топология / Tom Dieck T. Algebraic topology (EMS, 2008)

468 Chapter 19. Characteristic Classes

Suppose the theory is C-oriented. Then an n-dimensional complex vector bundle over B has an Euler class e. / 2 h2n.B/ associated to t. /. Euler classes are natural, f e. / D e.f /, and multiplicative, e. ˚ / D e. / Y e. /.

The Thom classes have associated Thom homomorphisms. They are defined as before by cup product with the Thom class

These Thom homomorphisms are natural and multiplicative as we have explained earlier.

For an R-oriented theory we have natural and multiplicative Euler classes for real vector bundles.

A proof of (19.0.1) is based on a determination of characteristic classes. We present a construction of characteristic classes based on the cohomology of projective bundles. For this purpose, classifying spaces are not used. But they will of course appear and they are necessary for a more global view-point.

19.1 Projective Spaces

Let n W En ! CP n 1 be the canonical bundle with total space

En D Cn X 0 C C; .z; u/ . z; u/:

We have the embedding n W En ! CP n, .z; u/ 7! Œz; u . The image is the complement of the point D Œ0; : : : ; 0; 1 . Let t. n/ 2 h2.En; En0/ be a Thom class. The Thom class yields the element tn 2 h2.CP n/ as the image under

h2.En; En0/ Š h2.CP n; CP n X CP n 1/ Š h2.CP n; / ! h2.CP n/:

The first isomorphism is induced by n. Note that n sends the zero section to CP n 1, the image under the embedding W Œx1; : : : ; xn 7! Œx1; : : : ; xn; 0 . The total space En of n was denoted H.1/ in (14.2.6). The bundle n is the (complex) normal bundle of the embedding W CP n 1 ! CP n. The embeddingn is a tubular map; it also shows that CP n is the one-point compactification of En

(see the definition of a Thom space in the final chapter). The complement CP n X CP n 1 is the affine subset UnC1 D fŒx1; : : : ; xnC1 j xnC1 D6 0g. We obtain a homomorphism h Œtn ! h .CP n/ of graded h -algebras; it sends tnnC1

to zero (see (17.2.5)) and induces a homomorphism of the quotient by the principal ideal .tnnC1/.

(19.1.1) Lemma. Let t. n 1/ 2 h2.En 1; En0 1/ be the Thom class obtained from t. n/ by restriction along . Let tn 1 2 h2.CP n 1/ be obtained from t. n 1/ as explained above. Then:

(1)tn D tn 1.

(2)tn 1 is the Euler class associated to t. n/.

Proof. (1) The embedding is homotopic to the embedding 1 W Œx1; : : : ; xn 7! Œ0; x1; : : : ; xn . Thus it suffices to show 1 tn D tn 1. We have a bundle map2 W En 1 ! En which is compatible with the embeddings, i.e., n 2 D 1 n 1. We apply cohomology to this commutativity and obtain the desired result.

(19.1.2) Lemma. For singular homology and cohomology the Kronecker pairing relation h t1; ŒCP 1 i D 1 holds. The element t1 is by (19.1.1) also the Euler class of 2 and this is, by definition, the first Chern class.

Proof. 1 is a bundle over a point. We have the isomorphism ' W C ! E1, z 7!

that it sends tnk 1 t. n/ to tnkC1. Unraveling the definitions one shows that this claim is a consequence of the naturality of the cup product and the fact (19.1.1) that tn 1 is the Euler class of t. n/.

470 Chapter 19. Characteristic Classes

We denote by h ŒŒT the ring of homogeneous formal power series in T over the graded ring h . If T has the degree 2, then the power series in h ŒŒT of degree k consist of the series Pj aj T j with aj 2 hk 2j . If h is concentrated in degree zero, then this coincides with the polynomial ring h0ŒT .

Let t. 1/ 2 h2.E1; E10 / be a Thom class and t. n/ its restriction. Let t1 2 h2.CP 1/ and tn 2 h2.CP n/ be the corresponding elements. Since tn is the restriction of t1 let us write just t for all these elements. We have a surjective restriction homomorphism h .CP nC1/ ! h .CP n/. Thus the restrictions induce an isomorphism (see (17.1.6)and (17.1.7))

We extend the previous results by a formal trick to products X CP n. Let p W X CP n ! CP n be the projection. We set u D un D p .tn/.

