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

The element cn lies in the kernel of i . It therefore has a unique pre-image t. n/ 2 h2n.E. n/; E0. n//. For an n-dimensional numerable bundle W E. / ! B we define t. / 2 h2n.E. /E0. // to be the element t. n/ with a classifying

map W ! n. Then the elements t. / are natural with respect to bundle maps.

From the relation m;n.cmCn/ D cm0 cn00 we conclude (by naturality) t. m/ t. n/ D t. m n/ and then t. / t. / D t. / for arbitrary numerable bundles.

The element t. 1/ 2 h2.E. 1/; E0. 1// corresponds to the chosen element t1 2 h2.CP 1/. The restriction of t. n/ to 1 1 yields t. 1/ t. 1/. This

is a Thom class, since products of Thom classes are Thom classes. This shows that t. n/ is a Thom class.

19.4 Stiefel–Whitney Classes

The theory of Chern classes has a parallel theory for real vector bundles. Suppose given an element t1 2 h1.RP 1; / h1.RP 1/ such that its restriction to t1 2 h1.RP 1; / is a generator of this h0-module. Then there exists an isomorphism

h ŒT =.T nC1/ Š h .RP n/

which sends T to the restriction tn of t1. This is then used to derive isomorphisms

h .X RP n/ Š h .X/Œu =.unC1/; h .X RP 1/ Š h .X/ŒŒu ;

h ..RP 1/n/ Š h ŒŒT1; : : : ; Tn :

The projective bundle P . / of a real vector bundle over B yields a free h .B/- module h .P . // with basis 1; t ; : : : ; tn 1, and there exists a relation

P1 . 1/j wj . /tn j D 0

j D0

with elements wj . / 2 hj .B/ which satisfy the sum formula wr . ˚ / D PiCj Dr wi . /wj . /

where w0. / D 1 and wj . / D 0 for j > dim . These elements are natural with respect to bundle maps, hence characteristic classes. We obtain an injective

map W h .BO.n// ! h .BO.1/n/. The classes w1; : : : ; wn which belong to the universal n-dimensional bundle over BO.n/ yield

h .BO.n// Š h ŒŒw1; : : : ; wn :

The image of wj under is the j -th elementary symmetric polynomial in the

T1; : : : ; Tn. We pass to the limit n ! 1 and obtain h BO as a ring of graded power series in w1; w2; : : : with wj of degree j . The wj are the universal Stiefel–

Whitney classes. The Stiefel–Whitney classes are natural, stable, and satisfy the sum formula. There holds a splitting principle for real bundles. We write w. / D 1 C w1. / C w2. / C for the total Stiefel–Whitney class of ; then the sum formula reads w. ˚ / D w. / Y w. /.

The existence of the universal element has the consequence that the unit element 1 2 h0 is of order 2, so that the cohomology groups consist of Z=2-vector spaces and signs can be ignored. This result is due to the fact that multiplication by 1 in the fibres of vector bundles is a bundle map. If we apply this to the universal onedimensional bundle, then we see that this bundle map preserves the Thom class t1. On the other hand, if we restrict to t1 2 h1.RP 1; /, this bundle map is of degree 1 and changes the sign of t1. Since t1 corresponds under suspension to a unit of h0, we conclude that 1 D 1 2 h0.

One shows as in the complex case that the theory is R-oriented. One can apply these results to singular cohomology with coefficients in Z=2. There is a unique

choice for the universal element t1 2 H 1.RP 1I Z=2/. The resulting characteristic classes are the classical Stiefel–Whitney classes.

(19.4.1) Example. The tangent bundle of RP n satisfies ˚ " Š .n C 1/ with the canonical line bundle . The total Stiefel–Whitney class of is therefore

.1 C w/nC1 2 H .RP n/ Š Z=2Œw =.w/nC1. Suppose RP n has an immersion into RnCk . Then has an inverse bundle of dimension k, the normal bundle of this immersion. Suppose n D 2r . Properties of binomial numbers modulo 2 show that w. / D 1 C w C wn. If is inverse to , then w. /w. / D 1, and this implies in

19.5 Pontrjagin Classes

We now discuss characteristic classes for oriented bundles. Suppose W E. / ! B is an oriented n-dimensional real vector bundle. The orientation determines a Thom

definition implies e.m/ e.n/ D . 1/mne.mCn/ and has the following consequences:

480 Chapter 19. Characteristic Classes

(19.5.1) Lemma. Let and be oriented bundles and give the sum orientation. Then t. / D . 1/j jj jt. / t. / and e. ˚ / D . 1/j jj je. /e. /. Here

j j D dim .

