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-equiva*
*riant complex
vector bundles are orientable over a complex orientable equivariant cohom*
*ology 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-equivaria*
*nt coho-
mology theory E*A(.) is orientable if complex line bundles are well behaved. Mo*
*re precisely,
we let CP (U) denote the space of lines in a complete complex A-universe, and f*
*fl denote
the trivial representation. We say that E*A(.) is a complex stable ring theory*
* if there are
suspension isomorphisms
~= n+|V |V
oeV : "EnA(X) -! "EA (S ^ X)
for all complex representations V , where SV is the one-point compactification *
*of V , and |V |
is the real dimension of V ; these are required to be transitive, and given by *
*multiplication
with a generator of "E|V(|SV ). We say the theory is complex orientable if in a*
*ddition, there
is a cohomology class x(ffl) 2 E*A(CP (U); CP (ffl)) which restricts to a gener*
*ator of
-1
E*A(CP (ff ffl); CP (ffl)) ~="E*A(Sff )
for all one dimensional representations ff. Thus if the complex orientation is *
*in cohomological
degree 2, it determines a complex stable structure. Many important theories ar*
*e complex
orientable, for instance equivariant K-theory, tom Dieck's equivariant bordism *
*MU*A(.), and
Borel cohomology for non-equivariantly complex orientable theories.
The purpose of this article is to show that this good behaviour is sufficient*
* to ensure good
behaviour of complex vector bundles of any dimension. In particular all comple*
*x 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 MU*A(.) is universal for cohomolog*
*y theories
with Thom classes. Combined with our main result this implies the following uni*
*versality
statement.
Theorem 1.2. If E*A(.) is a complex oriented cohomology theory with orientati*
*on in coho-
mological degree 2 then there is a unique ring map MU -! E of A-spectra under w*
*hich the
orientation of E is the image of the canonical orientation. Conversely, a map M*
*U -! E
of ring A-spectra endows E with the structure of a complex oriented cohomology *
*theory with
orientation in cohomological degree 2.
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 u*
*seful we
have the following calculation.
Theorem 1.3. If E*A(.) is a complex orientable theory then EA*(MU) is a polyn*
*omial E*A-
algebra and
RingA(MU; E) = EA*-alg(EA*(MU); E*A):
This is proved in Section 8. We shall be more specific about the polynomial g*
*enerators
in due course. The good behaviour of EA*(MU) should be contrasted with the fac*
*ts that
E*A(CP (U)) is not a power series ring in general, and MU*Ais 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 import*
*ant 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 filtrations are not so simple *
*equivariantly.
Accordingly, geometric arguments are required as substitutes for the use of the*
* Atiyah-
Hirzebruch 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 ) = CP (V ).
Lemma 2.1. The A-space Grn(U) is a classifying space for A-equivariant comple*
*x 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
(CP (U))xn = Gr1(U)xn -! Grn(Un ) ~=Grn(U);
where the final homeomorphism arises from an isometric isomorphism Un ~=U. Thi*
*s sum
map induces maps
EA*(CP (U))n = EA*((CP (U))xn) -! EA*(Grn(U))
and
E*A(Grn(U)) -! E*A((CP (U))xn) = E*A(CP (U))^n;
where the completed tensor producte ^ refers to the skeletal filtration topolo*
*gy. The
K"unneth theorems implicit in this statement are corollaries of Cole's Splittin*
*g 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 invari*
*ants for the
symmetric group n.
Theorem 2.2. For any complex orientable cohomology theory E*A(.), the direct *
*sum of lines
induces isomorphisms
EA*(Grn(U)) ~={EA*(CP (U))n }n
and
E*A(Grn(U)) ~={E*A(CP (U))^n}n :
3
Theorem 2.2 will be proved in Section 5 below. We pause to remark that this *
*gives us
specific generators, and hence all the structure of the homology and cohomology*
* of Grass-
mannians follows from that of CP (U) made explicit in [4].
Indeed, Cole showed that EA*(CP (U)) is additively free over E*Aand E*A(CP (U*
*)) is a
product of suspensions of E*A. Furthermore, he showed how an orientation x(ffl)*
* of E, together
with a complete flag
F = 0 = V 0 V 1 V 2 . . .
in U determines a topological basis 1 = y(V 0); y(V 1); y(V 2); : :o:f E*A(CP (*
*U)), and we
may let fi0(F); fi1(F); fi2(F); : :d:enote the dual basis of EA*(CP (U)). The n*
*otation for the
homology generators reflects the fact that fii(F) depends on the initial segmen*
*t V 0 V 1
. . .V iof the flag. Furthermore, a K"unneth theorem holds for the homology or *
*cohomology
of products of CP (U).