(19.1.5) Proposition. Consider h .X CP n/ as a graded h .X/-algebra. Then h .X CP n/ Š h .X/Œu =.unC1/ and h .X CP 1/ Š h .X/ŒŒu .

Proof. The cohomology theory k . / D h .X / is additive and multiplicative, and the coefficient algebra is h .X/. The multiplicative structure in k . / is induced by the -product of h . / and the diagonal of X. The element u1 now plays the role of t1.

Let pi W .CP 1/n ! CP 1 be the projection onto the i-th factor, and set Ti D

This statement uses algebraic identities of the type h ŒŒx; y Š .h ŒŒx /ŒŒy for graded formal power series rings.

Problems

19.2 Projective Bundles

Let W E. / ! B be an n-dimensional complex vector bundle. (For the moment we work with bundles over spaces of the homotopy type of a CW-complex.) The group C acts fibrewise on E0. / by scalar multiplication. Let P . / be the orbit

each fibre P . b/ we have a bundle canonically isomorphic to n.

The construction of the projective bundle is compatible with bundle maps. LetW E. / ! C be a further bundle and ' W ! a bundle map over f W B ! C . These data yield an induced bundle map

We now assume that we are given a Thom class t1 2 h2.CP 1/ of the universal line bundle 1 over CP 1. A classifying map k W P . / ! CP 1 of the line bundle Q. / ! P . / provides us with the element

t D k .t1/ 2 h2.P . //:

We consider h .P . // in the standard manner as left h .B/-module, x y D p .x/ Y y.

(19.2.1) Example. Let D nC1 W EnC1 ! CP n. Then Q. / ! P . / is canon-

In particular p W h .B/ ! h .P . // is injective.

Proof. This is a consequence of the Leray–Hirsch theorem (17.8.4) and the computation (19.1.3).

(19.2.3) Corollary. There exist uniquely determined elements cj . / 2 h2j .B/ such that

and 1 D 1. We use that h .BU.1/n/ Š h ŒŒT1; : : : ; Tn , see (19.1.6). Let us recall the ring of formal graded power series h ŒŒc1; : : : ; cn . The indeterminate cj has degree 2j . The degree of a monomial in the cj is the sum of the degrees of

the factors

degree.c1k.1/c2k.2/ : : : / D 2k.1/ C 4k.2/ C :

A homogeneous power series of degree k is the formal sum of terms of the form

(19.3.1) Lemma. A classifying map ˇ W BU.n 1/ BU.1/ ! BU.n/ of the product n 1 1 is the projective bundle of n.

Proof. Let U.n 1/ U.1/ U.n/ be the subgroup of block diagonal matrices. We obtain a map

˛ W B.U.n 1/ U.1// D EU.n/=.U.n 1/ U.1//

DEU.n/ U.n/ .U.n/=U.n 1/ U.1//

!BU.n/:

A model for the universal vector bundle is n W EU.n/ U.n/ Cn ! BU.n/. The U.n/-matrix multiplication on Cn induces a U.n/-action on the corresponding

projective space P .Cn/. The projective bundle associated to the universal bundle n

is EU.n/ U.n/ P .Cn/ ! BU.n/. We now use the U.n/-isomorphism P .Cn/ Š U.n/=U.n 1/ U.1/. Hence ˛ is the projective bundle of n. We compose with

a canonical h-equivalence j W BU.n 1/ BU.1/ ! B.U.n 1/ U.1//. It remains to show that ˇ D ˛ ı j is a classifying map for n 1 1.

Let EU.n 1/ EU.1/ ! EU.n/ be a U.n 1/ U.1/-map. From it we obtain a bundle map

E. n 1/ E. 1/ D .EU.n 1/ EU.1// U.n 1/ U.1/ .Cn 1 C1/

! EU.n/ U.n 1/ U.1/ Cn

! EU.n/ U.n/ Cn D E. n/:

It is a bundle map over ˛ ı j .

(19.3.2) Theorem. Let W BU.1/n ! BU.n/ be a classifying map of the n-fold Cartesian product of the universal line bundle. Then the following holds: The induced map W h .BU.n// ! h .BU.1/n/ is injective. The image consists of the power series which are symmetric in the variables T1; : : : ; Tn. Let ci 2 h2i .BU.n// be the element such that .ci / is the i-th elementary symmetric polynomial in

T1; : : : ; Tn. Then

The elements c1; : : : ; cn are those which were obtained from the projective bundle associated to n by the methods of the previous section.