Let W E ! B be a real vector bundle. It has a complexification C D ˝R C. We take ˚ with complex structure J.x; y/ D . y; x/ on each fibres as a model for C.

Let be a complex n-dimensional bundle and R the underlying 2n-dimensional real bundle. If v1; : : : ; vn is a basis of a fibre, then v1; iv1; : : : ; vn; ivn is a basis of the fibre of R, and it defines the canonical orientation.

(19.5.2) Lemma. Let be an oriented n-dimensional real bundle. Then . C/R in our model for C above is isomorphic to . 1/n.n 1/=2 ˚ as an oriented bundle. The factor indicates the change of orientation, and ˚ carries the sum orientation.

(19.5.3) Proposition. Let W E ! B be a complex n-dimensional bundle. Consider it as a real bundle with orientation and canonical Thom class induced by the complex structure. Then cn. / D e. / 2 H 2n.BI Z/.

Proof. This holds for 1-dimensional bundles by definition of c1. The general case

and call this characteristic class the i -th Pontrjagin class of . The bundle C is

general for conjugate bundles. Hence the odd Chern classes of C are elements of order 2. This is a reason why we ignore them for the moment. The Pontrjagin classes are by definition compatible with bundle maps (naturality) and they do not change by the addition of a trivial bundle (stability). The next proposition justifies the choice of signs in the definition of the pj .

(19.5.4) Proposition. Let be an oriented 2k-dimensional real bundle. Then pk . / D e. /2.

One can remove elements of order 2 if one uses the coefficient ring R D ZΠ12 of rational numbers with 2-power denominator (or, more generally, assumes that 12 2 R). The next theorem shows the universal nature of the Pontrjagin classes.

(19.5.5) Theorem. Let pj denote the Pontrjagin classes of the universal bundle and e its Euler class. Then

H .BSO.2nC1/I R/ Š RŒp1; : : : pn ; H .BSO.2n/I R/ Š RŒp1; : : : ; pn 1; e :

Proof. Induction over n. Let n W ESO.n/ SO.n/ Rn ! BSO.n/ be the universal oriented n-bundle and p W BSO.n 1/ ! BSO.n/ the classifying map of n 1 ˚ ".

As model for p we take the sphere bundle of n. Then we have a Gysin sequence at our disposal. Write Bn D BSO.n/ for short.

Suppose n is even. Then, by induction, H .Bn 1/ is generated by the Pontrjagin classes, and p is surjective since the classes are stable. Hence the Gysin sequence decomposes into short exact sequences. Let Hn denote the algebra which is claimed to be isomorphic to H .Bn/. And let n W Hn ! H .Bn/ be the homomorphism which sends the formal elements pj , e onto the cohomology classes with the same name. We obtain a commutative diagram

By induction, n 1 is an isomorphism. By a second induction over i the left arrow is an isomorphism. Now we apply the Five Lemma. In order to start the induction, we note that by the Gysin sequence n W Hni ! H i .Bn/ is an isomorphism for i < n.

Suppose n D 2m C 1. The Euler class is zero, since we use the coefficient ring R. Hence the Gysin sequence yields a short exact sequence

0 H j .B / p H j .B / H j 2m.B / 0:

! n ! n 1 ! n !

Therefore H .Bn/ is a subring of H .Bn 1/ via p . The image of p contains the subring P generated by p1; : : : ; pm. We use pm D e2. The induction hypothesis

implies

rank H j .Bn 1/ D rank P j C rank P j 2m:

The Gysin sequence yields

rank H j .Bn 1/ D rank H j .Bn/ C rank H j 2m.Bn/:

The equality rank P j D rank H j .Bn/ is a consequence. If p H j .Bn/ 6DP j then the image would contain elements of the form x C ey, x 2 P j , y 2 P j 2m. Such an element would be linearly independent of the basis elements of P . This contradicts the equality of ranks.

