THE UNIVERSALITY OF EQUIVARIANT COMPLEX BORDISM
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Abstract. We show that if A is an abelian compact Lie group, all A-equivariant complex
vector bundles are orientable over a complex orientable equivariant cohomology theory. In
the process, we calculate the complex orientable homology and cohomology of all complex
Grassmannians.
1. Introduction
Suppose A is an abelian compact Lie group. We recall [3] that an A-equivariant cohomology theory EA () is orientable if complex line bundles are well behaved. More precisely,
we let C P (U) denote the space of lines in a complete complex A-universe, and denote
the trivial representation. We say that EA () is a complex stable ring theory if there are
suspension isomorphisms
= ~ n+jV j V
V : E~An (X ) ?!
EA (S ^ X )
for all complex representations V , where S V is the one-point compacti cation of V , and jV j
is the real dimension of V ; these are required to be transitive, and given by multiplication
with a generator of E~ jV j(S V ). We say the theory is complex orientable if in addition, there
is a cohomology class x() 2 EA (C P (U); C P ()) which restricts to a generator of
EA (C P ( ); C P ()) = E~A (S ? )
for all one dimensional representations . Thus if the complex orientation is in cohomological
degree 2, it determines a complex stable structure. Many important theories are complex
orientable, for instance equivariant K -theory, tom Dieck's equivariant bordism MUA (), and
Borel cohomology for non-equivariantly complex orientable theories.
The purpose of this article is to show that this good behaviour is sucient to ensure good
behaviour of complex vector bundles of any dimension. In particular all complex vector
bundles have Thom classes.
Theorem 1.1. If E is complex orientable then any A-equivariant complex vector bundle is
E -orientable.
This is proved in Section 6.
Okonek [6] has proved that by construction MUA () is universal for cohomology theories
with Thom classes. Combined with our main result this implies the following universality
statement.
Theorem 1.2. If EA () is a complex oriented cohomology theory with orientation in cohomological degree 2 then there is a unique ring map MU ?! E of A-spectra under which the
orientation of E is the image of the canonical orientation. Conversely, a map MU ?! E
of ring A-spectra endows E with the structure of a complex oriented cohomology theory with
orientation in cohomological degree 2.
1
1
2
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
We include a proof of this in Section 7. To make this result somewhat more useful we
have the following calculation.
Theorem 1.3. If EA () is a complex orientable theory then EA(MU ) is a polynomial EA algebra and
RingA (MU; E ) = EA -alg(EA(MU ); EA ):
This is proved in Section 8. We shall be more speci c about the polynomial generators
in due course. The good behaviour of EA(MU ) should be contrasted with the facts that
EA (C P (U)) is not a power series ring in general, and MUA is not a polynomial algebra.
The main step for all the proofs is the calculation of the cohomology of the universal
Grassmannian for any complex oriented cohomology theory. This is also an important step
in the calculational exploitation of complex orientable cohomology theories, as is familiar
from the non-equivariant case. The traditional methods for the non-equivariant case (see
[1] for example) do not apply since the cellular ltrations are not so simple equivariantly.
Accordingly, geometric arguments are required as substitutes for the use of the AtiyahHirzebruch spectral sequence, and these are illuminating even in the classical case.
2. Equivariant Grassmannians.
Let Grn (V ) be the complex Grassmannian of complex n-dimensional subspaces of V . For
example Gr1(V ) = C P (V ).
Lemma 2.1. The A-space Grn (U) is a classifying space for A-equivariant complex n-plane
bundles:
Grn (U) = BU (n):
We retain the Grassmannian notation, because it will be useful to display the universe
explicitly at various points. The direct sum of lines gives a map
(C P (U))n = Gr1(U)n ?! Grn (Un ) = Grn (U);
where the nal homeomorphism arises from an isometric isomorphism Un = U. This sum
map induces maps
EA(C P (U)) n = EA ((C P (U))n ) ?! EA (Grn (U))
and
EA (Grn (U)) ?! EA ((C P (U))n ) = EA (C P (U)) ^ n;
where the completed tensor producte ^ refers to the skeletal ltration topology. The
Kunneth theorems implicit in this statement are corollaries of Cole's Splitting Theorem
[3].
Since any permutation of the copies of U in Un is homotopic to the identity through
isometric isomorphisms, the induced maps factor through coinvariants and invariants for the
symmetric group n.
Theorem 2.2. For any complex orientable cohomology theory EA (), the direct sum of lines
induces isomorphisms
EA (Grn(U)) = fEA (C P (U)) n gn
and
EA (Grn (U)) = fEA (C P (U)) ^ n gn :
3
Theorem 2.2 will be proved in Section 5 below. We pause to remark that this gives us
speci c generators, and hence all the structure of the homology and cohomology of Grassmannians follows from that of C P (U) made explicit in [4].
