Nishida Relations in Bordism and Homology
(June 1995. To Appear, C.R.Math.Rep.Acad.Sci.Canada))
Terrence Bisson Andre Joyal
bisson@canisius.edu joyal@math.uqam.ca
This is the second of a series of Compte Rendus. In the first [1] we have p*
*resented
a theory of Dyer-Lashof operations in unoriented bordism. Here we shall dis*
*cuss
the (Nishida) relations between Dyer-Lashof and Landweber-Novikov operation*
*s.
They are used to represent the algebra N* of covering manifolds in terms of
their homology characteristic numbers. The proofs are based on the properti*
*es
of the covering space operations and the notions of D-ring and Q-ring intro*
*duced
in [1].
1. The Nishida relations in homology.
In homology mod 2 the Nishida relations are commutation relations between Dyer-*
*Lashof
and Steenrod operations (see [4] for instance). We shall express the Nishida r*
*elations
as a commutative square which involves the Milnor coaction and the Q-structure *
*on the
homology of any E1 -space discussed in [1].
Recall that the Milnor Hopf algebra A* is the dual of the Steenrod algebra;*
* see [6]
for instance. As a graded algebra it is A* = Z2[0 ; 1; 2; : :]:= -10Z2[0; 1; 2;*
* : :]:with
grade(i) = 2i - 1; the diagonal ffi : A* ! A*P A* is the unique ring homomorphi*
*sm
such that ffi() = ( 1) O (1 ) where (x) = ix2i. We are diverging from the u*
*sual
convention that puts 0 = 1.
The homology of any space has a natural (left) coaction
ff : H*(X) ! A* H*(X) = H*(X)[0 ; 1; 2; : :]:
which restricts to the usual (left) coaction when we put 0 = 1 and to the "grad*
*ing"
coaction ff(xn) = -n0xn for x 2 Hn(X) when we put 0 = 1 = 2 = . ... P
For example, for X = RP 1 we have ff(b)(x) = b((-1)(x)) where b(x) = ibix*
*i and
b0; b1; : :i:s the canonical basis of H*RP 1 and (-1)(x) is the composition inv*
*erse of the
power series (x).
Proposition. If R is a Q-ring, then there is a Q-structure on A* R determined *
*by
Qt()(x(x + t)) = (x)(x + t). In particular, there is a unique Q-structure on A*
** such
that Qt()(x(x + t)) = (x)(x + t).
Remark: Since the definition forces
X i
Qt(0) = 0 it2 -1
i
we see that the Q-structure on A* cannot survive if we try to insist that 0 = 1.
Suppose now that R is both a Q-ring and has a Milnor coaction. We give A* *
*R the
Q-structure from the proposition.
1
Definition: The Nishida relations hold for a Q-ring R if the following diagram
Qt
R __________>R[[t]]
| |
| |
|ff |ff
| |
_| Qt _|
A* R _____>A* R[[t]]
commutes, where ff is extended to R[[t]] by putting ff(t) = (t).
Theorem. If X is an E1 -space, then H*(X) is a Q-ring with a Milnor coaction, a*
*nd the
Nishida relations hold for H*(X).
This is a complete description of the Nishida relations in mod 2 homology. *
*We want
to give a similar description in unoriented bordism
2. The Nishida relations in unoriented bordism.
For any Z2-algebra R let B*(R) be the group of inversible formal power series f*
*(x) 2
xR[[x]] under substitution. The functor R 7! B*(R) is representable by a Hopf *
*algebra
B* called the Faa di Bruno Hopf algebra (see example 49 in [2] for instance). *
*We have
B* = Z2[h0 ; h1; : :]:= h-10Z2[h0; h1; : :]:and the diagonal of B* is given by
ffi(h) = (h 1) O (1 h)
P
where h(x) = n hnxn+1. We want to consider left comodules over B*. Recall tha*
*t the
tensor product of left comodules over a bialgebra is a left comodule.
Example: Let N* denote the Lazard ring for formal power series of order two (se*
*e [5]
for example). There is a natural (left) coaction OE : N* ! B* N* encoding cha*
*nge of
parameters in formal group laws. We have OE(F )(x; y) = h(F (h-1x; h-1y)) where*
* OE : N* !
N*[h0 ; h1; : :]:.
Notice that N*N* ! N* is a morphism of left B* comodules, so that N* is a m*
*onoid
in the category of left B* comodules.
Definition: A Landweber-Novikov coaction is a module over N* in the category of*
* B*-
comodules. More explicitly, this is a module over N* with a comodule structure *
*over B*
such that the module structure map N* M ! M is a map of B*-comodules.