An isomorphism from R ˚ R with the complex structure .x; y/ 7!. y; x/ is

(19.5.7) Example. Let D T CP n denote the complex tangent bundle of CP n. Then R is the real tangent bundle. In order to determine the Pontrjagin classes we

Problems

1. The Pontrjagin classes are stable. Under the hypothesis of (19.5.5) we obtain in the limit

H .BSOI R/ Š RŒp1; p2; : : : . The sum formula pk . ˚ / D PiCj Dk pi . /pj . / holds (p0 D 1).

19.6 Hopf Algebras

We fix a commutative ring R and work in the category R-MOD of left R-modules. The tensor product of R-modules M and N is denoted by M ˝ N . The natural isomorphism W M ˝ N ! N ˝ M , m ˝ n 7!n ˝ m expresses the commutativity of the tensor product. We have canonical isomorphisms l W R˝M ! M , ˝m 7! m and r W M ˝ R ! M , m ˝ 7! m. Co-homology will have coefficients in

R, if nothing else is specified.

An algebra .A; m; e/ in R-MOD consists of an R-module A and linear maps m W A ˝ A ! A (multiplication), e W R ! A (unit) such that m.e ˝ 1/ D l, m.1 ˝ e/ D r. If m.m ˝ 1/ D m.1 ˝ m/ holds, then the algebra is associative, and if m D m holds, the algebra is commutative. Usually we write m.a ˝ b/ D a b D ab. We use similar definitions in the category of Z-graded R-modules (with its tensor product and interchange map).

(19.6.1) Example. Let X be a topological space. Then the graded R-module H .X/ becomes a (graded) associative and commutative algebra with multiplica-

tion

m W H .X/ ˝ H .X/ ! H .X X/ ! H .X/;

where the first map is the -product and the second map is induced by the diagonal d W X ! X X. The unit is the map induced by the projection X ! P onto a point P . Þ

A coalgebra .C; ; "/ in R-MOD consists of an R-module C and linear mapsW C ! C ˝ C (comultiplication), " W C ! R (counit) such that ." ˝ 1/ D l 1,

.1 ˝ "/ D r 1. If . ˝ 1/ D .1 ˝ / holds, the coalgebra is coassociative, and if D holds, the coalgebra is cocommutative.

(19.6.2) Example. Let X be a topological space. Suppose H .X/ is a free R-module. The graded R-module H .X/ becomes a (graded) coassociative and cocommutative coalgebra with comultiplication

W H .X/ ! H .X X/ Š H .X/ ˝ H .X/

where the first map is induced by the diagonal d and the isomorphism is the Künneth isomorphism. The counit is induced by X ! P . Þ

A homomorphism of algebras ' W .A; m; e/ ! .A0; m0; e0/ is a linear map

. ˝ / D 0 and "0 D ". A continuous map f W X ! Y induces a homomorphism f W H .Y / ! H .X/ of the algebras (19.6.1) and a homomorphism f W H .X/ ! H .Y / of the coalgebras (19.6.2).

The tensor product of algebras .Ai ; mi ; ei / is the algebra .A; m; e/ with A D

A1 ˝A2 and m D .m1 ˝m2/.1˝ ˝1/ and e D e1 ˝e2 W R Š R˝R ! A1 ˝A2. The multiplication m is determined by .a1 ˝ a2/.b1 ˝ b2/ D a1b1 ˝ a2b2 (with the appropriate signs in the case of graded algebras). The tensor product of coalgebras .Ci ; i ; "i / is the coalgebra .C; ; "/ with C D C1 ˝ C2, comultiplication

D .1 ˝ ˝ 1/. 1 ˝ 2/ and counit " D "1"2 W C1 ˝ C2 ! R ˝ R Š R.

Let .C; ; "/ be a coalgebra. Let C D Hom.C; R/ denote the dual module. The data

and " define the dual algebra of the coalgebra. (The e

W

R

Š

R

!

C

.C

;

m;

e/

first map is the tautological homomorphism. It is an isomorphism if C is a finitely generated, projective R-module.)

Let .A; m; e/ be an algebra with A a finitely generated, projective R-module. The data

484 Chapter 19. Characteristic Classes

(19.6.3) Example. Let W H .X/ ! Hom.H .X/; R/ be the map in the universal coefficient sequence. Then is an isomorphism of the algebra (19.6.1) onto the dual algebra of the coalgebra (19.6.2). Þ

(19.6.4) Proposition. Let C be a coalgebra and A an algebra. Then Hom.C; A/ carries the structure of an algebra with product ˛ ˇ D m.˛ ˝ ˇ/ , for ˛; ˇ 2 Hom.C; A/, and unit e". The product is called convolution.

