Note on the characteristic classes oî the elements of the unitary bordism module

R O CZN IK I POLSKIEGO TOW ARZYSTW A MATEMATYCZNEGO Séria I: PEA C E MATEMATYCZNE X V (1971)

A N N A L E S SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)

A. J

(Warszawa)

Note on the characteristic classes oî the elements of the unitary bordism module

Let (Y, f : Y -> X), where Y , X are 17-manifolds of dimensions n and к respectively, represent an element a e Un(X). For the definitions of 17-manifolds and unitary bordism group we refer to [2].

We recall that U*(-) is the generalized homology theory represented by the spectrum MU. The corresponding generalized cohomology theory U*(') is called cobordism. If X is a 17-manifold of dimension к there is defined a homomorphism

D : Un(X) Vk~'{X),

being an isomorphism and called the Thom-Atiyah duality.

Let the element aeUn(X) be represented by the map f a: Y -> X and assume n < к/2. In this case the map f a can be taken to be an em­

bedding. Let v be the normal bundle of Y in X. Then, for some r, r+(r) has the structure of 17-bundle determined by the 17-structures on the tangent bundles rx and rY• r + {r) can be considered as the normal bundle

of Y in X x I f . We define a map

g : X x D r M U { k - n + r) as the composition

X x D r ^ M( v + { r ) ) ^ M U ( k - n + r),

where g is induced by the classifying map of the bundle r + (r) and у takes the complement of the normal neighbourhood into the point. It is clear that this map can be factorized by the identification map

A: X x D r - +SrX.

The element D(a) of Uk~n(X) is represented by the map ga such that the diagram

д

MU (k—n + r)

<Ja/

SrX /

is commutative.

92 A. J a n k o w s k i

In this note we will introduce the characteristic classes св ( Г , / ) е Я

depending only on the class of ( Y , f ) in Un(X) and we will prove thaty under some conditions on H*(X), those classes determine the element of Un{X). For X = P (a point) those classes are in fact the dual Chern numbers. Thus the well-known result that the dual Chern numbers de­

termine the 77-bordism class of manifold follows from our result.

We will determine also the action of the algebra A u of stable unitary cobordism operations on the classes cR.

1. Homology and cohomology of the spectrum 31U. In the sequel we will use the homology and cohomology groups with integer coeffi­

cients. We recall, following Adams [1], that H*(BU) has a Z-base con­

sisting of the monomials

where deg bk = 21c and the elements bk are images of the elements bkeH2k[BU {!)) = H2k{CP{ oo)). There is multiplication in H*(BU) in­

duced by the Whitney sum map

and 1I*(BU) is the polynomial ring Z[bly ..., bk, ...]. The Thom iso­

morphism

is the multiplicative transformation if the multiplication in H*(31U) is induced by the usual multiplication in the spectrum MU and thus H*(31U) is the polynomial.ring Z[b[, ..., b'k, ...], where bk = <pbk. There are dual bases cR in H*(BU) and cR in H*(MU), where cR is dual to bR and cR is dual to b'R. The Whitney sum map and the multiplication in 31U induce the comultiplications A* in II*{BU) and H*(31U) such that the Thom isomorphism

B U {n) X BU(m) -> BU{n + m)

<p: H*(BU) -+H*{MU)

<p : H*(BU) -+H*{MU) is comultiplicative. We have

A C r — cR f ê cR2i

R= R±+R

2

the addition of sequences is done term by term.

U n i t a r y b o r d i s m m o d u l e 93

It was proved by Novikov that each element of A u can be written in the form У tR

R, where tR is the multiplication by the element

R

xReü< ^ 2[R[ and SR belongs to some subalgebra

of the A u. The operation

8

r is the homotopy class of the map sR : M U S2lRtMU (for X a spectrum

8

r X is a spectrum such that {

r X )m = X m+r) and induces the homo­

morphism

sR*: Hq(MU) -> Hq_

{R{(MU).

This homomorphism has been computed by Adams [1] as follows*

oo

Let b' = £ % and x, yeH*{ 3I U) . Then

i = 0 •

and

sir(b') = 2 W »<)(!>')<+1.

г> 0 We will use this result later.

