Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Bounding Tori

As experimental conditions or control parameters

change, strange attractors can undergo ‘‘grosser’’

perestroikas than those that can be described by a

change in the basis set of orbits. This occurs when new

orbits are created that cannot be contained on the initial

branched manifold – for example, when orbits are

created that must be described by a new symbol. This is

seen experimentally in the transition from horseshoe

type dynamics to gateau roule´ type dynamics. This

involves the addition of a third branch to the branched

manifold with two branches, as shown in Figures 7a

and 7b. Strange attractors can undergo perestroikas

described by the addition of new branches to, or

deletion of old branches from, a branched manifold.

These perestroikas are in a very real sense ‘‘grosser’’

than the perestroikas that can be described by changes

in the basis sets of orbits on a fixed branched manifold.

There is a structure that provides constraints on

the allowed bifurcations of branched manifolds

(creation/annihilation of branches), which is analo-

gous to the constraints that a branched manifold

provides on the bifurcations and topological organi-

zation of the periodic orbits that can exist on it. This

structure is called a bounding torus.

Bounding tori are constructed as follows. The semi-

flow on a branched manifold is ‘‘inflated’’ or ‘‘blown

up’’ to a flow on a thin open set in R

containing this

branched manifold. The boundary of this open set is a

two-dimensional surface. Such surfaces have been

classified. They are uniquely tori of genus g; g = 0

(sphere), g = 1 (tire tube), g = 2, 3, .... The torus of

genus g has Euler characteristic  = 2  2g. The flow is

into this surface. The flow, restricted to the surface,

exhibits a singularity wherever it is normal to the

surface. At such singularities the stability is determined

by the local Lyapunov exponents: 

> 0and

< 0,

since the flow direction (

= 0) is normal to the

surface. As a result, all singularities are saddles; so, by

the Poincare´–Hopf theorem, the number of singularities

is strongly related to the genus. The number is 2(g  1).

The flow, restricted to the genus-g surface, can be

put into canonical form and these canonical forms can

be classified. The classification involves projection of

the genus-g torus onto a two-dimensional surface. The

planar projection consists of a disk with outer

boundary and g interior holes. All singularities can be

placed on the interior holes. The flow on the interior

holes without singularities is in the same direction as

the flow on the exterior boundary. Interior holes with

singularities have an even number, 4, 6, ....Some

canonical forms are shown in Figure 9.

Poincare´ sections have been used to simplify the

study of flows in low-dimensional spaces by effec-

tively reducing the dimension of the dynamics. In

three dimensions, a Poincare´ surface of section for a

strange attractor is a minimal two-dimensional sur-

face with the property that all points in the attractor

intersect this surface transversally an infinite number

of times under the flow. The Poincare´ surface need

not be connected and in fact is often not connected.

The Poincare´ section for the flow in a genus-g torus

consists of the union of g  1 disjoint disks (g  3) or

is a single disk (g = 1). The locations of the disks are

determined algorithmically, as shown in Figure 9.The

interior circles without singularities are labeled by

capital letters A, B, C, ... and those with singularities

are labeled with lowercase letters a, b, c , ... The

components of the global Poincare´ surface of section

are numbered sequentially 1, 2, ..., g  1, in the order

they are encountered when traversing the outer

boundary in the direction of the flow, starting from

any point on that boundary. Each component of the

global Poincare´ surface of section connects (in the

projection) an interior circle without singularities to

the exterior boundary. There is one component

between each successive encounter of the flow with

ABCBDED ABCDCBE ABCBDBE

abbccaaabccbaaabbacca

(a) (b) (c)

c bac

A E

Figure 9 Three inequivalent canonical forms of genus 8 are shown. Each is identified by a ‘‘period-7 orbit’’ and its dual. Reprinted

figure with permission from Physical Review E, 69, 056206, 2004. Copyright (2004) by the American Physical Society.

Chaos and Attractors 485

holes that have singularities. Heavy lines are used to

show the location of the seven components of the

global Poincare´ surface of section for each of the three

inequivalent genus-8 canonical forms shown in

Figure 9. The structure of the flow is summarized by

a transition matrix. For the canonical form shown in

Figure 9c the transition matrix is

T ¼

1100000

0011000

0000110

0100001

1000001

where T

i, j

= 1 if the flow can proceed directly from

component i to component j, 0 otherwise.

Bounding tori, dressed with flows, can be labeled. In

fact, two dual labeling schemes are possible. Following

the outer boundary in the direction of the flow, one

encounters the g  1 components of the global Poin-

care´ surface of section sequentially, the interior holes

without singularities at least once each, and the interior

holes with singularites at least twice each. The

canonical form (genus-g torus dressed with a flow) on

the genus-8 bounding torus shown in Figure 9a can be

labeled by the sequence in which the holes without

singularities are encountered (ABCBDED) or the order

in which the holes with singularities are encountered

(abbacca). Both sequences contain g  1symbols.

