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

.1 /C .1 ˝ a1/ˇi C .1 ˝ a2/ˇ2 C

i

DP .a ˝ 1/b.n/ P .1 ˝ a /b.n/ D P .a ˝ a /b.n/b.n/

DP .a ˝ b / P b b.n/:

Now we compare coefficients and obtain P a b.n/ D P b b.n/. The sum is

finite in each degree. We pass to the stable values b and compare again coefficients.

Let Hom.RŒa ; R/ be the graded dual of RŒa . We can view this as the module of R-linear maps RŒa ! R which have non-zero value only at a finite number of monomials. Let a be the dual of a , i.e., a .a / D ı . The Hopf algebra structure of RŒa induces a Hopf algebra structure on Hom.RŒa ; R/. The basic algebraic result of this section is that the dual Hopf algebra is isomorphic to the original Hopf algebra.

Proof. The dual basis element of a is mapped to b . Therefore ˛ is an R-linear isomorphism. It remains to show that ˛ is compatible with the multiplication and the comultiplication.

.f g/.a / D .f ˝g/. a / D .f ˝g/ P; a a ˝a D P; a f .a /g.a /:

Now we use the equality (19.7.2).

The definition of the comultiplication in Hom.RŒa ; R/ gives for the element a which is dual to a the relation

P

erators of the algebras Hom.RŒa ; R/ and RŒb have the same coproduct. Since we know already that ˛ is a homomorphism of algebras, we conclude that ˛ pre-

The Hopf algebras which we have discussed have other interesting applications, e.g., to the representation theory of symmetric groups, see [113].

(19.7.4) Remark. If we define ˛ in (19.7.3) on the R-module of all R-linear maps, then the image is the algebra RŒŒb of formal power series. The homogeneous components of degree n in RŒb and RŒŒb coincide. Þ

(19.7.5) Remark. The duality isomorphism (19.7.3) can be converted into a sym-

with the homogeneous part RŒb n of RŒb .

A graded group-like element K of RŒb is defined to be a sequence of polynomials .Kn.b1; : : : ; bn/ j n 2 N0/ with K0 D 1 and Kn 2 RŒb n of degree n such

Since is a homomorphism of algebras, the relation

Kn.b1; : : : ; bn/ D Kn. b1; : : : ; bn/

(19.7.6) Proposition. The sequence .'n/ is an R-algebra homomorphism if and only if the sequence .Kn/ with Kn D ˛.'n/ is a graded group-like element.

'n.xy/ D h Kn; xy i D h Kn; x ˝ y i D h Ki ; x ih Kj ; y i D 'i .x/'j .y/:

Hence .'n/ is an algebra homomorphism.

(19.7.7) Remark. The algebra homomorphisms ' W RŒa ! R correspond to fam-

We now return to classifying spaces and apply the duality theorem (19.7.3). We have the Kronecker pairing W H .BOI F2/˝H .BOI F2/ ! F2; x ˝y 7! hx; y i and the duality pairing (19.7.5) ˛Q W F2Œw ˝ F2Œu ! F2, now with variables w; u in place of a; b. We also have the isomorphism W F2Œw Š H .BOI F2/ from the determination of the Stiefel–Whitney classes. We obtain an isomorphism of Hopf algebras W F2Œu ! H .BOI F2/ determined via algebraic duality by the compatibility relation h x; y i D ˛Q .x ˝ y/. The generators of a polynomial algebra are not uniquely determined. Our algebraic considerations produce from the universal Stiefel–Whitney classes as canonical generators of H .BOI F2/ canonical generators of H .BOI F2/ via .

In a similar manner we obtain isomorphisms H .BUI Z/ Š ZŒd1; d2; : : : (variables c; d ) and H .BSOI R/ Š RŒq1; q2; : : : (variables p; q).

2. The assignment RŒa ˝ RŒb ! R, a ˝ b 7! .a / .b / D ı is a symmetric pairing.

P

(The formal element U D a b could be called a symmetric copairing.)