Let £ be the vector bundle on X and : X -> B ü the classifying map. The Chern classes of the bundle ^ с й( |) е Я 1К,(Х) are defined as

C r (£) — fs CR‘

For the Z7-manifold X we define the Chern classes and the dual Chern classes (we denote those classes by cR{X) and cR(X) respectively) as the Chern classes of the tangent and normal (in Rk+r) bundles of X respectively.

2. A piece of homotopy theory. Let

be the Postnikov tower of (ç—1)-connected space X and

J £ (n )

l(n)

9

\ X

X

94 A. J a n k o w s k i

the Whitehead tower of X. We recall that

(i) : nq+k(X) лд+к( Х{п)) are isomorphisms for к < n.

(ii) ла+к(Х{п]) = 0 for к > n.

(iü) 9n* : ™ Q-rk{X{n)) -* nQ+k(X) are isomorphisms for k ^ n . (iv) Trff+A.(X(w)) = 0 for к < n.

(v) j n : X (w+1) -> X (n) is the fibration with fibre К(ла+п{ Х) , q + n ) in­

duced by k(n)€Hn+q+l(X(n), ng+n{X)).

(vi) i (n) : X (w+1) X (w) is the fibration with fibre К(ла+п(Х), q~\~n—l) induced by &(w>eHn+9(X(n>, яв+я(Х)).

It is well known that there is the homotopy commutative diagram X (n)- °in)

/( - X > л.()

1 “ 1 Й

i y F -> E - -> В

such that the lower rows are the fibrations and a, fi, y are homotopy equivalences. Let

fi* ■ В ^ Ц Х {п), nt+n(X)) -* Н ^ ( Х {п),жм (Щ, fj# . я » +«(Х<*>, лд+п(Х)) -* ля+„(Е)) be the coefficient homomorphism induced by

ft* . Hgj-niX) — > 7lq^_nÇE) j

ÇeHn+9(F, nn+q(F)) the fundamental class of the fibre and £' e eHn+9(F1 nn+q{E)) the image of | under the coefficient homomorphism

9* : Я "+3(Я, nMq(F)) - Я»+«(Я, Я„+„(Я)) induced by

<P* : ™n+q(F) ^n+Q(E).

We have

(**) a* f ’ = /»*#*>, where «/ ~ r£' and

t : Hn+9(F, nn+q(E)) -> Hn+9+1(B, nn+q{E)) is the transgression.

L emma 2.1. For each integer X, Nk(n^ — 0 if and only if Nk(w) is in the image of

Æ) = Я 5+”(Х, nq+n(X)) Я«+»(Х<">, *„+„(X)).

U n i t a r y b o r d i s m m o d u l e 9 5

P roof. It follows by the truncated exact sequence that Nco* = 0 if and only if N f is in the image of q>*. The lemma follows from the homotopy commutativity of diagrams (*) and (**) since a*, y* and /S# are isomorphisms.

C orollary 2.2 I f r(n) is the order of the class k(n), then r(n)k(n) г>

in the image of g*ny

3. The orders of the Postnikow invariants ikw for the spectrum MTJ, We will follow [5]. Since n ^MU (q)] in the stable range is zero for i odd we have to consider only the invariants

л ^ +%п(Ш1 7(e))), n < q . Let R = (fj, . . . , r if ...). We define the number

oo

ÜR = [ ] & } ,

i=*l

where

I ^{p ri} if i — p k—i for some кФ 0 and prime p , 1 otherwise.

P roposition 3.1. The order of the invariant fy2n) is the l.c.m. of the numbers dR, |_R| = n.

In order to prove Proposition 3.1 we will follow [5]. The only change is that, instead of the Theorem C of [5], p. 596, we have to prove

P roposition 3.2. There exists a Milnor basis {T2fe} such that if 1c

— р г—1, p —a prime all the dual Ghern numbers of Y 2k are multiples of p i 1).

Proof. We will follow [3], p. 114. Let I(p) be the ideal of Qu con­

sisting of those bordism classes all of whose dual Chern numbers are di­

visible by p. It follows from the computation of Milnor and Novikov that

where d(h) is the number of partitions of к which does not contain the numbers of the form p s—1. Similarly

where d'{k) is the number of the partitions of к which contains at least one number of the form p

—1.