In order to describe the natural Landweber-Novikov coaction on unoriented b*
*ordism,
we first need to recall the theory of cobordism characteristic classes for vect*
*or bundles,
as sketched by Quillen in [5] for instance; we use a slightly unstable version *
*of the usual
definitions, however. For any space X the total characteristic class of a real *
*vector bundle
V on X is the element c(V ) 2 N*(X)[b0; b1; :::] having the following propertie*
*s:
1) the map V ! c(V ) is a natural transformation c : V ect(X) ! N*(X)[b0; b*
*1; :::]
2) c(V1 V2) = c(V1)c(V2)
2
P
3) c(L) = bie(L)i if L is a line bundle with euler class e(L) 2 N1(X):
We have an expansion X
c(V ) = cR (V )bR
R
where bR = br00br11. .f.or R = (r0; r1; : :):. For virtual bundles the total ch*
*aracteristic class
is the element of N*(X)[b0 ; b1; :::] defined by putting c(V - W ) = c(V )c(W )*
*-1.
Example: The unoriented bordism group N*(X) of any space X has a Landweber-
Novikov coaction given by the Landweber-Novikov (total) operation
OE : N*(X) ! B* N*(X) = N*(X)[h0 ; h1; :::]:
It can be defined via the following explicit formula. If f : M ! X represents a*
*n element
of N*(X) then X
OE(M; f) = f*(cR (M ) \ M )hR
R
where M = -oM is the normal bundle of M and M is the fundamental class of M.
Example: The Landweber-Novikov coactionPon N*(RP 1) is characterised by the ide*
*ntity
OE(b)(x) = b(h(-1)(x)) where b(x) = ibixi.
Proposition. If R is a D-ring, then there is a D-structure on B* R = R[h0 ; h1*
*; :::]
determined by Dt(h)(x(x + t)) = h(x)h(F (x; t)).
Suppose now that R is a D-ring which also has a Landweber-Novikov coaction.*
* We
give B* R the D-structure from the proposition.
Definition: The Nishida relations hold for a D-ring R if the diagram
Dt
R _________>R[[t]]
| |
| |
|OE |OE
| |
_| Dt _|
B* R _____>B* R[[t]]
commutes, where OE is extended to R[[t]] by putting OE(t) = h(t).
Suppose that X is an E1 -space. Then N*X is an D-ring together with a Landw*
*eber-
Novikov coaction.
Theorem 2. If X is an E1 -space, then the Nishida relations hold for N*X in tha*
*t the
following diagram commutes:
Dt
N*(X) _________>N*(X)[[t]]
| |
| |
|OE |OE
| |
_| Dt _|
B* N*(X) _____>B* N*(X)[[t]]
3
Here OE has been extended to N*(X)[[t]] by putting OE(t) = h(t).
Recall from [1] that for any space X we have N*E1 (X) = D, the free *
*D-ring
generated by N*(X). Of course, N*(X) also has a Landweber-Novikov coaction.
Theorem 3. Suppose that M has a Landweber-Novikov coaction, and let D be the
free D-ring generated by M. Then there is a unique Landweber-Novikov coaction o*
*n D
which satisfies all of the following:
i)the canonical map M ! D is a comodule map;
ii)D is an algebra over its Landweber-Novikov coaction;
iii)D satisfies the Nishida relations.
The theorem shows that the Landweber-Novikov coaction on N*E1 (X) is fully *
*deter-
mined by its values on N*X. In particular, the Landweber-Novikov coaction on N**
*(B*) =
D is determined by the relation OE(x) = x.
It is well known that the Thom reduction ffl : N*(X) ! H*(X) induces an iso*
*morphism
N*(X) N* Z2 ' H*(X)
We shall write ffl : B* ! A* for the ring homomorphism such that ffl(h) = . We *
*will char-
acterize the behavior of the Thom reduction with respect to the usual Landweber*
*-Novikov
operations in unoriented cobordism and the Steenrod operations in mod 2 homolog*
*y by
showing that the Thom reduction gives rise to a simple equivalence of categorie*
*s.
Let [B?N? ] denote the category of Landweber-Novikov coactions (left B*-com*
*odules
which are modules over N*) and let [A? ] denote the category of Milnor coaction*
*s (left
A*-comodules).
For any M 2 [B?N? ] let us put T (M) = M N* Z2. By composing the coaction
M ! B* M with ffl : B* ! A* we obtain a coaction M ! A* M. By further
tensoring with Z2 we obtain a coaction T (M) ! A* T (M). This defines a funct*
*or
T : [B?N? ] ! [A? ].
Proposition. The functor T defines an equivalence of categories [B?N? ] ' [A? ].
In fact, we get a similar result even when we take the Dyer-Lashof operatio*
*ns into
account, which we do by using monads which encode the Dyer-Lashof operations in*
* bordism
and homology. For background on monads and their category of actions see [3] fo*
*r instance.