Proof. The map .˛; ˇ/ 7!˛ ˝ ˇ is bilinear by construction. The (co-)associativity of m and is used to verify that is associative. The unit and counit axioms yield

˛.e"/ D m.˛ ˝ e"/ D m.1 ˝ e/.˛ ˝ 1/.1 ˝ "/ D ˛:

Hence e" is a right unit.

A bialgebra .H; m; e; ; "/ is an algebra .H; m; e/ and a coalgebra .H; ; "/ such that and " are homomorphisms of algebras. (Here H ˝ H carries the tensor product structure of algebras.) The equality m D .m ˝ m/.1 ˝ ˝ 1/. ˝ / expresses the fact that is compatible with multiplication. The same equality says that m is compatible with comultiplication. This and a similar interpretation of the identities id D "e, e D .e ˝ e/ , m." ˝ "/ D "m is used to show that a bialgebra can, equivalently, be defined by requiring that m and e are homomorphisms of coalgebras. A homomorphism of bialgebras is an R-linear map which is at the

same time a homomorphism of the underlying algebras and coalgebras.

An antipode for a bialgebra H is an s 2 Hom.H; H / such that s is a two-

sided inverse of id.H / 2 Hom.H; H / in the convolution algebra. A bialgebra with antipode is called Hopf algebra.

(19.6.5) Example. Let X be an H -space with multiplication W X X ! X and neutral element x. Then

m W H .X/ ˝ H .X/ ! H .X X/ ! H .X/

is an algebra structure on H .X/ with unit induced by fxg X. Suppose H .X/ is a free R-module. Then the algebra structure m and the coalgebra structure (19.6.2) define on H .X/ the structure of a bialgebra. An inverse for the multiplication induces an antipode.

Suppose H .X/ is finitely generated and free in each dimension. Then

H .X/ H .X X/ H .X/ H .X/

W ! Š ˝

is a coalgebra structure and together with the algebra structure (19.6.1) we obtain a bialgebra. Again an inverse for induces an antipode. The duality isomorphism H .X/ ! Hom.H .X/; R/ is an isomorphism of the bialgebra onto the dual bialgebra of H .X/.

This situation was studied by Heinz Hopf [91]. The letter for the comultiplication (and even the term “diagonal”) has its origin in this topological context. For background on Hopf algebras see [1], [182], [142], [138]. Þ (19.6.6) Example. The space CP 1 is an H -space with multiplication the classifying map of the tensor product of the universal line bundle. The algebra structure is H .CP 1I Z/ D ZŒc with c the universal Chern class c1. Let ŒCP i 2

basis xi D ŒCP ; i 2 N0; by dualization of the cohomological coalgebra structure we obtain the multiplicative structure xi xj D .i; j /xiCj with .i; j / D

.i C j /Š=.iŠj Š/. Geometrically this means that the map CP i CP j ! CP iCj ,

We generalize the Hom-duality of Hopf algebras and define pairings. Let A and B be Hopf algebras. A pairing of Hopf algebras is a bilinear map A B ! R,

.a; b/ 7! ha; b i with the properties: For x; y 2 A and u; v 2 B

The bilinear form h ; i on A B induces a bilinear form on A ˝ A B ˝ B by h x ˝ y; u ˝ v i D . 1/jyjjujh x; uih y; v i. This is used in the first two axioms.

A pairing is called a duality between A; B, if h x; ui D 0 for all u 2 B implies x D 0, and h x; ui D 0 for all x 2 A implies u D 0. An example of a pairing is the

Kronecker pairing H .X/ H .X/ ! R in the case of an H -space X.

An element x of a bialgebra H is called primitive, if .x/ D x ˝ 1 C 1 ˝ x. Let P .H / H be the R-module of the primitive elements of H . The bracket

.x; y/ 7!Œx; y D xy yx defines the structure of a Lie algebra on P .H /. The inclusion P .H / H yields, by the universal property of the universal enveloping algebra, a homomorphism W U.P .H // ! H . For cocommutative Hopf algebras over a field of characteristic zero with an additional technical condition, is an isomorphism [1, p. 110].