These labels are unique up to cyclic permutation.

Symbol sequences for canonical forms for bounding

tori act in many ways like symbol sequences for

periodic orbits on branched manifolds. Although there

is a 1:1 correspondence between bounded closed two-

dimensional surfaces in R

and genus g, the number of

canonical forms grows rapidly with g, as shown in

Table 4. In fact, the number, N(g), grows exponen-

tially and can even be assigned an entropy:

lim

g !1

lnðNðgÞÞ

g  1

¼ ln 3 ½5

In some sense, canonical forms that constrain

branched manifolds within them behave like branched

manifolds that constrain periodic orbits on them.

Every strange attractor that has been studied in R

has been described by a canonical bounding torus that

contains it. This classification is shown in Table 5.

Branched manifold perestroikas are constrained

by bounding tori as follows. Each branch line of any

branched manifold can be moved into one of the

g  1 components of the global Poincare´ surface of

section. Any branched manifold contained in a

genus-g bounding torus (g  3) must have at least

one branch between each pair of components of the

global Poincare´ surface of section between which the

flow is allowed, as summarized by the canonical

form’s transition matrix. New branches can only be

added in a way that is consistent with the canonical

form’s transition matrix, continuity requirements,

and the no intersection condition.

Table 4 Number of canonical bounding tori as a function of

genus g

gN(g) gN(g) gN(g)

3 1 9 15 15 2 211

4 1 10 28 16 5 549

5 2 11 67 17 14 290

6 2 12 145 18 36 824

7 5 13 368 19 96 347

8 6 14 870 20 252 927

Table 5 All known strange attractors of dimension d

< 3 are bounded by one of the standard dressed tori. Dual labels for the

bounding tori depend on g  1 symbols describing holes with or without singularities

Strange attractor Holes w/o singularites Holes with singularities Genus

Rossler, Duffing, Burke, and Shaw A 1

Various lasers, gateau roule

A 1

Neuron with subthreshold oscillations A 1

Shaw–van der Pol A 1

Lorenz, Shimizu–Morioka, Rikitake AB aa 3

covers of Rossler AB a

cover of Lorenz

ABCD a

cover of Lorenz

ABCB abba 5

2 ! 1 Image of figure-8 branched manifold ABCB ab(ab)

1

Figure-8 branched manifold AEBECEDE a

covers of Rossler AB Na

n þ 1

cover of Lorenz

AB (2N) a

2n þ 1

cover of Lorenz

(AZ )(BZ ) (NZ ) a

n

2n þ 1

Multispiral attractors A(B M)N(B M)

1

(ab m)(ab m)

1

2m þ 1

Rotation axis through origin.

Rotation axis through one focus.

486 Chaos and Attractors

In the simplest case, g = 1, a third branch can be

added to a branched manifold with two branches only

if its local torsion differs by 1fromtheadjacent

branch. In addition, the ordering of the new branch

must be consistent with the continuity and no

intersection (ODE uniqueness theorem) requirements.

Embeddings of Bounding Tori

The last level of topological structure needed for the

classification of strange attractors in R

describes

their embeddings in R

. The classification using

genus-g bounding tori is intrinsic – that is, the

canonical form shows how the flow looks from

inside the torus. Strange attractors, and the tori that

bound them, are actually embedded in R

. For a

complete classification, we must specify not only the

canonical form but also how this form sits in R

This program has not yet been completed, but we

illustrate it with the genus-1 bounding torus in

Figure 10. Figure 10a shows the canonical form, and

two different embeddings of it in R

. The embedding

on the left is unknotted. The embedding on the right is

knotted like a figure-8 knot. Extrinsic embeddings of

genus-1 tori are described by tame knots in R

,and

tame knots can be used as ‘‘centerlines’’ for extrinsi-

cally embedded genus-1 tori. Higher-genus (g  3)

canonical forms – intrinsic genus-g tori dressed with a

canonical flow – have a larger (but discrete) variety of

extrinsic embeddings in R

The Embedding Question

The mechanism that nature uses to generate chaotic

behavior in physical systems is not directly observable,

and must be deduced by examining the data that are

generated. Typically, the data consist of a single scalar

time series that is discretely recorded: x

, i = 1, 2, ....

In order to exhibit a strange attractor, a mapping of the

data into R

must also be constructed. If the attractor

is low dimensional (d

< 3), one can hope that a

mapping into R

can be constructed that exhibits no

self-intersections or other degeneracies. Such a map is

called an embedding. Once an embedding in R

available, a topological analysis can be carried out. The

analysis reveals the mechanism that underlies the

creation of the embedded strange attractor.