Now, let {T2*} be the Milnor base. That means

c ( y 2*) p for к —p s—1, p a prime, 1 otherwise.

I1) After this note has been written this result appeared in: R. S tron g, Notes

on eobordism theory, Princeton 1968.

A. J a n k o w s k i 96

Let Tc be not of the form p s —1. Then we put Y 2k = Y 2k. We proceed to prove the existence of the elements Y 2p ~2 by induction. For Tc

= 1 we put Y 2p~2 = CP(p —1). Suppose Y 2pr~2 exists for r < Tc and consider the ideal I'{p) generated by Y 2pV~2 for r < Tc, where we put

Y° = p x (point). We have

. IL(P) = hniP) for n < pk i 1 - 1 and

1*2n(P) <= hni-P) for n = p k+1- 1.

Let M2n represent the element of I 2n{p) which is not in l'2p(p). We will show that cAr<[ M2n] = ap, where а Ф Omodp. Suppose on the other hand that cAn[M2n] = bp2. Then [M2n] — b p [Y 2n] is decomposable and has all dual Chern numbers divisible on p. Thus

[ i r 2<- ]

and every non-zero term on the right-hand side has ж > 2 . It follows that in each term there is r for which ir = p s—1 and s^CTc. Then [M2n] — b p [ Y 2n] e еГ (p). Since bp [ Y 2n] = b [ Y°] [ Y 2n] , [ilf2w] is contained in Г {p) contrary to the hypothesis. Hence cAn[ M2n] = а р , а Ф Omodp. There exist integers c and d such that ac-{~pkd = 1. Consider now

Y2pk+1~2 = cM2n- d C P ( p k+l- l ) . It follows that Y2pfc4~1-2 belongs to I(p) and

cAn( Y 2pk+1^2) = cap-\-dpk+1 — p .

This completes the proof of Proposition 3.2. JSTow, Proposition 3.1 follows by repetition of the arguments in [5].

4. Characteristic classes of singular manifolds. Let f: Y n -» X k be a map of F-manifolds and let f ~ : H * ( Y ) - > H * ( X ) be the forward ho­

momorphism induced by /. We recall that f = D ^ f * D Y, where D T : Hs( Y) -> Hn_s(Y) and Dx : НЦХ) Hk_s{X) are the duality homo- morphisms.

D efinition 4.1. Let x e Hs(X). We define the class cB( Y , f , x ) eHs+2lR\+k-n(X) as

cR{ y j , o c ) = f (cR ( Y)f*œ) .

D efinition 4.2. We define the dual Chern class of the singular manifold ( Y , f ) in I

гй( Г , Я £я а д + *-”(Х) as

C r ( Y , f ) = ^ Сщ( Y , f , Сщ(X)).

R=R^ + R2

U n i t a r y b o r d i s m m o d u l e 97

We are going to prove that the dual Chern classes defined above depend only on the class of the singular manifold (P , / ) in U*(Xk). We will consider first the characteristic classes of the elements of U*(Xk).

Let a e Vs(Xk) be represented by the map f a :

2r~sX -> MU(r).

We define

mR(a) =

where cx2r~s is the suspension isomorphism.

It is clear that the classes mR{a) are well defined since {a

r'~s)~

f* cR depends only on the stable homotopy class of f a and thus only on the element a.

Let D : Un( Xk) -> TJk~n{Xk) be the Thom-Atiyah duality. We have the following

P roposition 4.3. Let ( Y , / ) be the singular manifold in X and let a be the class of ( Y , f ) in Un{X). Then

cB( Y J ) = мл (Х>(а».

Proof. Consider first the case n < hj2. We can assume that /: ¥ X is an embedding. In this case the map / representing D ( a)

can be defined as the composition

& X Л M( v+{t ) ) -4 MUf k— n + t)

and mR{a) = (a1) C o n s i d e r the commutative diagram H*(SlX) ^*— H(M( v+( t ) ) ) <-— H* ( MU ( k - n + t))

<£ I I <p ~ I <p

H*(X) --- H*(T) <---H ( B U ( k - n + t))

where / is the classifying map of the U-bundle v-\- (t ), v the normal bundle of ¥ in X. It follows that mR(Da) = /* (cR(v)j. Since [v]-f[rF] = f [ r x ] in К ( 7) we have

m R ( D a ) = f

(сд(/'тх+ ( — tf ))) = / (

tf ) cr 2(/’ tv ))

r = r 1+ r 2

= Г ( У гВ

[ 7 ) Г е щ ( Х ) ) = У f (cRl( Y ) f c R

(X))

= у cBl{ J , f , c Rl(X)) = cB( Y , f ) . i?2

R oczniki PTM — P ra ce M a tem a ty czn e X V

7

98 A. J a n k o w s k i

Now, the general case follows since the map / : Y X can be re­

placed by embedding into X x D r for some r.

C oeollaey 4.4. The dual Ghern classes cR( Y , f ) depend only on the class of ( Y , f ) in U*{X).

Consider now the Hurewicz homomorphism y \ -> Hk(MTJ). We have the following

L emma 4.5. Let the manifold Y k represent the element aeQff • Then

cr } —

Proof: There is the commutative diagram H*(Sk+2m) J — H*(MU(m))

U' U

H*(Yk) H*(BU(m))

where q> is the Thom isomorphism and cp' the forward homomorphism induced by the embedding of Yk into Sk+2m. It follows that for aeH* (BTJ{m)) we have

< 7 V , [$Wml> = < r ' f a , [A’Mm]) = ( f a , Thus

(x(a)> 4 > = <7.[S*+S“ ], cRy = <[Я*+Я“], /% >

= < [ V ] , / V B> = cB(Y*).

C oeollaey 4.6. Let £*+2m be the generator of Hk+2m(Sk+2m) dual to [Sk+2m]. Then

7*4 = cR( Y k)£k+2m.

C oeollaey 4.7. For a, (3eU*{X) and weQu we ham

<Ьг(а + /5) = <Ьг(а) + C r (P) and

cR{coa) = ^ сВ1{а)Сщ{М), Ej -f i?2

where M is a U-manifold representing со.

Proof. The first part is clear. In order to prove the second one let us put f : X MU(m) a map representing D{a) and g : S* -> MU(r}

a map representing M. It is clear that JD(coa) is represented by the compo­

sition

SlX °M MU(r) 4P MU(m) 4 MTJ{r+m),

U n i t a r y b o r d i s m m o d u l e 99

where zl is the multiplication in the spectrum MV. It follows that cR(coa) = mR(Bœa) = % f)*cR = (<r*)_1(gr % f)*A*cR

= {otr

( g # f ) * X сВх®с'щ = {<уг)~

д*сщГ с

r = r

+ r

r = r

+ r

— сщ {М){о*) 1(£lf gr 2) — ^ cRi{M)mR

{Da)

= ^ GR

(M)CR

(a) .

r = r

+ r

The proof is completed.

5. Application. Let ns denote by г(и) the order of the Postnikov invariant k(2n)e H

t+

n+

(MU(q),

n),TcW n (MU(q))). It follows from Pro- position 3.1 that

r{n) =l . c. m. ( dR,\R\ = n).

T

5.1. Let X be U-manifold of the dimension к such that there is no f(n) torsion in Hs+

n{X). Then a e Vk_s(X) is zero if and only if cR(a) =

for all R.

Proof. We will nse the standard arguments. Since the Thom-Atiyah duality D : Vk_s{X) -> US{X) is the isomorphism in order to prove our theorem it is sufficient to prove В {a) = 0 . Now, let / : S2m~sX -> MU (m) represents the element В (a). It follows from our assumption that

f c R = o

m~smR {Ba) = o

m~scR(a) = 0

and it follows from Corollary 2.2 that if we denote by f n the lift of / to MU{ m) (2n)

$

m-sx J ^ MU{

n f n)

\

M V {m)

thenf*r(n)U2n> = r{n)f*nk(2n) = 0, where/*fe(2n)eH2m+2n(i82m- sA) = Hs+

n{X).

Hence from our assumption/* k(2n) is zero and f n can be lifted to M V ( mf 2n+1)

= MV{ m) ^ n+1)f It follows by induction that / is null-homotopic and thus В (a) = 0. The proof is completed.

C

5 .2 . If X is a V-manifold such that there is no torsion in H*( X), then aeU^(X) is zero if and only if cR(a) — 0 for all R.

