mathematical Tools. Bristol: IoP Publishing. with subthreshold oscillations. In: Moss F and Gielen S (eds.) Tufillaro NB, Abbott T, and Reilly J (1992) An Experimental Handbook of Biological Physics, Neuro-informatics, Neural Approach to Nonlinear Dynamics and Chaos. Reading, MA: Modeling, vol. 4, pp. 155–203. Amsterdam: North-Holland. Addison-Wesley. Characteristic Classes P B Gilkey, University of Oregon, Eugene, OR, USA Frames R Ivanova, University of Hawaii Hilo, Hilo, HI, USA A frame s := (s , . , s ) for V Vect (M, F) over an S Nikcˇ evic´ , SANU, Belgrade, Serbia and Montenegro 1 k k open set M is a collection2of k smooth sections ª 2006 Elsevier Ltd. All rights reserved. O to V so that {s1(P), . , sk(P)} is a basis for the jO fiber VP of V over any point P . Given such a frame s, we can construct a local trivializat2 O ion which Vector Bundles identifies Fk with V by the mapping O  jO Let Vectk(M, F) be the set of isomorphism classes of P; 1; . ; 1s1 P s P real (F = R) or complex (F = C) vector bundles of ð kÞ ! ð Þ þ Á Á Á þ k kð Þ rank k over a smooth connected m-dimensional Conversely, given a local trivialization of V, we can manifold M. Let take the coordinate frame Vect M; F Vect M; F si P P 0; . ; 0; 1; 0; . ; 0 ð Þ ¼ kð Þ ð Þ ¼  ð Þ k [ Thus, frames and local trivializations of V are equivalent notions. Principal Bundles – Examples Let H be a Lie group. A fiber bundle Simple Covers : M An open cover { } of M, where ranges over some P ! indexing set A, Ois said to be a simple cover if any with fiber H is said to be a principal bundle if there finite intersection 1 k is either empty or is a right action of H on which acts transitively on contractible. O \ Á Á Á \ O P the fibers, that is, if =H = M. If H is a closed Simple covers always exist. Put a Riemannian P subgroup of a Lie group G, then the natural metric on M. If M is compact, then there exists a projection G G=H is a principal H bundle over uniform 0 so that any geodesic ball of radius is ! > the homogeneous space G=H. Let O(k) and U(k) geodesically convex. The intersection of geodesically denote the orthogonal and unitary groups, respec- convex sets is either geodesically convex (and hence k k 1 tively. Let S denote the unit sphere in R þ . Then contractible) or empty. Thus, covering M by a finite we have natural principal bundles: number of balls of radius yields a simple cover. The argument is similar even if M is not compact O k O k 1 Sk ð Þ ð þ Þ ! where an infinite number of geodesic balls is used 2k 1 U k U k 1 S þ and the radii are allowed to shrink near . ð Þ ð þ Þ ! 1 Let RPk and CPk denote the real and complex k 1 Transition Cocycles projective spaces of lines through the origin in R þ k 1 and C þ , respectively. Let Let Hom(F, k) be the set of linear transformations of Fk and let GL(F, k) Hom(F, k) be the group of all Z2 Id O k ¼ fÆ g ð Þ invertible linear transformations. 1 S Id : 1 U k Let {s } be frames for a vector bundle V over some ¼ f Á j j ¼ g ð Þ open cover { } of M. On the intersection , 1 One has Z2 and S principal bundles: O O \ O one may express s = s , that is k 1 k 1 Z2 S À RP À j ! ! s ;i P ; P s ;j P 1 2k 1 k 1 ð Þ ¼ i ð Þ ð Þ S S À CP À 1 j k ! ! X Characteristic Classes 489 The maps : GL(F, k) satisfy Spin Structures O \ O ! For k 3, the fundamental group of SO(k) is Z2. Id on ¼ O 1 Let Spin(k) be the universal cover of SO(k) and let on ½ ¼ O \ O \ O : Spin k SO k ð Þ ! ð Þ Let G be a Lie group. Maps belonging to a be the associated double cover; set Spin(2) = S1 and collection { } of smooth maps from to G let () = 2. An oriented bundle V is said to be spin which satisfy eqn [1] are said to Obe \transO ition if the transition functions can be lifted from SO(k) cocycles with values in G; if G GL(F, k), they