But how do you know that the mechanism that

generates the observed, embedded strange attractor

has anything to do with the mechanism nature used

to generate the experimental data?

If the embedding is contained in a genus-1 bounding

torus, then the topological mechanism that generates

the data, as defined by some unknown branched

manifold BM

EXP

, and the topological mechanism that

is identified from the embedded strange attractor

EMB

, are identical up to three degrees of freedom:

parity, global torsion, and the knot type. As a result, in

this case (genus-1) a topological analysis of embedded

data does reveal nature’s hidden secrets.

See also: Ergodic theory; Fractal dimensions in

dynamics; Generic Properties of Dynamical Systems;

Gravitational N-body Problem (Classical);

Homeomorphisms and Diffeomorphisms of the Circle;

Homoclinic phenomena; Inviscid Flows; Lyapunov

Exponents and Strange Attractors; Nonequilibrium

Statistical Mechanics (Stationary): Overview; Random

Algebraic Geometry, Attractors and Flux Vacua; Random

Matrix Theory in Physics; Regularization for Dynamical

Zeta Functions; Singularity and Bifurcation Theory;

Symmetry and Symmetry Breaking in Dynamical

Systems; Synchronization of Chaos.

Further Reading

Abraham R and Shaw CD (1992) Dynamics: The Geometry of

Behavior, Studies in Nonlinearity, 2nd edn. Reading, MA:

Addison-Wesley.

Eckmann J-P and Ruelle D (1985) Ergodic theory of chaos and

strange attractors. Reviews of Modern Physics 57(3): 617–656.

Gilmore R (1998) Topological analysis of chaotic dynamical

systems. Reviews of Modern Physics 70(4): 1455–1529.

Gilmore R and Lefranc M (2002) The Topology of Chaos, Alice

in Stretch and Squeezeland. New York: Wiley.

(b) (c)

(a)

Figure 10 (a) Canonical form for genus-1 bounding torus.

Extrinsic embeddings of the torus into R

that are (b) unknotted

and (c) knotted like the figure-8 knot.

Chaos and Attractors 487

Gilmore R and Letellier C (2006) The Symmetry of Chaos Alice

in the Land of Mirrors. Oxford: Oxford University Press.

Gilmore R and Pei X (2001) The topology and organization of

unstable periodic orbits in Hodgkin–Huxley models of receptors

with subthreshold oscillations. In: Moss F and Gielen S (eds.)

Handbook of Biological Physics, Neuro-informatics, Neural

Modeling, vol. 4, pp. 155–203. Amsterdam: North-Holland.

Ott E (1993) Chaos in Dynamical Systems. Cambridge: Cambridge

University Press.

Solari HG, Natiello MA, and Mindlin GB (1996) Nonlinear

Physics and Its Mathematical Tools. Bristol: IoP Publishing.

Tufillaro NB, Abbott T, and Reilly J (1992) An Experimental

Approach to Nonlinear Dynamics and Chaos. Reading, MA:

Addison-Wesley.

Characteristic Classes

P B Gilkey, University of Oregon, Eugene, OR, USA

R Ivanova, University of Hawaii Hilo, Hilo, HI, USA

S Nikc

evic

, SANU, Belgrade, Serbia and Montenegro

Vector Bundles

Let Vect

(M, F) be the set of isomorphism classes of

real (F = R) or complex (F = C) vector bundles of

rank k over a smooth connected m-dimensional

manifold M. Let

VectðM; FÞ¼

[

Vect

ðM; FÞ

Principal Bundles – Examples

Let H be a Lie group. A fiber bundle

 : P!M

with fiber H is said to be a principal bundle if there

is a right action of H on P which acts transitively on

the fibers, that is, if P=H = M.IfH is a closed

subgroup of a Lie group G, then the natural

projection G ! G =H is a principal H bundle over

the homogeneous space G=H. Let O(k) and U(k)

denote the orthogonal and unitary groups, respec-

tively. Let S

denote the unit sphere in R

kþ1

. Then

we have natural principal bundles:

OðkÞOðk þ 1Þ!S

UðkÞUðk þ 1Þ!S

2kþ1

Let RP

and CP

denote the real and complex

projective spaces of lines through the origin in R

kþ1

and C

kþ1

, respectively. Let

¼fIdgOðkÞ

¼f  Id : jj¼1gUðkÞ

One has Z

and S

principal bundles:

! S

k1

! RP

k1

! S

2k1

! CP

k1

Frames

A frame s := (s

, ..., s

) for V 2 Vect

(M, F) over an

open set OM is a collection of k smooth sections

to V j

so that {s

(P), ..., s

(P)} is a basis for the

fiber V

of V over any point P 2O. Given such a

frame s, we can construct a local trivialization which

identifies OF

with Vj

by the mapping

ðP; 

; ...;

Þ!