474 Chapter 19. Characteristic Classes

Proof. Let 2 Sn be a permutation and also the corresponding permutation of the factors of BU.1/n. Then is covered by a bundle automorphism of the n-fold product 1n D 1 1. Hence ı is another classifying map of 1n and therefore homotopic to . The permutation induces on h .BU.1/n/ D h ŒŒT1; : : : ; Tn the corresponding permutation of the Tj . Hence the image of is contained in the symmetric subring, since ı ' . Let prj W BU.1/n ! BU.1/ be the projection onto the j -th factor. We write .j / D prj . 1/ so that 1n D .1/ ˚ ˚ .n/ and Tj D c1. .j //. We have the relation (naturality)

ci . n/ D ci . n/ D ci . 1n/ D ci . .1/ ˚ ˚ .n//:

By the sum formula (19.2.6) this equals

c1. .1//ci 1. .2/ ˚ ˚ .n// C ci . .2/ ˚ ˚ .n//:

This is used to show by induction that this element is the i-th elementary symmetric polynomial i in the variables Tj . We now use the algebraic fact that the symmetric part of h ŒŒT1; : : : ; Tn equals the ring of graded power series h ŒŒ 1; : : : ; n in the elementary symmetric polynomials i . This shows that the image of is as claimed.

It remains to show that is injective. From (19.3.1), (19.2.2) and (19.1.5) we obtain an injective map

ˇ W h .BU.n// ! h .BU.n 1/ BU.1// Š h .BU.n 1//ŒŒ .n/ :

This fact yields, by induction on n, the claimed injectivity.

Elements of h .BU.n// are called universal h . /-valued characteristic classes for n-dimensional complex vector bundles. Given c 2 h .BU.n// and a classifying map f W B ! BU.n/ of the bundle over B we set c. / D f .c/ and call c. / a characteristic class. With this definition, the naturality ' c. / D c. / holds for

each bundle map ' W ! . Theorem (19.3.2) shows that it suffices to work with ci . The corresponding characteristic class ci . / is called the i -th Chern class of

with respect to the chosen Thom class t1. It is sometimes useful to consider the total Chern class c. / D 1 C c1. / C c2. / C 2 h .B/ of a bundle over B; the sum formula then reads c. ˚ / D c. / Y c. /.

The preceding results can in particular be applied to integral singular cohomology. Complex vector bundles have a canonical orientation and a canonical Thom class. There are two choices for the element t1, they differ by a sign.

complex structure.

Chern classes are stable characteristic classes, i.e., cj . / D cj . ˚ "/ if " denotes the trivial 1-dimensional bundle; this follows from the sum formula (19.2.6) and ci ."/ D 0 for i > 0. This fact suggests that we pass to the limit n ! 1.

(19.3.3) Example. The complex tangent bundle T CP n of the complex manifold CP n satisfies T CP n ˚ " Š .n C 1/ nC1, see (15.6.6). Therefore the total Chern class of this bundle is .1 C c1. nC1//nC1. Þ

Let ! W BU.n/ ! BU.n C 1/ be a classifying map for n ˚ ". Then ! ci D ci for i n and ! cnC1 D 0. Let U D colimn U.n/, with respect to the inclusions

be the stable unitary group. The classifying space BU is called the classifying space for stable complex vector bundles. We think of this space as a homotopy colimit (telescope) over the maps BU.n/ ! BU.n C 1/. By passage to the limit we obtain (since the lim1-term vanishes by (17.1.7)):

(19.3.4) Theorem. h .BU/ Š lim h BU.n/ Š h ŒŒc1; c2; : : : .

(19.3.5) Example. Let m;n W BU.m/ BU.n/ ! BU.m C n/ be a classifying

The map m;n is continuous in the sense that the effect on a formal power series in the variables c1; : : : ; cmCn is obtained by inserting for ck the value (2).

For the proof of (1) one can use the theory h .BU.m/ / and proceed as for (19.3.2). In the case of integral singular cohomology one has the Künneth isomorphism and m;n becomes the homomorphism of algebras

P

determined by ck !7 iCj Dk ci ˝cj . Since formal power series are not compatible with tensor products, one has to use a suitably completed tensor product for general theories if one wants a similar statement.