Indeed, Cole showed that EA (C P (U)) is additively free over EA and EA (C P (U)) is a
product of suspensions of EA . Furthermore, he showed how an orientation x() of E , together
with a complete ag
?
F = 0 = V 0 V 1 V 2 in U determines a topological basis 1 = y(V 0); y(V 1); y(V 2); : : : of EA (C P (U)), and we
may let 0(F); 1(F); 2(F); : : : denote the dual basis of EA(C P (U)). The notation for the
homology generators re ects the fact that i(F) depends on the initial segment V 0 V 1 V i of the ag. Furthermore, a Kunneth theorem holds for the homology or cohomology
of products of C P (U).
Lemma 2.3. An EA -basis of the coinvariants fEA(C P (U)) n gn is given by the images of
all products i (F) i (F) in (F) so that 0 i1 i2 in . A topological
EA -basis of the invariants fEA (C P (U)) ^ ngn corresponds to the collection of sequences 0 i1 i2 in; the basis elements are the symmetric sums
0y(V i ) y(V i ) y(V i n )
where 0 denotes the sum over the orbit of (i1; i2; : : : ; in).
Proof: The result is clear once we remark that the n action arises from an action on the
basis of the homology or cohomology of (C P (U))n . In the case of cohomology, we pass to
limits from the case of C P (V )n .
1
2
(1)
(2)
( )
Corollary 2.4. If E is a complex oriented cohomology theory then E ^ Grn (U) splits as a
wedge of copies of E indexed by sequences 0 i1 i2 in:
_
E ^ Grn (U) ' 2j jE;
i
i
where jij = i1 + i2 + + in . The splitting depends on the orientation.
Proof: In the usual way, from the basis of EA (Grn (U)), we may construct a map
_
s : 2j jE ?! E ^ Grn (U)
i
i
of A-spectra, using the product on E .
By construction it induces an isomorphism in A(). Now suppose B A, and consider
the B -equivariant situation. The restriction map EA (C P (U)) ?! EB (C P (U)) takes an Aorientation to a B -orientation, and the resulting basis corresponding to a complete ag to
the basis corresponding to the same ag regarded B -equivariantly. It therefore follows that
the restriction EA (Grn (U)) ?! EB (Grn(U)) takes the sequence basis to another basis. Thus
s induces an isomorphism of B (). Since this applies to all subgroups B of A, the map s is
an A-equivalence by the Whitehead theorem.
It may be useful to record the calculation relative to a speci c orientation in very concrete
terms.
4
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Corollary 2.5. If E is a complex oriented cohomology theory then
EA (Grn (U)) = EAf (V i ) (V i ) (V in ) j 0 i1 i2 ing
1
2
and
EA (Grn (U)) = EA-mod(EA(Grn (U)); EA)
= EA ff0y(V i ) y(V i ) y(V i n ) j 0 i1 i2 in gg:
(1)
(2)
( )
Finally we may consider the extra structure on the system
fEA (Grn (U))gn0 or fEA (Grn (U))gn0:
The relevant extra structure is given by the conjugation map
: Grn (U) ?! Grn (U) = Grn (U);
the action maps
: Grn (U) ?! Grn (U ) = Grn (U)
for 2 A , the direct sum maps
: Grm(U) Grn (U) ?! Grm+n (U U) = Grm+n (U)
and the tensor product maps
: Grm (U) Grn (U) ?! Grmn (U U) = Grmn (U):
The structure constants for all these maps in homology and cohomology can be deduced from
those for n = 1 and for m = n = 1. Furthermore the coproduct on EA(Grn (U)) and
the product on EA (Grn (U)) may be deduced from the corresponding structure for C P (U).
These facts, like all our methods, depend crucially on the fact that the group A is abelian,
so that all representations are sums of one dimensional representations.
3. The equivariant Schubert cells of a Grassmannian.
Consider the decomposition of Grn (V ) into Schubert cells, as described for example in [5].
We choose a complete A-invariant ag
0 = V 0 V 1 V 2 V 3 V m = V;
and let i = V i=V i?1 as usual. It is well known that Grn (V ) admits a non-equivariant
CW-structure in which the cells are indexed by sequences of integers
1 1 < 2 < < n m;
the sequence = (1; 2; : : : ; n) is called a Schubert symbol. It is convenient to use to
select a complete ag
0 = V ()0 V ()1 V ()2 V ()3 V ()n
of length n where
V ()i = i :
The cell e() corresponding to the Schubert symbol consists of all n-planes X with
dim(X \ V i ) = i and dim(X \ V i?1) = i ? 1
for i = 1; 2; : : : ; n. Such an n-plane X admits a basis x1; x2; : : : ; xn with xi 2 V i and nonzero in V i =V i?1 . This is usually represented as an m n-matrix with rows x1; x2; : : : ; xn,
and columns indexed by 1; 2; : : : ; m. Dividing xi by its i coordinate, we may assume
1
2
5
the last entry in each row is 1, and then subtracting a suitable multiple of xi from the other
rows we may assume the matrix is in row-reduced echelon form.