to Spin(k); this is possible if and only if the second can be used to define a vector bundl e by making Stiefel–Whitney class of V, which is defined later, appropriate identifications. vanishes. There can be inequivalent spin structures, which are parametrized by the cohomology group 1 Reducing the Structure Group H (M; Z2). If G is a subgroup of GL(F, k), then V is said to have a G-structure if we can choose frames so the The Tangent Bundle of Projective Space transition cocycles belong to G; that is, we can The tangent bundle TRPm of real projective space is reduce the structure group to G. orientable if and only if m is odd; TRPm is spin if Denote the subgroup of orientation-preserving and only if m 3 mod 4. If m 3 mod 4, there are linear maps by two inequivalen t spin structures on this bundle as 1 m H (RP ; Z2) = Z2. GLþ R; k : GL R; k : det > 0 ð Þ ¼ f 2 ð Þ ð Þ g The tangent bundle TCPm of complex projective space is always orientable; TCPm is spin if and only If V Vectk(M, R), then V is said to be orientable if we can2 choose the frames so that if m is odd. GLþ R; k Principal and Associated Bundles 2 ð Þ Let H be a Lie group and let Not every real vector bundle is orientable; the first Stiefel–Whitney class sw (V) H1(M; Z ), which is 1 2 : H defined later, vanishes if and 2only if V is orientable. O \ O ! In particular, the Mo¨ bius line bundle over the circle be a collection of smooth functions satisfying the is not orientable. compatibility conditions given in eqn [1]. We define Similarly, a real (resp. complex) bundle V is a principal bundle by gluing H to H P O  O  said to be Riemannian (resp. Hermitian) if we can using : reduce the structure group to the orthogonal group P; h P; P h for P O(k) GL(R, k) (resp. to the unitary group ð Þ ð ð Þ Þ 2 O \ O U(k) GL(C, k)). Because right multiplication and left multiplication We can use a partition of unity to put a positive- commute, right multiplication gives a natural action definite symmetric (resp. Hermitian symmetric) fiber of H on : metric on V. Applying the Gram–Schmidt process P then constructs orthonormal frames and shows that P; h h~ : P; h h~ the structure group can always be reduced to O(k) ð Þ Á ¼ ð Á Þ (resp. to U(k)); if V is a real vector bundle, then the The natural projection =H = M is an H fiber structure group can be reduced to the special bundle. P ! P orthogonal group SO(k) if and only if V is Let be a representation of H to GL(F, k). For orientable. , Fk, and h H, define a gluing 2 P 2 2 1 ; hÀ ; h Lifting the Structure Group ð Þ ð Á ð Þ Þ The associated vector bundle is then given by Let be a representation of a Lie group H to GL(F, k). One says that the structure group of V can k k F : F = be lifted to H if there exist frames {s } for V and P  ¼ P  smooth maps : H, so = Clearly, { } are the transition cocycles of the where eqn [1] holds forO \.O ! vector bundle Fk. P  490 Characteristic Classes Frame Bundles Homotopy If V is a vector bundle, the associated principal Two smooth maps f0 and f1 from N to M are GL(F, k) bundle is the bundle of all frames; if V is said to be homotopic if there exists a smooth map given an inner product on each fiber, then the F : N I M so that f0(P) = F(P, 0) and so that  ! associated principal O(k) or U(k) bundle is the bundle f1(P) = F(P, 1). If f0 and f1 are homotopic maps from of orthonormal frames. If V is an oriented Riemannian N to M, then f1ÃV is isomorphic to f2ÃV. vector bundle, the associated principal SO(k) bundle is Let [N, M] be the set of all homotopy classes the bundle of oriented orthonormal frames. of smooth maps from N to M. The association V f V induces a natural map ! à Direct Sum and Tensor Product N; M Vect M; F Vect N; F ½  kð Þ ! kð Þ Fiber-wise direct sum (resp.