The maps m;n combine in the colimit to a map W BU BU ! BU. It is associative and commutative up to homotopy and there exists a unit element. A classifying map for the “inverse bundle” yields a homotopy inverse for . With these structures BU becomes a group object in h-TOP. The precise definition and the detailed verification of these topological results are not entirely trivial. We are content with the analogous algebraic result that we have a homomorphism

476 Chapter 19. Characteristic Classes

h .BU/ ! h .BU BU/ determined by the sum formula and continuity. The inverse W h .BU/ ! h .BU/ is determined by the formal relation

.1 C c1 C c2 C /.1 C .c1/ C .c2/ C / D 1:

It allows for an inductive computation c1 D c1; c2 D c12 c2 etc. We also have the homomorphism of algebras (3) for BU. It will be important for the discussion of the Hopf algebra structure. Þ

(19.3.6) Example. We know already K.Z; 2/ D CP 1 D BS1 D BU.1/ '

BGL1.C/. A numerable complex line bundle over X is determined by its classi-

fying map in

H 2.XI Z/ D ŒX; CP 1 D ŒX; BU.1/ :

The corresponding element in c1. / 2 H 2.XI Z/ is the first Chern class of . Þ

(19.3.7) Proposition. The relation c1. ˝ / D c1. /Cc1. / holds for line bundlesand .

Proof. We begin with the universal situation. We know that H 2.CP 1I Z/ Š Z is generated by the first Chern class c of the universal bundle D 1. Let k W CP 1 CP 1 ! CP 1 be the classifying map of ˝O . Let prj W CP 1 CP 1 ! CP 1 be the projection onto the j -th factor. Then H 2.CP 1 CP 1I Z/ has the Z-basis

T1; T2 with Tj D prj .c/. There exists a relation k c1. / D a1e1 C a2e2 with certain ai 2 Z. Let i1 W CP 1 ! CP 1 CP 1; x 7!.x; x0/ for fixed x0. Then

i1 e1 D c1. /, since pr1 i1 D id, and i1 e2 D 0 holds, since pr2 i1 is constant. We compute

a1c1. / D i1 k c1. / D c1.i1 k / D c1.i1 .pr1 ˝ pr2 // D c1. /;

since i1 pr1 D and i1 pr2 is the trivial bundle. Hence a1 D 1, and similarly we see a2 D 1.

We continue with the proof. Let k ; k W B ! CP 1 be classifying maps ofand . Then c1. / D k c1. / and similarly for . With the diagonal d the equalities ˝ D d . ˝O / D d .k k / . ˝O / hold. This yields

(19.3.8) Proposition. Let W E. / ! B be an n-dimensional vector bundle and p W P . / ! B the associated projective bundle. The induced bundle splits p . / D 1 ˚ 0 into the canonical line bundle 1 over the projective bundle and another .n 1/-dimensional bundle 0.

Proof. Think of as associated bundle E U.n/ Cn ! B. Let H be the subgroup U.n 1/ U.1/ of U.n/. We obtain the pullback

We now iterate this process: We consider over P . / the projective bundle P . 0/, et cetera. Finally we arrive at a map f . / W F . / ! B with the properties:

(1)f . / splits into a sum of line bundles.

(2)The induced map f . / W h .B/ ! h .F . // is injective.

Assertion (2) is a consequence of (19.2.2).

A model for f . / is the flag bundle. The flag space F .V / of the n-dimensional vector space V consists of the sequences (D flags)

f0g D V0 V1 Vn D V

of subspaces Vi of dimension i. Let V carry a Hermitian form. Each flag has an orthonormal basis b1; : : : ; bn such that Vi is spanned by b1; : : : ; bi . The basis vectors bi are determined by the flag up to scalars of norm 1. The group U.n/ acts transitively on the set of flags. The isotropy group of the standard flag is the

maximal torus T.n/ of diagonal matrices. Hence we can view F .V / as U.n/=T.n/. The flag bundle associated to E U.n/ Cn is then

f . / W F . / D E U.n/ U.n/=T.n/ Š E=T.n/ ! B:

We can apply this construction to a finite number of bundles.

(19.3.9) Theorem (Splitting Principle). Let 1; : : : ; k be complex vector bundles over B. Then there exists a map f W X ! B such that f W h .B/ ! h .X/ is injective and f . j / is for each j a sum of line bundles.

We now prove (19.0.1). Consider the exact cohomology sequence of the pair

.E. n/; E0. n//. We can use E0. n/ as a model for BU.n 1/. The projection E. n/ ! BU.n/ is an h-equivalence. Our computation of h .BU.n// shows that