ðPÞþþ

ðPÞ

Conversely, given a local trivialization of V, we can

take the coordinate frame

ðPÞ¼P ð0; ...; 0; 1; 0; ...; 0Þ

Thus, frames and local trivializations of V are

equivalent notions.

Simple Covers

An open cover {O



}ofM, where  ranges over some

indexing set A, is said to be a simple cover if any

finite intersection O



\\O



is either empty or

contractible.

Simple covers always exist. Put a Riemannian

metric on M.IfM is compact, then there exists a

uniform >0 so that any geodesic ball of radius  is

geodesically convex. The intersection of geodesically

convex sets is either geodesically convex (and hence

contractible) or empty. Thus, covering M by a finite

number of balls of radius  yields a simple cover.

The argument is similar even if M is not compact

where an infinite number of geodesic balls is used

and the radii are allowed to shrink near 1.

Transition Cocycles

Let Hom(F, k) be the set of linear transformations of

and let GL(F, k)  Hom(F, k) be the group of all

invertible linear transformations.

Let {s



} be frames for a vector bundle V over some

open cover {O



}ofM. On the intersection O





one may express s







, that is

;i

ðPÞ¼

1jk

;

ðPÞs

;j

ðPÞ

488 Characteristic Classes

The maps



: O





! GL(F, k) satisfy



¼ Id on O









on O







½1

Let G be a Lie group. Maps belonging to a

collection {



} of smooth maps from O





to G

which satisfy eqn [1] are said to be transition

cocycles with values in G;ifG  GL(F, k), they

can be used to define a vector bundle by making

appropriate identifications.

Reducing the Structure Group

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

transition cocycles belong to G; that is, we can

reduce the structure group to G.

Denote the subgroup of orientation-preserving

linear maps by

ðR; kÞ :¼f 2 GLðR; kÞ: detð Þ > 0g

If V 2 Vect

(M, R), then V is said to be orientable if

we can choose the frames so that



2 GL

ðR; kÞ

Not every real vector bundle is orientable; the first

Stiefel–Whitney class sw

(V) 2 H

(M; Z

), which is

defined later, vanishes if and only if V is orientable.

In particular, the Mo¨ bius line bundle over the circle

is not orientable.

Similarly, a real (resp. complex) bundle V is

said to be Riemannian (resp. Hermitian) if we can

reduce the structure group to the orthogonal group

O(k)  GL(R, k) (resp. to the unitary group

U(k)  GL(C, k)).

We can use a partition of unity to put a positive-

definite symmetric (resp. Hermitian symmetric) fiber

metric on V. Applying the Gram–Schmidt process

then constructs orthonormal frames and shows that

the structure group can always be reduced to O(k)

(resp. to U(k)); if V is a real vector bundle, then the

structure group can be reduced to the special

orthogonal group SO(k) if and only if V is

orientable.

Lifting the Structure Group

Let  be a representation of a Lie group H to

GL(F, k). One says that the structure group of V can

be lifted to H if there exist frames {s



} for V and

smooth maps 



: O





! H,so



where eqn [1] holds for .

Spin Structures

For k  3, the fundamental group of SO(k)isZ

Let Spin(k) be the universal cover of SO(k) and let

 : SpinðkÞ!SOðkÞ

be the associated double cover; set Spin(2) = S

and

let () = 

. An oriented bundle V is said to be spin

if the transition functions can be lifted from SO(k)

to Spin(k); this is possible if and only if the second

Stiefel–Whitney class of V, which is defined later,

vanishes. There can be inequivalent spin structures,

which are parametrized by the cohomology group

(M; Z

The Tangent Bundle of Projective Space

The tangent bundle TRP

of real projective space is

orientable if and only if m is odd; TRP

is spin if

and only if m  3 mod 4. If m  3 mod 4, there are

two inequivalent spin structures on this bundle as

(RP

; Z

) = Z

The tangent bundle TCP

of complex projective

space is always orientable; TCP

is spin if and only

if m is odd.

Principal and Associated Bundles

Let H be a Lie group and let





: O





! H

be a collection of smooth functions satisfying the

compatibility conditions given in eqn [1]. We define

a principal bundle P by gluing O



 H to O



 H

using :

ðP; hÞ



ðP;



ðPÞhÞ



for P 2O





Because right multiplication and left multiplication

commute, right multiplication gives a natural action

of H on P:

ðP; hÞ



h :¼ðP; h 

hÞ



The natural projection P!P=H = M is an H fiber

bundle.

Let  be a representation of H to GL(F, k). For

 2P,  2 F

, and h 2 H, define a gluing

ð; Þð  h

1

;ðhÞÞ

The associated vector bundle is then given by

P



:¼PF

=

Clearly, {



} are the transition cocycles of the

vector bundle P



Characteristic Classes 489

Frame Bundles

If V is a vector bundle, the associated principal

GL(F, k) bundle is the bundle of all frames; if V is

given an inner product on each fiber, then the

associated principal O(k)orU(k) bundle is the bundle

of orthonormal frames. If V is an oriented Riemannian

vector bundle, the associated principal SO(k) bundle is

the bundle of oriented orthonormal frames.