Note that since V j is A-invariant, the cell e() is an A-subspace. Accordingly, these
same Schubert cells e() give a decomposition as an equivariant Rep(A)-CW-complex, in
the sense that the cells are unit discs in complex representations of A, and attached to cells
corresponding to proper summands. Indeed, the ith row gives the representation ?i1
(V i =V ()i), so that the cell e() is of Rep(A)-dimension
n
M
i=1
?1
i
(V i =V ()i):
4. Thom complexes and the Schubert filtration.
There is a convenient ltration associated to the Schubert cells, which we shall need to
use. We let Grn (U)[k] be the subcomplex of Grn (U) corresponding to Schubert cells with
1 k + 1; the indexing is chosen since k + 1 is the lowest complex dimension of a cell not
in Grn (U)[k]. The resulting ltration
Grn (U)[0] Grn (U)[1] Grn (U)[2] Grn (U)
will be called the Schubert ltration. Of course the Schubert ltration depends on the
complete ag F; when we use the universe U ? V k , we use the cell structure associated to
the complete ag F=V k .
To obtain sucient naturality we need to interpret the ltration in terms of Thom complexes. For this we let n denote the tautological n-plane bundle over Grn (U) as usual.
The dimensions of the Rep(A)-cells to suggest the plausibility of the following result.
Theorem 4.1. There is a homotopy equivalence
Grn (U)=Grn (U)[k?1] ' (Grn (U ? V k ))Hom( n;V k ):
This may be chosen natural as k varies in the sense that the diagram
Grn (U)=Grn(U)[k?1] '
(Grn (U ? V k ))Hom( VnV;V k )
/
VVVV
VVVVjV
VVVV
VVVV
Grn (hU ? V k ))Hom( n;V k
+
'
hhhh
hhhhh
h
h
h
h '
hhhhh
+1
)
3
(Grn(U ? V k+1 ))Hom( n;V k )
where the left hand vertical is the quotient map, and j is induced by the inclusion V k V k+1 .
Remark 4.2. Since U is a complete A-universe, for any nite dimensional subspace V U,
the inclusion U ? V U induces the stabilization equivalence
Grn (U ? V ) ' Grn(U):
Proof: The proof is by induction on k, using the following two equivalences.
Lemma 4.3. There is an equvialence
Grn (U ? V k )=Grn (U ? V k )[0] ' Grn (U ? V k+1)Hom( n ; k );
Grn (U)=Grn (U)[k]
+1
/
+1
6
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Remark 4.4. In view of the fact that Grn (U ? V k )[0] = Grn?1 (U ? V k+1) the lemma may
be viewed as giving a Gysin co bre sequence
k Grn?1 (U ? V k+1) ?!
Grn (U ? V k ) ?! Grn (U ? V k+1)Hom( n;
+1
k+1 ) :
Lemma 4.5. There is a homeomorphism
Grn (U)[k]=Grn (U)[k?1] = (Grn (U ? V k )[0])Hom( n;V k ):
Using these we may complete the proof. First note that taking k = 0 in 4.3 gives the
base of the induction. Now suppose that Grn (U)=Grn (U)[k?1] has been identi ed as in the
statement of the theorem. We then
calculate
?
?
[
k
]
Grn (U)=Grn (U) = Grn (U)=Grn (U)[k?1] = Grn (U)[k]=Grnk(U)[k?1]
' fGrn (U ? V k )=Grn (U ? V k )[0]gHom( kn;V )
' fGrn (U ? V k+1)Hom( n ; k k )gHom( n;V )
= Grn (U ? V k+1)Hom( n;V );
where the rst equivalence uses the inductive hypothesis and 4.5, and the second uses 4.3.
It therefore remains to prove the two lemmas.
+1
+1
Proof of 4.3: We view the Thom space as the mapping cone of the projection of the unit
sphere bundle. It therefore suces to construct a homotopy commutative square
S (Hom( n; k+1 )) ?! Grn (U ? V k+1 )
'#
#'
k
[0]
Grn (U ? V )
?! Grn (U ? V k ):
The right hand equivalence is the stabilization equivalence induced by the inclusion of complete universes. The right hand vertical S (Hom( n ; k+1)) ?! Grn (U ? V k )[0] is de ned
by (z; X ) 7?! k+1 ker(z), where z is a unit vector in Hom(X; k+1 ). To see it is an
equivalence, we note that the bre over Y is the unit sphere in U ? (V k Y ).
To see the square commutes up to homotopy, note that the two routes round the square can
both be interpreted as sending (z; X ) to the kernel of a suitable map f : k+1 X ?! k+1.