Lemma 2.3. An E*A-basis of the coinvariants {EA*(CP (U))n }n is given by the*
* images of
all products fii1(F) fii2(F) . . .fiin(F) so that 0 i1 i2 . . .in. A topo*
*logical
E*A-basis of the invariants {E*A(CP (U))^n}n corresponds to the collection of *
*sequences 0
i1 i2 . . .in; the basis elements are the symmetric sums
0y(V ioe(1)) y(V ioe(2)) . . .y(V ioe(n))
where 0denotes the sum over the orbit of (i1; i2; : :;:in).
Proof: The result is clear once we remark that the n action arises from an acti*
*on on the
basis of the homology or cohomology of (CP (U))xn. In the case of cohomology, w*
*e pass_to_
limits from the case of CP (V )xn. *
* |__|
Corollary 2.4. If E is a complex oriented cohomology theory then E ^ Grn(U) spl*
*its as a
wedge of copies of E indexed by sequences 0 i1 i2 . . .in:
_
E ^ Grn(U) ' 2|i|E;
i
where |i| = 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 : 2|i|E -! E ^ Grn(U)
i
of A-spectra, using the product on E.
By construction it induces an isomorphism in ssA*(.). Now suppose B A, and c*
*onsider
the B-equivariant situation. The restriction map E*A(CP (U)) -! E*B(CP (U)) tak*
*es an A-
orientation to a B-orientation, and the resulting basis corresponding to a comp*
*lete flag to
the basis corresponding to the same flag regarded B-equivariantly. It therefore*
* follows that
the restriction E*A(Grn(U)) -! E*B(Grn(U)) takes the sequence basis to another *
*basis. Thus
s induces an isomorphism of ssB*(.). 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 specific 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)) = EA*{fi(V i1) fi(V i2) . . .fi(V in) | 0 i1 i2 . . .in}
and
E*A(Grn(U)) = EA*-mod(EA*(Grn(U)); EA*) *
* __
= E*A{{0y(V ioe(1)) y(V ioe(2)) . . .y(V ioe(n)) | 0 i1 i2 *
*. . .in}}: |__|
Finally we may consider the extra structure on the system
{EA*(Grn(U))}n0 or{E*A(Grn(U))}n0 :
The relevant extra structure is given by the conjugation map
_.: Gr __ ~
n(U) -! Grn(U ) = Grn(U);
the action maps
ff : Grn(U) -! Grn(U ff) ~=Grn(U)
for ff 2 A*, the direct sum maps
: Grm (U) x Grn(U) -! Grm+n (U U) ~=Grm+n (U)
and the tensor product maps
: Grm (U) x Grn(U) -! Grmn(U U) ~=Grmn(U):
The structure constants for all these maps in homology and cohomology can be de*
*duced from
those ff for n = 1 and for m = n = 1. Furthermore the coproduct on EA*(Grn(U))*
* and
the product on E*A(Grn(U)) may be deduced from the corresponding structure for *
*CP (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 e*
*xample in [5].
We choose a complete A-invariant flag
0 = V 0 V 1 V 2 V 3 . . .V m = V;
and let ffi = V i=V i-1as usual. It is well known that Grn(V ) admits a non-eq*
*uivariant
CW-structure in which the cells are indexed by sequences of integers
1 oe1 < oe2 < . .<.oen m;
the sequence oe = (oe1; oe2; : :;:oen) is called a Schubert symbol. It is conve*
*nient to use oe to
select a complete flag
0 = V (oe)0 V (oe)1 V (oe)2 V (oe)3 . . .V (oe)n
of length n where
V (oe)i= ffoe1 ffoe2 . . .ffoei:
The cell e(oe) corresponding to the Schubert symbol oe consists of all n-planes*
* X with
dim(X \ V oei) = i and dim(X \ V oei-1) = i - 1
for i = 1; 2; : :;:n. Such an n-plane X admits a basis x1; x2; : :;:xn with xi2*
* V oeiand non-
zero in V oei=V oei-1. This is usually represented as an m x n-matrix with rows*
* x1; x2; : :;:xn,
and columns indexed by ff1; ff2; : :;:ffm . Dividing xi by its ffoeicoordinate,*
* we may assume
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 jis A-invariant, the cell e(oe) is an A-subspace. Accordin*
*gly, these
same Schubert cells e(oe) give a decomposition as an equivariant Rep(A)-CW-comp*
*lex, in
the sense that the cells are unit discs in complex representations of A, and at*
*tached to cells
corresponding to proper summands. Indeed, the ith row gives the representation*
* ff-1oei
(V oei=V (oe)i), so that the cell e(oe) is of Rep(A)-dimension
Mn
ff-1oei (V oei=V (oe)i):
i=1
4.Thom complexes and the Schubert filtration.