Direct Sum and Tensor Product

Fiber-wise direct sum (resp. tensor product) defines the

direct sum (resp. tensor product) of vector bundles:

 : Vect

ðM; FÞVect

ðM; FÞ

! Vect

kþn

ðM; FÞ

 : Vect

ðM; FÞVect

ðM; FÞ

! Vect

ðM; FÞ

The transition cocycles of the direct sum (resp.

tensor product) of two vector bundles are the direct

sum (resp. tensor product) of the transition cocycles

of the respective bundles.

The set of line bundles Vect

(M, F) is a group

under . The unit in the group is the trivial line

bundle l := M  F; the inverse of a line bundle L is

the dual line bundle L



:= Hom(L, F) since

L  L



¼ l

Pullback Bundle

Let  : V ! M be the projection associated with

V 2 Vect

(M, F). If f is a smooth map from N to M,

then the pullback bundle f



V is the vector bundle

over N which is defined by setting



V :¼fðP; vÞ2N  V : f ðPÞ¼ðvÞg

The fiber of f



V over P is the fiber of V over f (P).

Let {s



} be local frames for V over an open cover



}ofM. For P 2 f

1



), define





gðPÞ :¼ðP; s



ðf ðPÞÞÞ

This gives a collection of frames for f



V over the

open cover {f

1



)} of N. Let





:¼



 f

be the pullback of the transition functions. Then





gðPÞ¼ðP;



ðf ðPÞÞs



ðf ðP ÞÞÞ

¼fðf





Þðf





ÞgðPÞ

This shows that the pullback of the transition

functions for V are the transition functions of the

pullback f



(V).

Homotopy

Two smooth maps f

and f

from N to M are

said to be homotopic if there exists a smooth map

F : N  I ! M so that f

(P) = F(P, 0) and so that

(P) = F(P, 1). If f

and f

are homotopic maps from

N to M, then f



V is isomorphic to f



Let [N, M] be the set of all homotopy classes

of smooth maps from N to M. The association

V ! f



V induces a natural map

½N; MVect

ðM; FÞ!Vect

ðN; FÞ

If M is contractible, then the identity map is

homotopic to the constant map c. Consequently,

V = Id



V is isomorphic to c



V = M  F

. Thus, any

vector bundle over a contractible manifold is trivial.

In particular, if {O



} is a simple cover of M and if

V 2 Vect(M, F), then Vj



is trivial for each . This

shows that a simple cover is a trivializing cover for

every V 2 Vect(M, F).

Stabilization

Let l 2 Vect

(M, F) denote the isomorphism class of

the trivial line bundle M  F over an m-dimensional

manifold M. The map V ! V  l induces a stabili-

zation map

s : Vect

ðM; FÞ!Vect

kþ1

ðM; FÞ

which induces an isomorphism

Vect

ðM; RÞ¼Vect

kþ1

ðM; RÞ for k > m

Vect

ðM; C Þ¼Vect

kþ1

ðM; C Þ for 2k > m

½2

These values of k comprise the stable range.

The K-Theory

The direct sum  and tensor product  make

Vect(M, F) into a semiring; we denote the associated

ring defined by the Grothendieck construction by

KF(M). If V 2 Vect(M, F), let [V] 2 KF(M) be the

corresponding element of K-theory; KF(M) is gener-

ated by formal differences [V

]  [V

]; such formal

differences are called virtual bundles.

The Grothendieck construction (see K-theory)

introduces nontrivial relations. Let S

denote the

standard sphere in R

mþ1

. Since

TðS

Þl ¼ðm þ 1Þl

we can easily see that [TS

] = m[ l ]inKR(S

despite the fact that T(S

) is not isomorphic to ml

for m 6¼ 1, 3, 7.

Let L denote the nontrivial real line bundle over

. Then TRP

 l = (k þ 1)L,so

½TRP

¼ðk þ 1Þ½L½l 

490 Characteristic Classes

The map V ! Rank(V) extends to a surjective

map from KF(M)toZ. We denote the associated

ideal of virtual bundles of virtual rank 0 by

KFðMÞ :¼ kerðRankÞ

In the stable range, V ! [V]  k[ l ] identifies

Vect

ðM; RÞ¼

KRðMÞ if k > m

Vect

ðM; C Þ¼

KCðMÞ if 2k > m

½3

These groups contain nontrivial torsion. Let L be the

nontrivial real line bundle over RP

. Then

KRðRP

Þ¼Z f½L½l g=2

ðkÞ

Zf½L½l g

where (k) is the Adams number.