(19.6.7) Example. A coalgebra structure on the algebra of formal powers series RŒŒx is, by definition, a (continuous) homomorphism W RŒŒx ! RŒŒx1; x2 with . ˝ 1/ D .1 ˝ / and ".x/ D 0. Here RŒŒx1; x2 is interpreted as a completed tensor product RŒŒx1 ˝O RŒŒx2 . Then is given by the power series.x/ D F .x1; x2/ with the properties

486 Chapter 19. Characteristic Classes

Problems

1. The Group algebra. Let G be a group and RG the group algebra. The R-module RG is the free R-module on the set G, and the multiplication RG ˝ RG Š R.G G/ ! RG is the linear extension of the group multiplication. This algebra becomes a Hopf algebra, if we define the comultiplication by .g/ D g ˝ g for g 2 G, the counit by ".g/ D 1, and the antipode by s.g/ D g 1.

Let G be a finite group and O.G/ the R-algebra of all maps G ! R with pointwise

An element g in a Hopf algebra H is called group-like if .g/ D g ˝ g and ".g/ D 1. The set of group-like elements in H is a group under multiplication. The inverse of g is s.g/.

2.Let D be a Hopf algebra and A a commutative algebra. The convolution product induces on the set A Hom.D; A/ of algebra homomorphisms D ! A the structure of a group.

3.Let H be a Hopf algebra with antipode s. Then s is an anti-homomorphism of algebras and coalgebras, i.e., s.xy/ D s.y/s.x/, se D e, "s D s, .s ˝ s/ D s. If H is commutative or cocommutative, then s2 D id.

4.Let H1 and H2 be Hopf algebras and ˛ W H1 ! H2 a homomorphism of bialgebras. Then ˛ commutes with the antipodes.

5.Let R be a field of characteristic p > 0. Let A D RŒx =.xp/. The following data define a Hopf algebra structure on A W .x/ D x ˝ 1 C 1 ˝ x, ".x/ D 0, s.x/ D x.

19.7 Hopf Algebras and Classifying Spaces

The homology and cohomology of classifying spaces BU, BO, BSO lead to a Hopf algebra which we will study from the algebraic view-point in this section. The polynomial algebra RŒa D RŒa1; a2; : : : becomes a Hopf algebra with coassociative

P

and cocommutative comultiplication determined by .an/ D pCqDn ap ˝ aq

and a0 D 1. We consider the algebra as a graded algebra with ai of degree i. (In the following we disregard the signs which appear in graded situations. Another device would be to assume that the ai have even degree, say degree 2i, or that R has characteristic 2.) Let D . 1; : : : ; r / 2 Nr0 be a multi-index with r components. We use the notation a D a11 : : : ar r . The monomials of type a (for arbitrary r) form an R-basis of RŒa . The homogeneous component RŒa n of degree n is spanned by the monomials a with k k D 1 C 2 2 C C r r D n.

We have an embedding RŒa1; : : : ; an ! RŒ˛1; : : : ; ˛n where aj is the j -th elementary symmetric polynomial in the ˛1; : : : ; ˛n. The embedding respects the grading if we give ˛j the degree 1. The image is the subalgebra of symmetric functions. The a with 2 Nn0 form an R-basis of the symmetric polynomials.

Another, more obvious, R-basis is obtained by starting with a monomial

polynomial in the a1; : : : ; an and denote it by I .a1; : : : ; an/. The monomials a which are summands of I have degree k k D jI j. Thus I .a1; : : : ; an/ D

I .a1; : : : ; an 1; 0/ D I .a1; : : : ; an 1/ for n > jI j.

(19.7.1) Lemma. If I is a partition of k and n k, then I .a1; : : : ; ak / is independent of n. We consider it as a polynomial in RŒa . In this way we obtain another R-basis of RŒa which consists of the polynomials I . The homogeneous component of degree n is spanned by the I with I an (unordered) partition of n.

Consider the formal power series

where bk is the k-th elementary symmetric polynomial in the variables ˇ1; : : : ; ˇn. For D . 1; : : : ; r / let I. / denote the multi-index .i1; : : : ; im/ with i D j

of n for n jI. /j. We denote this stable version by b . The b form an R-basis of the symmetric polynomials in RŒˇ . In this sense we can write formally

product itself is not defined). The reader may verify

U Œ1 D a1b1

U Œ2 D a12b2 C a2b12 2a2b2

U Œ3 D a13b3 C a3b13 C a1b1a2b2 C 3a3b3 3a1a2b3 3a3b1b2: