The Dyer-Lashof Algebra in Bordism
(June 1995. To Appear, C.R.Math.Rep.Acad.Sci.Canada)
Terrence Bisson Andre Joyal
bisson@canisius.edu joyal@math.uqam.ca
We present a theory of Dyer-Lashof operations in unoriented bordism (the ca*
*non-
ical splitting N*(X) ' N*H*(X), where N*( ) is unoriented bordism and H*( )
is homology mod 2, does not respect these operations). For any finite cover*
*ing
space we define a "polynomial functor" from the category of topological spa*
*ces
to itself. If the covering space is a closed manifold we obtain an operati*
*on de-
fined on the bordism of any E1 -space. A certain sequence of operations cal*
*led
squaring operations are defined from two-fold coverings; they satisfy the C*
*artan
formula and also a generalization of the Adem relations that is formulated *
*by us-
ing Lubin's theory of isogenies of formal group laws. We call a ring equipp*
*ed with
such a sequence of squaring operations a D-ring, and observe that the bordi*
*sm
ring of any free E1 -space is free as a D-ring. In particular, the bordism *
*ring of
finite covering manifolds is the free D-ring on one generator. In a second *
*compte-
rendu we discuss the (Nishida) relations between the Landweber-Novikov and *
*the
Dyer-Lashof operations, and show how to represent the Dyer-Lashof operations
in terms of their actions on the characteristic numbers of manifolds.
1. The algebra of covering manifolds.
We begin with the observation that a covering space p : T ! B can be used to de*
*fine a
functor X 7! p(X) from the category of topological spaces to itself, where
p(X) = {(u; b) | b 2 B; u : p-1(b) ! X}:
Then p(X) is the total space of a bundle over B with fibers Xp-1(b), and any co*
*ntinuous
map f : X ! Y induces a continuous map p(f) : p(X) ! p(Y ). We shall say that p*
*( )
is a polynomial functor. For functors F and G from the category of topological *
*spaces to
itself, we have functors F + G, F x G and F O G given by (F + G)(X) = F (X) + G*
*(X),
(F x G)(X) = F (X) x G(X), and (F O G)(X) = F (G(X)). Polynomial functors happen
to be closed under these operations, and we obtain well-defined operations p + *
*q, p x q and
p O q on coverings. These operations satisfy the kinds of identities that one s*
*hould expect
for an algebra of polynomials.
We define the derivative p0 of a covering p : T ! B to be the covering whos*
*e base
space is T and whose fiber over t 2 T is the set p-1(p(t)) - {t}. The rules of *
*differential
calculus apply: (p + q)0= p0+ q0, (p x q)0= p0x q + p x q0 and (p O q)0= (p0O q*
*) x q0. If
we observe that the total space of p is p0(1) (where 1 denotes a single point) *
*and that its
base space is p(1) the formula (p x q)0(1) = p0(1) x q(1) + p(1) x q0(1) expres*
*ses the total
space of (p x q) in terms of the total and based spaces of p and q. Similarly f*
*or the formula
(p O q)0(1) = p0(q(1)) x q0(1).
Remark 1: There is a parallel between this algebra of covering spaces and the a*
*lgebra of
combinatorial species developed in [9] and [10].
1
Remark 2: By using the Euler-Poincare characteristic one can associate a polyno*
*mial
O(p) to any covering p of a finite complex. We have O(p + q) = O(p) + O(q), O(p*
* x q) =
O(p) x O(q), O(p O q) = O(p) O O(q), and O(p0) = O(p)0.
Remark 3: It is also possible to define various kinds of higher differential op*
*erators on
coverings. For example, the group 2 acts on any second derivative p00by permuti*
*ng the
order of differentiation, and we can define
_1_d2p_= p00= :
2!dx2 2
Higher divided derivatives can be handled similarly.
Remark 4: Polynomial functors of n variables are easily defined. They are obtai*
*ned from
n-tuples (p1; : :;:pn) where pi: Ti! B is a finite covering for every i.
Let us now consider coverings of smooth compact manifolds. We say that two *
*coverings
of closed manifolds are cobordant if together they form the boundary of a cover*
*ing. Let
N* denote the set of cobordism classes of closed coverings. Let Ndn denote the *
*set of
cobordism classes of degree n (i.e. n-fold) coverings over closed manifolds of *
*dimension d.
Proposition 1. The operations of sum +, product x, and composition O are compat*
*ible
with the cobordism relation on closed coverings. They define on N* the structur*
*e of a
commutative Z2 algebra, graded by dimension.
Notice that if p 2 Nkm and q 2 Nrn then p O q 2 Nmr+k mn . This defines in
particular an action of N* on N*0 = N*. More generally, let us see that N* acts*
* on
the bordism ring of any E1 -space.
Recall (see [1], [18]) that an E1 -space X has structure maps En xn Xn ! X*
* for
each n. These structure maps give rise to structure maps p(X) ! X for every de*
*gree
n covering space p : T ! B. To see this it suffices to express p as a pull bac*
*k of the
tautological n-fold covering un of Bn along some map B ! Bn: This furnishes a m*
*ap
p(X) ! un(X) = En xn Xn and the structure map p(X) ! X is then obtained by
composing with un(X) ! X.
Recall (see [6] for instance) that an element of N*X is the bordism class o*
*f a pair
(M; f) where f : M ! X and M is a compact manifold; then p(M) is a compact mani*
*fold
and the structure map for X gives p(M) ! p(X) ! X, representing an element in N*
**X.
Proposition 2. Let X be an E1 -space. Each covering of degree n and dimension d*
* defines
an operation Nm X ! Nnm+d X. Cobordant covering spaces give the same operation.
Moreover, for double coverings these operations are additive.
It should be noted that tom Dieck [7] and Alliston [3] develop bordism Dyer*
*-Lashof
operations which agree with ours; the relationship will be clearer after sectio*
*n 2.
Example: The classifying space for finite coverings is B* the disjoint union of*
* the clas-
sifying spaces of the symmetric groups Bn. Then N*B* = N* and B* has a natural
E1 -space structure defined from disjoint sum. The covering operations on N*B* *
*corre-
spond to composition of coverings.
2
Remark: It is a classical result [19], [8], [12] that the inclusion i : n-1 n *
*defines a
split monomorphism i* : N*n-1 ! N*n: In our setting i* is the map p 7! x x p. I*
*t is
easy to see, by applying the rules of differential calculus, that the map
2q 1 d3q
q 7! dq_dx+ x_1_2!d_dx2+ x2__3!__dx3+ . . .
is a splitting [11].
For any space X let ffl : N*(X) ! H*(X) denote the Thom reduction, where H**
*( )
is mod 2 homology. If (M; f) 2 N*(X) we have ffl(M; f) = f*(M ) where M denote*
*s the
fundamental homology class of M. If X is an E1 -space then each covering of de*
*gree n
and dimension d defines an operation Hm X ! Hnm+d X which is the Thom reduction*
* of
the corresponding operation in bordism.
We now describe the sequence of cobordism class of double coverings that le*
*ads to the
concept of D-rings. It is a classical result that N*(RP 1) = N*[[t]]. Let qk in*
* N*B2 =
N*(RP 1) be represented by the canonical inclusion RP k,! RP 1. The sequence q0*
*; q1; : : :
is a basis of the N*-module N*(RP 1). The Kronecker pairing N*(RP 1) x N*(RP 1)*
* !
N* defines an exact duality between N*(RP 1) and N*(RP 1). Let d0; d1; : :b:e t*
*he basis
dual to the basis t0; t1; t2; : :u:nder the Kronecker pairing. The relation bet*
*ween the two
bases of N*(RP 1) can be expressed as an equality of generating series
X X X
( [RP i]xi)( dkxk) = ( qnxn);
i0 k0 n0
where x is a formal indeterminate. We have d0 = q0, and d1 = q1 since [RP 0] =*
* 1 and
[RP 1] = 0. It turns out (see [2] for instance) that dn can be represented by *
*the Milnor
hypersurface H(n; 1) ,! RP nx RP 1! RP n. The coverings dn and qn give operati*
*ons
which are distinct in bordism but agree in mod 2 homology.
2. D-rings and Dyer-Lashof operations
Recall that a formal group law over a commutative ring R is a formal power *
*series
F (x; y) 2 R[[x; y]] which satisfies identities corresponding to associativity *
*and unit; (see
Quillen [21] or Lazard [13] for instance). We say that a formal group law F has*
* order two
if F (x; x) = 0.
The Lazard ring (for formal group laws of orderPtwo) is the commutative rin*
*g with
generators ai;jand relations making F (x; y) = ai;jxiyj a formal group law of*
* order two.
Let us temporarily denote this Lazard ring by L. Then for any ring R and any f*
*ormal
group law G(x; y) 2 R[[x; y]] of order two there is a unique ring homomorphism *
*OE : L ! R
such that (OEF )(x; y) = G(x; y). Quillen [21] showed that L is naturally isom*
*orphic to
N* = N*(pt). This provides a beautiful interpretation of Thom's original calcu*
*lation of
the unoriented cobordism ring.
Let R be a commutative ring and let F 2 R[[x; y]] be a formal group law of *
*order two
(this implies that R is a Z2-algebra). According to Lubin [14] there exists a u*
*nique formal
group law Ftdefined over R[[t]] such that ht(x) = xF (x; t) is a morphism ht : *
*F ! Ft. The
3
kernel of ht is {0; t}, which is a group under the F -addition x+F y = F (x; y)*
*. We will refer
to Ft as the Lubin quotient of F by {0; t} and to ht as the isogeny. The constr*
*uction can be
iterated and a Lubin quotient Ft;sof Ft can be obtained by further killing ht(s*
*) 2 R[[t; s]].
The composite isogeny F ! Ft ! Ft;sis
ht;s(x) = ht(x)Ft(ht(x); ht(s)) = xF (x; t)F (x; s)F (x; F (s; t))
Its kernel consists of {0; t; s; F (s; t)}, which is an elementary abelian 2-gr*
*oup under the
F -addition. By doing the construction in a different order we obtain Fs;tbut i*
*t turns out
that Ft;s= Fs;t.
Definition: A D-ring is a commutative ring R together with a formal group law o*
*f order
two F defined over R and a ring homormorphism Dt : R ! R[[t]] called the total *
*square,
satisfying the following conditions:
i)D0(a) = a2 for every a in R;
ii)Dt(F ) = Ft;
iii)Dt O Ds is symmetric in t and s. Here we have extended Dt : R ! R[[t]] to*
* Dt :
R[[s]] ! R[[s; t]] by defining Dt(s) = ht(s) = sF (s; t).
A morphism of D-rings is a ring homomorphism which preserves the formal gro*
*up
laws and the total squares. A D-ring is also an algebra over the Lazard ring N*
**, and a
morphism of D-rings is a morphism of N*-algebras.
A D-ring is graded if R is graded and F is homogeneous in grade -1 and Dt(x*
*) has
grading 2i in R[[t]] for each element of grading i in R (where t and s have gra*
*ding -1).
Example: The Lazard ring N* has a unique ring homomorphism Dt : N* ! N*[[t]] su*
*ch
that Dt(F ) = Ft, and this defines a D-structure on N*. Thus N* is initial in t*
*he category
of D-rings.
Proposition. If X is an E1 -space then N*X is a commutative ring under Pontryag*
*in
product; it is also an N*-algebra. If d0; d1; : :a:re the double coverings des*
*cribed in the
previous section then the total squaring
X
Dt(x) = dn(x)tn
n
gives an D-structure on N*X.
Example: BO*, the disjoint union of the classifying spaces of the orthogonal gr*
*oups
BO(n), is an E1 -space with N*BO* = N*[b0; b1; : :]:. It forms a D-ring with F *
*given by
the cobordism formal group law over N* and with Dt determined by
Dt(b)(xF (x; t)) = b(x)b(F (x; t))
P
where b(x) = bixi.
We shall refer to any D-ring R with F = (+) as a Q-ring. The mod 2 homology*
* of
an E1 -space E is a Q-ring, and the Thom reduction ffl : N*(E) ! H*(E) is a mor*
*phism
of D-rings.
4
Proposition. A commutative ring R is a Q-ring if and only if it has a sequence *
*of additive
operations qn : R ! R which satisfy the following three conditions:
i)Squaring: q0(x) = x2 for all x 2 R.
P
ii)Cartan formula: qn(xy) = i+j=nqi(x)qj(y) for all x; y 2 R.
P i-n-1
iii)Adem relations: qm (qn(x)) = i 2i-m-n qm+2n-2i (qi(x)) for all x 2 R.
In the graded case, grade(qn(x)) = 2 . grade(x) + n.
This is exactly an action of the classical Dyer-Lashof algebra on R. This i*
*dea of writing
Adem relations via generating series is suggested by [4] and by Bullett and Mac*
*Donald [5].
See [17], [15], [16] for background on Dyer-Lashof operations.
Example: The Q-structure on H*BO* = Z2[b0; b1; : :]:is characterized by
Qt(b)(x(x + t)) = b(x)b(x + t)
P
where b(x) = bixi. This expresses via generating series a calculation of Prid*
*dy's in [20].
Notice that if A and B are Q-rings then A N* B = A Z2 B = A B is a Q-
ring. Let Q denote the free Q-ring generated by a Z2-vector space M. If M *
*has a
comultiplication, then Q has a comultiplication extending it which is a morp*
*hism of
Q-rings.
Recall that E1 (X) is the free E1 -space generated by X (see [18] or [1] fo*
*r back-
ground). The following is a classical result:
Theorem 1. (May [17]) For any space X the canonical map
Q ! H*E1 (X)
is an isomorphism which preserves the comultiplication. In particular, H*B* = Q*
* is
the free Q-ring on one generator.
If A and B are D-rings then A N* B is naturally a D-ring. Let us denote D<*
*M>
denote the D-ring freely generated by an N*-module M. If M is a coalgebra in th*
*e category
of N*-modules, then D has a comultiplication.
Theorem 2. The bordism of an E1 -space is an D-ring. Moreover, for any space X *
*the
canonical map
D ! N*E1 (X)
is an isomorphism which preserves the comultiplication. In particular, N* = N*(*
*B) =
D is the free D-ring on one generator.
Thus, both D and N* are algebras equipped with operations of substitutio*
*n; the
former because it is the set of unary operations in the theory of D-rings and t*
*he latter
because we have defined a substitution operation among coverings of manifolds. *
* The
above theorem says that the canonical isomorphism of D-rings D ! N* which se*
*nds
the generator x to the unique non-zero element x in N0(B1) preserves the operat*
*ions of
substitution.
5
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(*)Canisius College, Buffalo, N.Y. (U.S.A). e-mail: bisson@canisius.edu.
(**) Departement de Mathematiques, Universite du Quebeca Montreal, Montreal, Qu*
*ebec
H3C 3P8. e-mail: joyal@math.uqam.ca.
6