There is a convenient filtration associated to the Schubert cells, which we s*
*hall need to
use. We let Grn(U)[k]be the subcomplex of Grn(U) corresponding to Schubert cel*
*ls with
oe1 k + 1; the indexing is chosen since k + 1 is the lowest complex dimension *
*of a cell not
in Grn(U)[k]. The resulting filtration
Grn(U)[0] Grn(U)[1] Grn(U)[2] . . .Grn(U)
will be called the Schubert filtration. Of course the Schubert filtration depe*
*nds on the
complete flag F; when we use the universe U - V k, we use the cell structure as*
*sociated to
the complete flag F=V k.
To obtain sufficient naturality we need to interpret the filtration in terms *
*of Thom com-
plexes. For this we let fln denote the tautological n-plane bundle over Grn(U) *
*as usual.
The dimensions of the Rep(A)-cells to suggest the plausibility of the followi*
*ng result.
Theorem 4.1. There is a homotopy equivalence
k)
Grn(U)=Grn(U)[k-1]' (Grn(U - V k))Hom(fln;V:
This may be chosen natural as k varies in the sense that the diagram
' k Hom(fl ;V k)
Grn(U)=Grn(U)[k-1]______//(Grn(U - V )) n
| VVVVV
| VVVjVVV
| VVVVVV
| VV++
| k Hom(fln;V *
*k+1)
| Grn(U - V ))
| hhh33h
| hhhhhh
| hhhhh'h
fflffl| ' hhh
Grn(U)=Grn(U)[k]_____//_(Grn(U - V k+1))Hom(fln;V k+1)
where the left hand vertical is the quotient map, and j is induced by the inclu*
*sion V k V k+1.
Remark 4.2. Since U is a complete A-universe, for any finite dimensional subs*
*pace 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(fln;ffk+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 le*
*mma may
be viewed as giving a Gysin cofibre sequence
ffk+1 k k+1Hom(fl ;ff ) __
Grn-1(U - V k+1) - ! Grn(U - V ) -! Grn(U - V ) n k+1: |__|
Lemma 4.5. There is a homeomorphism
k)
Grn(U)[k]=Grn(U)[k-1]~=(Grn(U - V k)[0])Hom(fln;V:
Using these we may complete the proof. First note that taking k = 0 in 4.3 g*
*ives the
base of the induction. Now suppose that Grn(U)=Grn(U)[k-1]has been identified a*
*s in the
statement of the theorem. We then calculate
Grn(U)=Grn(U)[k] = Grn(U)=Grn(U)[k-1] = Grn(U)[k]=Grn(U)[k-1]
k)
' {Grn(U - V k)=Grn(U - V k)[0]}Hom(fln;V
k)
' {Grn(U - V k+1)Hom(fln;ffk+1)}Hom(fln;V
~= Grn(U - V k+1)Hom(fln;V k+1);
where the first equivalence uses the inductive hypothesis and 4.5, and the seco*
*nd uses 4.3.
It therefore remains to prove the two lemmas.
Proof of 4.3: We view the Thom space as the mapping cone of the projection of t*
*he unit
sphere bundle. It therefore suffices to construct a homotopy commutative square
S(Hom (fln; ffk+1))-! Grn(U - V k+1)
'# #'
Grn(U - V k)[0] -! Grn(U - V k):
The right hand equivalence is the stabilization equivalence induced by the incl*
*usion of com-
plete universes. The right hand vertical S(Hom (fln; ffk+1)) -! Grn(U - V k)[0*
*]is defined
by (z; X) 7-! ffk+1 ker(z), where z is a unit vector in Hom (X; ffk+1). To se*
*e it is an
equivalence, we note that the fibre 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 : ffk+1*
* X -! ffk+1.
For the lower route f(w; x) = z(x), and for the upper route f(w; x) = w. The i*
*nterval
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 compactific*
*ation of the
vector bundle. We may then define a map
k)
Grn(U)[k]=Grn(U)[k-1]-! (Grn(U - V k)[0])Hom(fln;V
by taking a representative n-plane X, expressed in standard form as X = (Xk |X>*
*k), to the
vector Xk in the fibre 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 contin*
*uous_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 Sc*
*hubert cell
filtration
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 comple*
*x dimen-
sion, and applying EA*(.) to the filtration we obtain a right half-plane homolo*
*gical spectral
sequence E(1)**;*concentrated in even filtration degrees with
E(1)12p;q= EA2p+q(Grn(U)[p]; Grn(U)[p-1]) =) EA2p+q(Grn(U)):
The spectral sequence is indexed so that a Rep(A)-s-cell in filtration 2p contr*
*ibutes to E12p;2s.