Classifying Spaces

Let Gr

(F, n) be the Grassmannian of k-dimensional

subspaces of F

. By mapping a k-plane  in F

to the

corresponding orthogonal projection on , we can

identify Gr

(F, n) with the set of orthogonal projec-

tions of rank k:

f 2 HomðF

Þ: 

¼ ; 



¼ ; trðÞ¼kg

There is a natural associated tautological k-plane

bundle

ðF; nÞ2Vect

ðGr

ðF; nÞ ; FÞ

whose fiber over a k-plane  is the k-plane itself:

ðF; nÞ :¼fð; xÞ2HomðF

ÞF

: x ¼ xg

Let [M,Gr

(F, n)] denote the set of homotopy

equivalence classes of smooth maps f from M to

(F, n). Since [f

] = [f

] implies that f



V is

isomorphic to f



V, the association

f ! f



ðF; nÞ2Vect

ðM; FÞ

induces a map

½M; Gr

ðF; nÞ ! Vect

ðM; FÞ

This map defines a natural equivalence of functors

in the stable range:

½M; Gr

ðR;þ kÞ ¼ Vect

ðM; RÞ for >m

½M; Gr

ðC;þ kÞ ¼ Vect

ðM; C Þ for 2>m

½4

The natural inclusion of F

in F

nþ1

induces natural

inclusions

ðF; nÞGr

ðF; n þ 1Þ

ðF; nÞV

ðF; n þ 1Þ

½5

Let Gr

(F, 1) and V

(F, 1) be the direct limit

spaces under these inclusions; these are the infinite-

dimensional Grassmannians and classifying bundles,

respectively. The topology on these spaces is the

weak or inductive topology. The Grassmannians are

called classifying spaces. The isomorphisms of

eqn [4] are compatible with the inclusions of eqn [5]

and we have

½M; Gr

ðF; 1Þ ¼ Vect

ðM; FÞ½6

Spaces with Finite Covering Dimension

A metric space X is said to have a covering

dimension at most m if, given any open cover {U



}

of X, there exists a refinement {O



} of the cover so

that any intersection of more than m þ 1 of the {O



}

is empty. For example, any manifold of dimension

m has covering dimension at most m. More

generally, any m-dimensional cell complex has

covering dimension at most m.

The isomorphisms of [2]–[4],and[6] continue to

hold under the weaker assumption that M is a metric

space with covering dimension at most m.

Characteristic Classes of Vector

Bundles

The Cohomology of Gr

(F, 1)

The cohomology algebras of the Grassmannians are

polynomial algebras on suitably chosen generators:



ðGr

ðR; 1Þ; Z

Þ¼Z

½sw

; ...; sw





ðGr

ðC; 1Þ; ZÞ¼Z½c

; ...; c



½7

The Stiefel–Whitney Classes

Let V 2 Vect

(M, R). We use eqn [6] to find

 : M ! Gr

(R, 1) which classifies V; the map 

is uniquely determined up to homotopy and, using

eqn [7], one sets

ðVÞ :¼ 



2 H

ðM; Z

The total Stiefel–Whitney class is then defined by

swðVÞ¼1 þ sw

ðVÞþþsw

ðVÞ

The Stiefel–Whitney class has the properties:

1. If f : X

! X

, then f



(sw(V)) = sw(f



V).

2. sw(V  W) = sw(V)sw(W).

3. If L is the Mo¨ bius bundle over S

, then sw

(L)

generates H

; Z

) = Z

The cohomology algebra of real projective space

is a truncated polynomial algebra:



ðRP

; Z

Þ¼Z

½x=x

kþ1

¼ 0

Characteristic Classes 491

Since TRP

 l = (k þ 1)L, one has

swðTRP

Þ¼ð1 þ xÞ

kþ1

¼ 1 þ kx þ

ðk þ 1Þk

þ ½8

Orientability and Spin Structures

The Stiefel–Whitney classes have real geometric

meaning. For example, sw

(V) = 0 if and only if V

is orientable; if sw

(V) = 0, then sw

(V) = 0 if and

only if V admits a spin structure. With reference to

the discussion on the tangent bundle or projective

space, eqn [8] yields

ðTRP

Þ¼

0ifk  0 mod 2

x if k  1 mod 2



Thus, RP

is orientable if and only if k is odd.

Furthermore,

ðTRP

Þ¼

0ifk  3 mod 4

x if k  1 mod 4



Thus, TRP

is spin if and only if k  3 mod 4.