19.8 Characteristic Numbers

Let W X ! BO.n/ be a classifying map of an n-dimensional bundle. It induces a ring homomorphism H .BO.n/I F2/ ! H .XI F2/. We can also pass to the stable classifying map X ! BO and obtain W H .BOI F2/ ! H .XI F2/. This homomorphism codifies the information which is obtainable from the Stiefel– Whitney classes. We use the isomorphism F2Œw Š H .BOI F2/ and the duality theorem (19.7.3). We use a slightly more general form. Let S be a graded R- algebra; the grading should correspond to the grading of RŒa , there are no signs. We obtain a graded algebra HomR.RŒa ; S / where the component of degree k consists of the homomorphisms of degree k. The product in this algebra is defined by convolution. Then we have:

(19.8.1) Theorem. There exists a canonical isomorphism

˛ W HomR.RŒa ; S / Š S ŒŒb

of graded R-algebras. Here S ŒŒb D S ŒŒb1; b2; : : : is the algebra of graded

we have v. / D 1 C w1. /u1 C w1. / u2 C . These properties characterize the assignment 7!v. /.

We can apply a similar process to oriented or complex bundles. In the case of a complex oriented theory h . / we obtain series v. / 2 h .X/ŒŒd which are

492 Chapter 19. Characteristic Classes

natural, multiplicative and assign to a complex line bundle the series v. / D

1 C c1. /d1 C c1. /2d2 C .

Interesting applications arise if we apply the process to the tangent bundle of a manifold. Let us consider oriented closed n-manifolds M with classifying mapM W M ! BSO of the stable oriented tangent bundle. We evaluate the homomorphism M on the fundamental class ŒM

By the Kronecker pairing duality Hn.BSOI R/ Š HomR.H n.BSOI R/; R/ this homomorphism corresponds to an element in Hn.BSOI R/, and this element isM ŒM , the image of the fundamental class ŒM 2 Hn.M I R/ under . M / , by the naturality h M .p/; ŒM i D h p; . M / ŒM i of the pairing. Under the isomorphismW RŒq1; q2; : : : Š H .BSOI R/ the element M ŒM corresponds to an element that we denote SO.M / 2 RŒq1; q2; : : : . From the definitions we obtain:

(19.8.2) Proposition. Let M denote the oriented tangent bundle of M . Then

If p 2 H n.M / is a polynomial of degree n in the Pontrjagin classes, then the element (number) pŒM is called the corresponding Pontrjagin number. In a similar manner one defines a Stiefel–Whitney number by evaluating a polynomial

in the Stiefel–Whitney classes on the fundamental class. A closed n-manifold M has an associated element O.M / 2 F2Œu1; u2; : : : Š Hn.BOI F2/, again the image of the fundamental class under the map induced by the stable classifying map M W M ! BO of the tangent bundle.

(19.8.3) Example. Let us consider M D CP 2k . The stable tangent bundle is2kC1 where is is the canonical complex line bundle, now considered as oriented bundle; see (15.6.6). By the multiplicativity of the v-classes, we have for the tangent bundle 2k of CP 2k the relation

v. 2k / D v. /2kC1 D .1 C p1. /q1 C p1. /2q2 C /2kC1

D .1 C c2q1 C c4q2 C /2kC1

where as usual H .CP 2k I R/ Š RŒc =.c2kC1/. Note that p1. / D c2, by (19.5.6). The evaluation on the fundamental class yields the coefficient of c2k in this series, since h c2k ; ŒCP 2k i D 1 (see (18.7.2)). Modulo decomposable elements in the

When we pass to rational coefficients R D Q we can divide by 2k C 1 and obtain:

(19.8.4) Proposition. The elements SO.CP 2k /; k 2 N are polynomial generators of QŒq .