For the lower route f (w; x) = z(x), and for the upper route f (w; x) = w. The interval
joining these two maps consists of nonzero linear maps and therefore provides a homotopy
as required.
Proof of 4.5: For this it is convenient to view the Thom space as a compacti cation of the
vector bundle. We may then de ne a map
Grn (U)[k]=Grn (U)[k?1] ?! (Grn (U ? V k )[0])Hom( n;V k )
by taking a representative n-plane X , expressed in standard form as X = (Xk jX>k ), to the
vector Xk in the bre over X>k . This is clearly a homeomorphism away from the basepoint,
and one may check it is equivariant and that both it and its inverse are continuous at the
basepoint.
This completes the proof of 4.1.
7
5. The spectral sequence of universal Thom complexes.
We now construct a spectral sequence for calculating EA (Grn (U)) using the Schubert cell
ltration
Grn (U)[0] Grn (U)[1] Grn (U)[2] Grn (U)
introduced in Section 4. We think of the superscript k in Grn (U)[k] as a complex dimension, and applying EA () to the ltration we obtain a right half-plane homological spectral
sequence E (1); concentrated in even ltration degrees with
E (1)12p;q = E2Ap+q (Grn (U)[p]; Grn (U)[p?1]) =) E2Ap+q (Grn (U)):
The spectral sequence is indexed so that a Rep(A)-s-cell in ltration 2p contributes to E21p;2s.
There is an analogous spectral sequence for cohomology obtained by applying EA () to the
ltration. It is a right half-plane cohomological spectral sequence concentrated in even
ltration degrees:
E (1)21p;q = EA2p+q (Grn (U)[p]; Grn (U)[p?1]) =) EA2p+q (Grn (U)):
Remark 5.1. In view of the equivalence
Grn (U)=Grn (U)[k?1] ' Grn (U ? V k )Hom( n;V k )
obtained from 4.1 by stabilization, the same spectral sequences may be obtained from the
sequence of maps
Grn (U)0 ?! Grn (U)Hom( n;V ) ?! Grn (U)Hom( n;V ) ?! Grn (U)Hom( n ;V ) ?! : : : :
This point of view is useful for establishing naturality.
Proposition 5.2. If E is a complex oriented cohomology
theory then the homological spectral
sequence E (1); collapses for all n. In fact EA(Grn (U)) is free over EA on the basis of all
products i (F) i (F) in (F) so that 0 i1 i2 in as described in 2.3.
Proof: We prove 2.2 and 5.2 by simultaneous induction on n. The case n = 1 is given by
Cole's Splitting Theorem [3], so we may suppose n 2 and that the results have been proved
for all smaller values. In particular we have Thom isomorphisms for all vector bundles of
dimension n ? 1 (see Section 6 for more details).
We consider the embedding
C P (U) Grn?1 (U) = Gr1 (U) Grn?1 (U) ?! Grn (U U) = Grn(U):
The proof proceeds by constructing a compatible spectral sequence E (2); for calculating
EA (C P (U) Grn?1 (U)). By induction the new spectral sequence E (2); calculates known
groups, and will be shown to collapse. On the other hand the map E (2); ?! E (1); of
spectral sequences will be shown to be surjective on E 1-terms, so it follows that E (1);
collapses as required.
We proceed to construct the new spectral sequence. The analogue of 4.4 is as follows.
1
1
2
2
3
8
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Lemma 5.3. There is a co bre sequence
X (k)+ ?! (C P (U?V k )Grn?1 (U?V k ))+ ?! (C P (U?V k+1)Grn?1 (U?V k+1 ))Hom( n; k
+1
)
where X (k) is a pushout
Grn?2 (U ? V k+1)
k+1
/
Grn?1 (U ? V k )
fc k+1 ;1g
C P (U ? V k ) Grn?2 (U ? V k+1 )
i
X (k);
where the top horizontal is induced by adding k+1 to the (n ? 1)-plane and universe as in
4.4, and the left hand vertical uses the inclusion of the A- xed line k+1 in C P (U ? V k ) in
/
the rst factor.
Proof: Over C P (U ? V k+1) Grn?1 (U ? V k+1 ) the bundle Hom( n ; k+1) is the product
Hom( 1; k+1)Hom( n?1 ; k+1). Thus the unit disc bundle is the product D(Hom( 1; k+1))
D(Hom( n?1 ; k+1 )) and the Thom space is the smash product
C P (U ? V k+1 )Hom( ; k ) ^ Grn?1 (U ? V k+1 )Hom( n? ; k ) :
This identi es the unit sphere bundle S (Hom( 1; k+1) Hom( n?1 ; k+1)) as the pushout
S (Hom( 1; k+1)) S (Hom( n?1 ; k+1))
S (Hom( 1; k+1)) D(Hom( n?1 ; k+1))
1
1
+1
+1
/
D(Hom( 1; k+1)) S (Hom( n?1;
The description of X (k) follows.
i
k+1 ))
/
S (Hom( 1;
k+1 ) Hom( n?1 ; k+1 )):
Corollary 5.4. There is a co bre sequence
X (k)Hom( n;V k ) ?!