6. The action of A u on the characteristic classes cR. The action

of the algebra A u on V*{X) has been defined by Novikov [4]. This action

100 A. J a n k o w s k i

is dual to the action on U*(X), that means we have the commutative diagram

U J X ) ---- — Uk~n{X)

i I

IS j s

Un- 2p( X ) - ? - + XJk-*+*°{X) for 8 e A u and 2p = degS.

Let the map s : MU(r) MU{ r AP ) represent the element 8 of A u (r large) and suppose that

S CR — aR,T (8) eT.

T We have the following

L

6 .1 . The classes cR{Sa) are related to the classes cT(a) as follows:

Cr ( So) — ^ a R ' T ( S ) c T ( g ) .

T

Proof. It follows from the above diagram and Proposition 4.3 that cR(Sa) = mR{D8a) — mR(SDa). Let / : $>г~к+пх -> MU(r) represents the element Da. Thus 8Da is represented by the composition

£ 2 r-k+nX Л MU{r) Д M U { r + p ) . We have

cR(Sa) = mB(SDa) = (o“- ‘+*)- \ S f ) * e R = (0г" ‘ +* Г 1/* « Ч

= (0»-*+*)-1/ * 2 ,«KJ.(S)ci. = У ав_т( ^ ' - к*пГ Т о т

T T

— а&,т{$)'т,т(Ва) — aR T(S)cT(a).

T T

We are going now to compute the coefficients aR>T(SK). Let Seq be the set of sequences with only the finite number of non-zero terms.

We will define the maps

/л, v : Seq Seq as follows

•••! 4 ; •••) == ifi? • • • ? • • •)?

where tj = the number of ik such that ik = j,

v (^1 ? • * • > ^ к ? * • * ) = (^1J " • f Q-ki • • • ) 1

<lj — 1 for j = 1, ...,

for j = rx+ l , ..., rx + ra,

where

U n i t a r y b o r d i s m m o d u l e 101

For PeSeq we will denote

i-Ki =

! ir*> Й =

Ti and for J ^-K j < г we denote by [-B]i the integer

_____ il

(i— I p I)!^! . . . rkl ...

D

6 .2 . Let P , B, Те Seq. We will call the decomposition P = P 1+ . . . + P r to be T , B-admissible if and only if:

(i) \B\ = r,

(ii) there exists Je Seq snch that

M I) = T , Pj ^ V + l i щ \ where Q =

We define the

..) = v(B).

number

К А Р ) = Л - р IPrX+i,

where the sum runs over all B, T-admissible decompositions of P.

We have the following

P

6.3 . Let sT: MU

2lTlMU be the map representing STeA u. Then

ST*(^ r ) = У! ^R,r(P)^P*

P

Proof. It follows from the Adams computation [1] that i m +A for t = 4 ,

0 otherwise,

sT*(bt) —

where the sum runs over all В such that | Р | < г + 1 and \B\ = t — i.

From the product formula we have

st * (Ьц) L v (*;>•••«* l K ) ... = ...

aii+...+/

,-

where the sum runs over all sequences I = {ilt . . . t ir) such that A^ -}- + . . . + Ai r —T, this being equivalent to p ( I ) = T, and all sequences Blt .. . , Br such that P? < \B.j\ = qj—ij with Q = {qf) = vB. The proposition follows.

C

6 .4 . The coefficients aPR(ST) are equal Proof. From Proposition 6.3 we have

^л,т(-Р) =

102 A. J a n k o w s k i

Hence

sTGp — JT" kB>T(P)cR.

R

C orollary 6.5. The operation ST acts on the dual Ghern classes as follows :

[1] J. P. A d a m s, 8. P . Novikov’s work on operations on complex cobordism (mimeo graphed notes).

[2] P. E. C on n er and E. E. F lo y d , Torsion in S U bordism, Mémoires Amer. Math.

Soc. 60 (1966).

[3] — Differentiable periodic maps, Ergebnisse der Mathematik, Band 33 (1964).

[4] S. P. N o v ik o r , Cobordism, Izviestia AN СССР 31 (1967), p. 855-951 (Russian).

[5] J. T ro u e, Orders o f Postnikov invariants for M 8 0 , 111. J. of Math. 10 (1966), p. 592-604.

P

References