In bordism theory it will be shown that the signature of an oriented 4k-manifold only depends on its image in H .BSOI Q/ Š QŒq . And from the multiplicativity of the signature it then follows that there exists an algebra homomorphism s W H .BSOI Q/ ! Q such that the signature .M / is obtained as the image of this homomorphism .M / D s. M / ŒM /. We know that the generators CP 2k have signature 1; see 18.7.2. The ring homomorphism s is determined by the values i D s.qi / 2 Q. Via the duality QŒp Š Hom.QŒq ; Q/ the homomorphism s corresponds to a group-like element .Ln.p1; : : : ; pn/ j n 2 N/ where Ln is a polynomial in the Pontrjagin classes of degree 4n such that the evaluation on the

property that the coefficient of c2n in H.c/2nC1 is 1. Hirzebruch [81, p. 14] has found this power series

From these data we obtain the polynomials Ln if we insert in the universal polynomials U Œn .p1; : : : ; pnI q1; : : : ; qn/ for qj the value j . We have already listed the polynomials U Œ1 , U Œ2 , U Œ3 , U Œ4 . The result is

494 Chapter 19. Characteristic Classes

The polynomials Ln are called the Hirzebruch L-polynomials.

Problems

1.Show that SO.M N / D SO.M / SO.N /.

2.Let M be the oriented boundary of a compact manifold. Then SO.M / D 0. (See the

bordism invariance of the degree.)

3.Show that O.RP 2n/ D u2n modulo decomposable elements. Therefore these elements can serve as polynomial generators of H .BOI F2/ Š F2Œu in even dimensions.

4.The convolution product of the homomorphisms defined at the beginning of the section

satisfies D ˚ .

5. Determine SO.CP 2/ and SO.CP 4/.

496 Chapter 20. Homology and Homotopy

We define natural homomorphisms, called Hurewicz homomorphisms,

commute (compatibility with exact sequences). For this purpose we use the definition n.X; / D ŒS.n/; X 0 and n.X; A; / D Œ.D.n/; S.n 1//; .X; A/ 0 of the homotopy groups (see (6.1.4)). We choose generators zn 2 Hn.S.n// and zQn 2 Hn.D.n/; S.n 1// such that @zQn D zn 1 and q .zQn/ D zn, where q W D.n/ ! D.n/=S.n 1/ D S.n/ is the quotient map. If we fix z1, then the other generators are determined inductively by these conditions. We define h W n.X; A; / ! Hn.X; A/ by Œf 7!f .zQn/ and h W n.X; / ! Hn.X/ by

Œf 7!f .zn/. With our choice of generators the diagram above is then commutative. From (10.4.4) and the analogous result for the relative homotopy groups we see that the maps h are homomorphisms. The singular simplex 1 ! I=@I D S.1/,

.t0; t1/ 7!t1 represents a generator z1 2 H1.S.1//. If we use this generator, then

via transport. We denote by n#.X; A; / the quotient of n.X; A; / by the normal subgroup generated by all elements of the form x x ˛ (additive notation in n). Recall from (6.2.6) that 2# is abelian. Representative elements in n which differ by transport are freely homotopic, i.e., homotopic disregarding the base point. Therefore the Hurewicz homomorphism induces a homomorphism

a1 to a2 induces an isomorphism of the n-groups, and this isomorphism is independent of the choice of the path. We can use this remark: An unpointed map

and then n D n. Since weak homotopy equivalences induce isomorphisms in homotopy and homology, we only need to prove the theorem for CW-complexes X. We can assume that X has a single 0-cell and no i-cells for 1 i n 1, see

(8.6.2). The inclusion XnC1 X induces isomorphisms n#.X/ Š n#.XnC1/ and Hn.X/ Š Hn.XnC1/. Since the Hurewicz homomorphisms h form a natural transformation of functors, it suffices to prove the theorem for .n C 1/-dimensional complexes. In this case X is h-equivalent to the mapping cone of a map of the form

with exact rows. The exactness of the top row is a consequence of the homotopy

Proof. (20.1.1) says, in different wording, that h W j .X/ Š Hj .X/ for the smallest j such that k .X/ D 0 for 1 k < j .

(20.1.3) Theorem. Let .X; A/ be a pair of simply connected CW-complexes. Sup-

pose Hi .X; A/ D 0 for i < n, n 2. Then i .X; A/ D 0 for i < n and h W n.X; A/ ! Hn.X; A/ is an isomorphism.

Proof. Induction over n 2. We use a consequence of the homotopy excision theorem: Let A be simply connected and i .X; A; / D 0 for 0 < i < n. Then

conclude i .X=A/ D 0 for i < n.

Let n D 2. Since X and A are simply connected, 1.X; A; / D 0 and the diagram

shows that h is an isomorphism.