(C P (U?V k )Grn?1 (U?V k ))Hom( n;V k ) ?! (C P (U?V k+1)Grn?1 (U?V k+1)Hom( n ;V k ):
+1
In view of the inductive hypothesis we know the homologies of all the spaces involved in the
co bre sequence of 5.3, and we have a basis corresponding to a Rep(A)-CW-decomposition
(where the correspondence depends on the orientation). We return to analysis of the maps
and the bases below.
Corollary 5.5. The map
C P (U ? V k+1 ) Grn?1 (U ? V k+1 ) ?! Grn (U ? V k+1 )
induces a retraction
r : X (k) = S (Hom( 1; k+1) Hom( n?1 ; k+1 )) ?! S (Hom( n ; k+1)) ' Grn?1 (U ? V k )
on unit sphere bundles.
9
Proof: The inclusion corresponding to the retraction r is the composite
i
Grn?1 (U ? V k ) ' S (Hom( 1; k+1 )) D(Hom( n?1 ; k+1)) ?!
S (Hom( 1; k+1) Hom( n?1 ; k+1)) = X (k):
It is easily checked that with the equivalence S (Hom( n ; k+1)) ' Grn?1 (U ? V k ) speci ed
in 4.3, we have ri = 1.
To guarantee a map of spectral sequences E (2); ?! E (1); converging to the map
EA(C P (U) Grn?1 (U)) ?! EA (Grn(U));
the spectral sequence for EA(C P (U) Grn?1 (U)) is constructed from the sequence of maps
(C P (U) Grn?1 (U))0 ?! (C P (U) Grn?1 (U))Hom( n;V ) ?! (C P (U) Grn?1 (U))Hom( n;V ) ?! as in 5.1. However, it may help if we relate this to a construction more closely analogous to
the construction of E (1); from the Schubert ltration. Note that over C P (U) Grn?1 (U)
the bundle Hom( n ; V k ) is Hom( 1; V k ) Hom( n?1 ; V k ), and hence
(C P (U) Grn?1 (U))Hom( n;V k ) ' C P (U)Hom( ;V k ) ^ Grn?1 (U)Hom( n? ;V k ):
We therefore de ne
(C P (U) Grn?1 (U))[k] := C P (U) (Grn?1 (U)[k]) [ (C P (U)[k] ) Grn?1 (U);
so as to ensure a co bre sequence
(C P (U)Grn?1 (U))[k?1] ?! C P (U)Grn?1 (U) ?! (C P (U?V k )Grn?1 (U?V k ))Hom( n;V k ):
Applying EA () to the resulting ltration we obtain a right half-plane homological spectral
sequence with
1
E (2)2p;q = E2Ap+q ((C P (U)Grn?1 (U))[p]; (C P (U)Grn?1 (U))[p?1]) =) E2Ap+q (C P (U)Grn?1 (U)):
By induction this spectral sequence is under complete control.
Proposition 5.6. If EA () is a complex
oriented cohomology theory then the spectral se1
quence E (2); above collapses at E (2); .
Proof: This follows from the inductive hypothesis, since E ^ Grn?1(U)+ splits as a wedge of
suspensions of E indexed by the Schubert cells of Grn?1 (U)+ . Smashing this with C P (U)+ ,
we again get splitting from Cole's Splitting Theorem [3]. The ltration of the new spectral
sequence is compatible with that of the splitting in view of the smash product decomposition
of the Thom spaces. The collapse of the spectral sequence follows.
1
2
1
1
To see that the map E (2)1; ?! E (1)1; is surjective, recall that
Grn (U)[k]=Grn (U)[k?1] ' Grn?1 (U ? V k )Hom( n;V k )
and
(C P (U) Grn?1 (U))[k]=(C P (U ? V k ) Grn?1 (U ? V k ))[k?1] ' X (k)Hom( n;V k ):
10
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
It therefore suces to show that for each k, the map
X (k)Hom( n;V k ) ?! Grn?1 (U ? V k )Hom( n;V k )
of subquotients is surjective in EA(). This follows for the unThomi ed map, since by 5.5 the
map r : X (k) ?! Grn?1 (U ? V k ) is a retraction. For k = 0 this is the required statement.
For k 1 we use the fact that over C P (U ? V k ) Grn?1 (U ? V k ) the bundle n splits as
the product 1 n?1 . Thus Hom( n ; V k ) splits as a sum of bundles of dimension n ? 1
for each of which we have a Thom isomorphism by induction. Surjectivity now follows from
that of the unThomi ed map for the universe U ? V k .