Chern Classes

Let V 2 Vect

(M, C). We use eqn [6] to find

 : M ! Gr

(C, 1) which classifies V; the map 

is uniquely determined up to homotopy and, using

eqn [7], one sets

ðVÞ :¼ 



2 H

ðM; ZÞ

The total Chern class is then defined by

cðVÞ :¼ 1 þ c

ðVÞþþc

ðVÞ

The Chern class has the properties:

1. If f : X

! X

, then f



(c(V)) = c(f



V).

2. c(V  W) = c(V)c(W).

3. Let L be the classifying line bundle over

= CP

. Then

(L) = 1.

The cohomology algebra of complex projective

space also is a truncated polynomial algebra



ðCP

; ZÞ¼Z½x=x

kþ1

where x = c

(L) and L is the complex classifying line

bundle over CP

= Gr

(C, k þ 1). If T

is the

complex tangent bundle, then

cðT

Þ¼ð1 þ xÞ

kþ1

The Pontrjagin Classes

Let V be a real vector bundle over a topological

space X of rank r = 2k or r = 2k þ1. The Pontrjagin

classes p

(V) 2 H

(X; Z) are characterized by the

properties:

1. p(V) = 1 þ p

(V) þþp

(V).

2. If f : X

! X

, then f



(p(V)) = p(f



V).

3. p(V  W) = p(V)p(W) mod elements of order 2.

(TCP

) = 3.

We can complexify a real vector bundle V to

construct an associated complex vector bundle V

We have

ðVÞ :¼ð1Þ

ðV

Conversely, if V is a complex vector bundle, we can

construct an underlying real vector bundle V

forgetting the underlying complex structure. Mod-

ulo elements of order 2, we have

pðV

Þ¼cðVÞcðV



Let TCP

be the real tangent bundle of complex

projective space. Then

pðTCP

Þ¼ð1  x

kþ1

Line Bundles

Tensor product makes Vect

(M, F) into an abelian

group. One has natural equivalences of functors

which are group homomorphisms:

: Vect

ðM; RÞ!H

ðM; Z

: Vect

ðM; C Þ!H

ðM; ZÞ

A real line bundle L is trivial if and only if it is

orientable or, equivalently, if sw

(L) vanishes. A

complex line bundle L is trivial if and only if

(L) = 0. There are nontrivial vector bundles with

vanishing Stiefel–Whitney classes of rank k > 1. For

example, sw

(TS

) = 0 for i > 0 despite the fact that

is trivial if and only if k = 1, 3, 7.

Curvature and Characteristic Classes

de Rham Cohomology

We can replace the coefficient group Z by C at the cost

of losing information concerning torsion. Thus, we

may regard p

(V) 2 H

(M; C)ifV is real or c

(V) 2

(M; C)ifV is complex. Let M be a smooth

manifold. Let C



M be the space of smooth

p-forms and let

d : C



M ! C



pþ1

492 Characteristic Classes

be the exterior derivative. The de Rham cohomology

groups are then defined by

deR

ðMÞ :¼

kerðd : C



M ! C



pþ1

MÞ

imðd : C



p1

M ! C



MÞ

The de Rham theorem identifies the topological

cohomology groups H

(M; C) with the de Rham

cohomology groups H

deR

(M) which are given

differential geometrically.

Given a connection on V, the Chern–Weyl theory

enables us to compute Pontrjagin and Chern classes in

de Rham cohomology in terms of curvature.

Connections

Let V be a vector bundle over M. A connection

r : C

ðVÞ!C

ðT



M  VÞ

on V is a first-order partial differential operator

which satisfies the Leibnitz rule, that is, if s is a

smooth section to V and if f is a smooth function

on M,

rðfsÞ¼df  s þ f rs

If X is a tangent vector field, we define

s ¼hX; rsi

where h, i denotes the natural pairing between the

tangent and cotangent spaces. This generalizes to the

bundle setting the notion of a directional derivative

and has the properties:

1. r

s = f r

2. r

(fs) = X(f )s þ f r

3. r

þX

s = r

s þr

4. r

þ s

) = r

þr

The Curvature 2-Form

Let !

be a smooth p-form. Then

r : C

ð

M  VÞ!C

ð

pþ1

M  VÞ

can be extended by defining

rð!

 sÞ¼d!

 s þð1Þ

^rs

In contrast to ordinary exterior differentiation, r

need not vanish. We set

ðsÞ : ¼r

This is not a second-order partial differential

operator; it is a zeroth-order operator, that is,

ðfsÞ¼ddf  s  df ^rs þ df ^rs þ f r

¼ f ðsÞ

The curvature operator  can also be computed

locally. Let (s

) be a local frame. Expand

 s

to define the connection 1-form !. One then has

¼ d!

 !

^ !



 s

and so



¼ d!

 !

^ !

s =

j is another local frame, we compute

! ¼ dgg

1

þ g!g

1

and

 ¼ gg

1

Although the connection 1-form ! is not tensorial, the

curvature is an invariantly defined 2-form-valued

endomorphism of V.