The functor M 7! D from [B?N? ] to itself is a monad that we shall denot*
*e D (it
is a monad since D is a free structure). The D-actions are exactly the D-rin*
*gs with
a Landweber-Novikov coaction which satisfies the Nishida relations. Let us den*
*ote this
category by [B?N? ]D . Similarly, the functor M 7! Q from [A? ] to itself i*
*s a monad
that we shall denote Q. The Q-actions are exactly the Q-rings with a Milnor co*
*action
which satisfies the Nishida relations. Let us denote this category by [A? ]Q .
Proposition 2. The functor T transforms the D-actions into the Q-actions and de*
*fines
an equivalence of categories
[B?N? ]D ' [A? ]Q :
This result shows that for any E1 -space X, the D-ring N*(X) can be recover*
*ed
entirely from the Q-ring H*(X) as long as the coaction of A* on H*(X) is known.
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3. Characteristic numbers and covering space operations
The bordism classes of manifolds are determined by their tangential characteris*
*tic numbers,
or equally by their normal characteristic numbers.
Recall the discussion in section 2 of characteristic classes in unoriented *
*cobordism.
The total Stiefel-Whitney class w(V ) 2 H*(X)[b0; b1; :::] of a vector bundle V*
* on X is the
Thom reduction of the total cobordism characteristicPclass. Then w( ) is multip*
*licative in
that w(V1 V2) = w(V1)w(V2), and w(L) = ibie(L)i for any line bundle over X, *
*where
e(L) 2 H1(X) is the euler class of L.
The tangential characteristic numbers can be grouped together as the coeffi*
*cients of
a polynomial fio(M) 2 Z2[h0; h1; :::]. We have
X
fio(M) = hR
R
P
where w(oM ) = R wR (oM )bR is the total Stiefel-Whitney class of the tangent*
* bundle of
M and M is the fundamental class of M.
In keeping with the Landweber-Novikov coaction, we can work instead with ch*
*arac-
teristic number polynomials for the normal bundle, the virtual bundle M = -oM *
*. More
generally, as in [5] for instance, we can define a Boardman map
fi : N*(X) ! B* H*(X) = H*(X)[h0 ; h1; :::]
by the explicit formula
X
fi(M; f) = f*(wR (M ) \ M )hR :
R
Let B* denote the classifying space for finite covering spaces. It is an E*
*1 -space.
Let N* denote N*B* and H* denote H*B. The Boardman map fi : N* ! B*
H* obviously preserves sums and products. In fact, it also preserves the opera*
*tion of
substitution, when substitution is properly defined on B* H*.
Proposition. If R is a Q-ring then there is a Q-structure on B* R determined by
Qt(h)(x(x + t)) = h(x)h(x + t). In particular, there is a unique Q-structure on*
* B* such
that Qt(h)(x(x + t)) = h(x)h(x + t).
From [1] we have that H* = Q is the free Q-ring on one generator. Since *
*B*H*
is the coproduct in the category of Q-rings, we can view B*H* as the collection*
* of unary
operations in the theory of Q-rings which are extensions of B*. Therefore H*[h0*
* ; h1; :::] =
B* H* admits an operation of substitution. Here is an explicit formula for sub*
*stituting
two elements of H*[h0 ; h1; :::]:
X R X R X S R
pR h O qR h = pR (qSh )h
R R R;S
where the operations pR are applied to the elements qShS of the Q-ring B* H*.
5
Theorem 4. The Boardman map fi : N* ! H*[h0 ; h1; :::] preserves sums, products
and substitutions.
We view this as describing for each covering p of closed manifolds how the*
* normal
characteristic numbers of p(M) are determined by the characteristic numbers of*
* M. The
complete description is summarized by the formula
fi(p(M)) = (fi(p))(fi(M));
where the right-hand substitution takes place in H*[h0 ; h1; : :]:.
References
[1]T. P. Bisson, A. Joyal, The Dyer-Lashof algebra in bordism, C. R. Math. Re*
*p. Acad.
Sci. Canada (to appear).
[2]A. Joyal, Une theorie combinatoire des series formelles, Adv. in Math. 42 *
*(1981), 1-82.
[3]S. MacLane. Categories for the Working Mathematician, Springer 1971.
[4]J. P. May, Homology operations on infinite loop spaces, in Proc. Symp. Pur*
*e Math.
22, A. M. S. 1971, 171-185.
[5]D. Quillen, Elementary proofs of some results of cobordism theory using St*
*eenrod
operations, Adv. in Math. 7 (1971), 29-56.
[6]N. E. Steenrod, D. B. A. Epstein, Cohomology operations. Ann. of Math. Stu*
*dies no.
50, Princeton 1962.
(*)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.
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