Finally, we may remark that the analysis identi es a basis. We may visualize the basis
of the homology (C P (U))n as the points of Nn , with the complex dimension of the cell
corresponding to i = (i1; i2; : : : ; in) being jij = i1 + i2 + + in. The homology of all other
relevant spaces have bases given by the images of these basis elements. These correspond
to subsets of Nn as follows. The basis for the space C P (U) = C P (U) corresponds to
the points (i1; 0; : : : ; 0), and that for Grn?1 (U) = Grn?1 (U) corresponds to the points
(0; i2; : : : ; in) with 0 i2 i3 in. The basis for the subspace Grn?2 (U) Grn?1 (U)
corresponds to the points (0; 0; i3; : : : ; in) with 0 i3 i4 in.
Applying this discussion to the universe U ? V k , we see that X (k) has homology in even
degrees, with basis corresponding to the points (i1; i2; : : : ; in) either (i) with i1 = 0 and
0 i2 i3 in or (ii) with i1 arbitrary, i2 = 0 and 0 i3 i4 in . Since the
map i occurs in the pushout description of X (k) we see that r : X (k) ?! Grn?1 (U ? V k )
maps the subspace on the basis elements elements corresponding to 0 i2 i3 in
isomorphically to the homology of Grn?1 (U ? V k ). The e ect of the Thom isomorphism is to
replace standard homology generators relative to the ag F=V k by those for F: the precise
meaning becomes clear from the special case C P (U).
Lemma 5.7.
Naming the cohomology EA (C P (U ? V k )Hom( ;V k )) by the equivalence C P (U ?
V k )Hom( ;V k ) ' C P (U)=C P (V k ), thekelement y(V k ) is a Thom class. The resulting Thom
isomorphisms EA (C P (U?V k )Hom( ;V )) =
= EA (C P (U?V k ) and EA (C P (U?V k )Hom( ;V k )) EA (C P (U ? V k ) are given by
y(V k+l =V k ) 7?! y(V k )y(V k+l=V k ) = y(V k+l )
in cohomology and
k
l (F=V ) ? k+l (F)
in homology for l 0.
1
1
1
1
Now we return to nd a basis for EA (Grn(U)) itself. By induction we know that the
elements i (F) i (F) i (F) in (F) with i1 arbitrary and 0 i2 i3 in
give a basis of EA(C P (U) Grn?1 (U)). The proof of this shows that the ltration is such
that the basis of E (2)12p; is given by the images of the elements either
(i)p with p = i1 i2 i3 in or
(ii)p with i1 p and p = i2 i3 in.
Since r is split by the map i from the pushout description of X (k), the induced map r
takes the basis elements (i)p to give a basis of E (1)22p;. Since the spectral sequences collapse at E 1, and since the E 1 terms are free over EA , it follows that the basis elements of
1
2
3
11
EA (C P (U) Grn?1 (U)) giving rise to the basis elements (i)p for some p (namely those with
i1 i2 in) map to a basis of EA (Grn (U)) as required.
6. Thom classes.
It follows from what we have proved that any complex oriented cohomology theory has
Thom classes for arbitrary bundles. It is the purpose of the present section to make this
explicit: we prove Theorem 1.1.
Following Okonek [6] we say that EA () has Thom classes if for any complex vector bundle
over a space X , there is an element () 2 EA (X ) so that this system of elements is
1. natural for bundle maps,
2. product preserving in the sense that ( ) = () ^ () and
3. normalized so that if X = A=B and the bre of over B is the representation W of B
then () = V (1)
Remark 6.1. (i) Okonek only requires the normalization when B = A.
(ii) To obtain Thom isomorphisms we would only require the normalization that () was
a unit multiple of W (1). However, we show that any complex oriented cohomology theory
has Thom classes in the present stricter sense.
To prove that a system of Thom classes gives Thom isomorphisms
= ~ EA (X ) ?!
EA(X )
(essentially by cup product with ()) or
=
EA (X )
E~A(X ) ?!
(essentially by cap product with ()) we proceed from the fact that V is an isomorphism.
We may then work our way up the cellular ltration of X using the normalization condition.
Alternatively, if has a complement so that = V we may use the product preserving
property.
Before proceeding we make a reduction.
Lemma 6.2. A cohomology theory EA () has a system of Thom classes if and only if it has
universal Thom classes,
n 2 E~A (Grn (U) n )
for each n, so that
1. the system is product preserving in the sense that under the direct sum map Grm (U) Grn(U) ?! Grm+n (U), the class m+n pulls back to m ^ n
2. Over the point W of Grn (U) with isotropy B A the class n restricts to W (1);
We may then de ne the Thom class for an n-plane bundle by pulling back n under the
Thomi cation of its classifying map.
Theorem 6.3. If E is a complex oriented theory with orientation in cohomological degree 2
and complex stable structure determined by the orientation, there is an associated system of
Thom classes.