Unitary Connections

Let ( , ) be a nondegenerate Hermitian inner product

on V.Wesaythatr is a unitary connection if

ðrs

; s

Þþðs

; rs

Þ¼dðs

; s

Such connections always exist and, relative to a

local orthonormal frame, the curvature is skew-

symmetric, that is,

 þ 



¼ 0

Thus,  can be regarded as a 2-form-valued element

of the Lie algebra of the structure group, O(V) in the

real setting or U(V) in the complex setting.

Projections

We can always embed V in a trivial bundle 1



dimension ; let 

be the orthogonal projection on

V. We project the flat connection to V to define a

natural connection on V. For example, if M is

embedded isometrically in the Euclidean space R



this construction gives the Levi-Civita connection on

the tangent bundle TM. The curvature of this

connection is then given by

 ¼ 

d

Let V

be the fiber of V over a point P 2 M. The

inclusion i : V  R

defines the classifying map

f : P ! Gr

(R, n) where we set

f ðPÞ¼iðV

Characteristic Classes 493

Chern–Weyl Theory

Let r be a Riemannian connection on a real vector

bundle V of rank k. We set

pðÞ :¼ det I þ

2





Let 

denote the transpose matrix of differential

form. Since  þ 

= 0, the polynomials of odd

degree in  vanish and we may expand

pðÞ¼1 þ p

ðÞþþp

ðÞ

where k = 2r or k = 2r þ 1 and the differential forms

() 2 C



(M) are forms of degree 4i.

Changing the gauge (i.e., the local frame) replaces

 by gg

1

and hence p() is independent of the

local frame chosen. One can show that dp

() = 0;

let [p

()] denote the corresponding element of de

Rham cohomology. This is independent of the

particular connection chosen and [p

()] represents

(V)inH

(M; C).

Similarly, let V be a complex vector bundle of

rank k with a Hermitian connection r. Set

cðÞ :¼ det I þ

ﬃﬃﬃﬃﬃﬃﬃ

1

2



¼ 1 þ c

ðÞþþc

ðÞ

Again c

() is independent of the local gauge and

() = 0. The de Rham cohomology class [c

()]

represents c

(V)inH

(M; C).

The Chern Character

The total Chern character is defined by the formal

sum

chðÞ :¼ trðe

ﬃﬃﬃﬃﬃ

1

=2



ﬃﬃﬃﬃﬃﬃﬃ

1



ð2Þ



!

trð



¼ ch

ðÞþch

ðÞþ

Let ch(V) = [ch()] denote the associated de Rham

cohomology class; it is independent of the particular

connection chosen. We then have the relations

chðV  WÞ¼chðVÞþchðWÞ

chðV  WÞ¼chðVÞchðWÞ

The Chern character extends to a ring isomorph-

ism from KU(M)  Q to H

(M; Q), which is a

natural equivalence of functors; modulo torsion,

K theory and cohomology are the same functors.

Other Characteristic Classes

The Chern character is defined by the exponential

function. There are other characteristic classes

which appear in the index theorem that are defined

using other generating functions that appear in

index theory. Let x := (x

, ...) be a collection of

indeterminates. Let s



(x) be the th elementary

symmetric function;



ð1 þ x



Þ¼1 þ s

ðxÞþs

ðxÞþ

For a diagonal matrix A := diag(

, ...), denote the

normalized eigenvalues by x

ﬃﬃﬃﬃﬃﬃﬃ

1



=2. Then

cðAÞ¼det 1 þ

ﬃﬃﬃﬃﬃﬃﬃ

1

2

¼ 1 þ s

ðxÞþ

Thus, the Chern class corresponds in a certain sense

to the elementary symmetric functions.

Let f (x) be a symmetric polynomial or more

generally a formal power series which is symmetric.

We can express f (x) = F(s

(x), ...) in terms of the

elementary symmetric functions and define

f () = F(c

(), ...) by substitution. For example,

the Chern character is defined by the generating

function

f ðxÞ :¼

¼1



The Todd class is defined using a different

generating function:

tdðxÞ :¼



ð1  e

x



1

¼ 1 þ td

ðxÞþ

If V is a real vector bundle, we can define

some additional characteristic classes similarly. Let

{

ﬃﬃﬃﬃﬃﬃﬃ

1



, ...} be the nonzero eigenvalues of a

skew-symmetric matrix A.Wesetx

=  

=2

and define the Hirzebruch polynomial L and the

genus by

LðxÞ :¼



tanhðx



¼ 1 þ L

ðxÞþL

ðxÞþ

AðxÞ :¼



2 sinhðð1=2Þx



¼ 1 þ

ðxÞþ

ðxÞþ

The generating functions

tanhðxÞ

and

2 sinhðð1=2ÞxÞ

494 Characteristic Classes