12
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Proof:
We shall construct classes n as speci ed in 6.2. Assume that the ag begins with
1
V = . By pullback along conjugation, we see that
Grn (U) n = Grn (U) n ' Grn (U)=Grn (U)[0];
and we have identi ed the cohomology of this space exactly. From the orientation we construct the cohomology class y() 2 EA (C P (U)), and hence by 2.2, the element y() n 2
EA (Grn (U)). By construction this restricts to zero in EA (Grn (U)[0]), and thus we obtain
n = y() n 2 EA (Grn (U); Grn(U)[0]) = E~A (Grn (U) n ):
The element n is identi ed with a map
n : Grn (U) n ?! E:
Lemma 6.4. The classes n are compatible under product in the sense that
Grm (U) m ^ Grn (U) n m^n E ^ E
/
Grm+n (U) m
n
commutes up to homotopy.
Proof: This follows from the fact that y()
E
+
m+n
m
/
y() n = y()
(m+n) .
Lemma 6.5. The class n is a Thom class for n .
Proof: An A- xed point of Grn (U) is a representation V = 1n 2 n, and such a
point lies in the image of the direct sum map from (C P (U)) . By naturality it therefore
suces to deal with the case n = 1. However, by de nition of an orientation 1 = y()
restricts to a generator of EA (C P ( ); C P ()). Since we have used the orientation to give
the complex stable structure, the generator is ? (1) as required.
For the bre over a non- xed point we use the fact that orientations behave well under
restriction to subgroups.
1
7. Universality of complex cobordism.
Let us turn now to the spectrum level statements, and the connection with maps MU ?!
E : we prove Theorem 1.2.
First we deduce that the existence of a ring map MU ?! E gives E a complex orientation.
This is not quite obvious because of the distinction between a map of spectra and a map
preserving the complex stable structure.
Lemma 7.1. If there is a ring map : MU ?! F of A-spectra, then F is complex orientable, and the image of the canonical orientation of MU is an orientation of F .
Remark 7.2. The proof makes no special use is made of the fact that the domain is MU
or the fact that we use its canonical orientation. However our results show that the general
case follows from this one.
13
~ (C P (U)) denote an orientation of MU . By hypothesis the restricProof: Let x() 2 MU
?
~ (S ) is a generator as an MUA -module. From the suspension
tion ( ?1 ) of x() to MU
~ ? ? , and ( ?1 ) is a unit in the RO(G)-graded coecient
~ (S ? ) isomorphism MU
= MU
ring. Hence (( ?1 )) is also a unit in the RO(G)-graded? sense, and therefore (x())
restricts to a generator of the integer graded module E~A (S ) as required.
1
1
1
1
In Section 6 we showed that any complex oriented theory has Thom classes for arbitrary
bundles. To complete the proof of 1.2 it therefore remains only to construct a map MU ?! E
from a system of Thom classes.
First recall that MU is the prespectrum given on a subspace V U of dimension n by
MU (V ) = Grn (V U) n :
If U V with U of dimension m, the structure map
S V ?U ^ MU (U ) = Grm (U U)(V ?U ) m ?! Grn (V U) n
is given by Thomifying the map adding V ? U . Thus
?V Gr (V U) n ;
MU ' holim
n
! V
and the cohomology of MU may be calculated from that of Thom spaces of Grassmannians
by the Milnor exact sequence.
It is useful to be more explicit about the homology.
Lemma 7.3.
Provided the complex stable structure is de ned by the orientation, the structure map V ?U Grn (U U) m ?! Grn (V U) n , takes the element
V ?U m ( i (F) i (F) im (F))
to
n( 0(F) 0(F) i (F)) i (F) im (F)):
1
2
1
2
Proposition 7.4. The classes n assemble to give a unique map
: MU ?! E
of ring spectra.
Proof: Since the structure maps are surjective in cohomology,
EA (MU ) = lim EA (?V Grn (U) n ):
V
The existence of the map therefore follows from compatibility of the elements n under
suspension.
Lemma 7.5. The classes n are compatible under suspension in the sense that
V ?U Grm (U U) m
V ?U m
Grn (V U) n
n
commutes up to homotopy.
/
E
E
/
14
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Proof: First note that the space of isometric isomorphisms U ?! U is contractible, so the
particular identi cation V U = U is not important. Now, since the structure maps in MU
arise from bundles, the lemma follows from 6.4.
Since the smash product MU ^ MU ?! MU is induced by the Thomi cation of the map
classifying direct sum of bundles, the compatibility of the elements n under products shows
that the map is a map of ring spectra. This completes the proof of 1.2.
8. Homology and cohomology of MU .
In this section we give an account of the relationship between orientations and ring maps
MU ?! E which makes no explicit reference to the orientation. The main new result is
Theorem 1.3.
First we need to calculate the homology of MU . We use the notation of Section 7. Of
course we have
A (?V n MU (V n )) = lim E A (?V n Gr (V n U) n ):
EA (MU ) = lim
E
n
!n
!n Thus we may construct elements using the maps
= A ?V n
EA(Grn (U)) ?!
E ( Grn (U) n ) = EA (?V n MU (V n)) ?! EA (MU ):
We identi ed the e ect of the maps in the direct system in 7.3. It is natural to suppress the
terms (V 0).
De nition 8.1. We write bi(F) for the image of i+1(F) in EA(MU ). It follows that the
image of i +1(F) i +1(F) in+1(F) is bi (F)bi (F) bin (F). In the nonequivariant
context, it is standard to write i+1 for i+1(F) and bi for bi(F).
Note that Whitney sum of bundles gives a product MU ^ MU ?! MU , and hence
A
E (MU ) is an EA -algebra. The simplicity of the following theorem is somewhat surprising.
Theorem
8.2. If EA () is a complex oriented theory, the homology of MU is polynomial
over EA:
EA (MU ) = EA [b1(F); b2(F); b3(F); ]:
Furthermore there is a Kunneth theorem in the sense that EA (MU ^s ) = EA (MU ) s .
Proof: From our construction of the generators of the homology of Grassmannians, their
behaviour under Whitney sum is obvious. The result for MU follows from the relation
( ) \ (
) = ( \ ) ( \ ).
1
2
1
2
It is now straightforward to deduce the global statements we require. The additive part is
no problem since the homology of MU is free over EA , and we have a Kunneth isomorphism.
Corollary 8.3. If E is complex orientable, passage to E -homology gives
[MU ^s; E ]A = EA-mod(EA(MU ^s ); EA):
15
We really want to understand the set RingA [MU; E ] of homotopy classes of ring maps
MU ?! E . Since EA (MU ) is a free EA-algebra, this too is immediate.
Corollary 8.4. If E is complex orientable, passage to E -homology gives
Ring[MU; E ]A = EA-alg(EA (MU ); EA ):
Combining this with 7.4 and being careful about normalization, we obtain a useful consequence.
Corollary 8.5. If x() is a complex orientation
of EA (), then there is a natural bijective
correspondence between orientations of EA () in cohomological degree 2 which give the same
complex stable structure as x() and
Ring[MU; E ]A = EA -alg(EA(MU ); EA ):
More precisely, suppose x0() is an orientation in cohomological degree 2 giving the same
complex stable structure as x(). If V 1 = and the associated parameter is
y0() = ii y(V i)
then 0 = 0 and 1 = 1. The associated algebra homomorphism
EA (MU ) = EA [b1(F); b2(F); b3(F); : : : ] ?! EA
is then determined by
bi(F) 7?! i+1
for i 1.
Proof: In general,
if x0() is an orientation of E , then the associated map MU ?! E
restricts to x0() n as a map
?2n((C P (U))n ) n ?! ?2n (Grn (U)) n ?! E:
Thus the algebra homomorphism associated to x0() may be calculated on an element z
arising from ?2nMU (V ) as x0() n(z).
Now suppose x0() is expressed in terms of the basis associated to the orientation x().
The coecient 0 = 0 since y0() restricts?trivially to C P (). The coecient 1 is obtained
by restricting to (C P (V 2); C P (V 1)) = S ; since the complex stable structures de ned by
x() and x0() agree, the coecient is 1. It is easy to see the images of bi(F) for i 1 are as
speci ed since they arise from ?V MU (V 1).
2
1
1
Remark 8.6. (i) For a complex stable structure arising from an orientation,
the element
(1) for any chosen determines the rest, because of the action of A . This explains why
consideration of ?2 1 alone was enough to show 1 = 1.
(ii) The statement of the corollary is consistent with the conjecture that MUA MU classi es
strict isomorphisms of A-equivariant formal group laws.
16
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
[1]
[2]
[3]
[4]
[5]
[6]
References
J.F.Adams \Stable homotopy and generalized homology." Chicago University Press (1974).
M.Cole \Complex oriented RO(G)-graded equivariant cohomology theories and their formal group laws."
Thesis, University of Chicago (1996)
M.Cole \Complex oriented equivariant cohomology theories and equivariant complex projective spaces"
(In preparation)
Michael Cole, J.P.C.Greenlees and I.Kriz \Equivariant formal group laws." Submitted for publication
(1997) pp 29
J.W.Milnor and J.Stashe \Characteristic classes." Princeton University Press (1974)
C.Okonek \Der Conner-Floyd-Isomorphismus fur Abelsche Gruppen." Math. Z. 179 (1982) 201-212.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-0001
E-mail address :
mmcole@math.lsa.umich.edu
School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address :
j.greenlees@sheffield.ac.uk
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003
E-mail address :
ikriz@math.lsa.umich.edu