There is an analogous spectral sequence for cohomology obtained by applying E*A*
*(.) to the
filtration. It is a right half-plane cohomological spectral sequence concentra*
*ted in even
filtration degrees:
E(1)2p;q1= E2p+qA(Grn(U)[p]; Grn(U)[p-1]) =) E2p+qA(Grn(U)):
Remark 5.1. In view of the equivalence
k)
Grn(U)=Grn(U)[k-1]' Grn(U - V k)Hom(fln;V
obtained from 4.1 by stabilization, the same spectral sequences may be obtained*
* from the
sequence of maps
1) Hom(fln;V 2) Hom(fln;V 3)
Grn(U)0 -! Grn(U)Hom(fln;V-! Grn(U) -! Grn(U) -! : :::
This point of view is useful for establishing naturality.
Proposition 5.2. If E is a complex oriented cohomology theory then the homologi*
*cal spectral
sequence E(1)**;*collapses for all n. In fact EA*(Grn(U)) is free over E*Aon th*
*e basis of all
products fii1(F) fii2(F) . . .fiin(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 g*
*iven 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
CP (U) x Grn-1(U) = Gr1(U) x Grn-1(U) -! Grn(U U) ~=Grn(U):
The proof proceeds by constructing a compatible spectral sequence E(2)**;*for c*
*alculating
EA*(CP (U) x Grn-1(U)). By induction the new spectral sequence E(2)**;*calculat*
*es 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 E1-terms, so it follows th*
*at E(1)**;*
collapses as required.
We proceed to construct the new spectral sequence. The analogue of 4.4 is as *
*follows.
8 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Lemma 5.3. There is a cofibre sequence
X(k)+ -! (CP (U-V k)xGrn-1(U-V k))+ -! (CP (U-V k+1)xGrn-1(U-V k+1))Hom(fln;ffk*
*+1)
where X(k) is a pushout
ffk+1
Grn-2(U - V k+1)___________//Grn-1(U - V k)
{cffk+1;1}|| |i|
fflffl| fflffl|
CP (U - V k) x Grn-2(U - V k+1)______//_X(k);
where the top horizontal is induced by adding ffk+1 to the (n - 1)-plane and un*
*iverse as in
4.4, and the left hand vertical uses the inclusion of the A-fixed line ffk+1 in*
* CP (U - V k) in
the first factor.
Proof: Over CP (U - V k+1) x Grn-1(U - V k+1) the bundle Hom (fln; ffk+1) is th*
*e product
Hom (fl1; ffk+1)xHom (fln-1; ffk+1). Thus the unit disc bundle is the product D*
*(Hom (fl1; ffk+1))x
D(Hom (fln-1; ffk+1)) and the Thom space is the smash product
CP (U - V k+1)Hom(fl1;ffk+1)^ Grn-1(U - V k+1)Hom(fln-1;ffk+1):
This identifies the unit sphere bundle S(Hom (fl1; ffk+1) x Hom (fln-1; ffk+1))*
* as the pushout
S(Hom (fl1; ffk+1)) x S(Hom (fln-1; ffk+1))//_S(Hom (fl1; ffk+1)) x D(Hom (fl*
*n-1; ffk+1))
| |
| i|
fflffl| fflffl|
D(Hom (fl1; ffk+1)) x S(Hom (fln-1; ffk+1))//_S(Hom (fl1; ffk+1) x Hom (fln-1*
*; ffk+1)):
*
* __
The description of X(k) follows. *
* |__|
Corollary 5.4. There is a cofibre sequence
k)
X(k)Hom(fln;V-!
k) k+1 k+1Hom(fln;V k+1)__
(CP (U-V k)xGrn-1(U-V k))Hom(fln;V-! (CP (U-V )xGrn-1(U-V ) : |__|
In view of the inductive hypothesis we know the homologies of all the spaces *
*involved in the
cofibre sequence of 5.3, and we have a basis corresponding to a Rep(A)-CW-decom*
*position
(where the correspondence depends on the orientation). We return to analysis of*
* the maps
and the bases below.
Corollary 5.5. The map
CP (U - V k+1) x Grn-1(U - V k+1) -! Grn(U - V k+1)
induces a retraction
r : X(k) = S(Hom (fl1; ffk+1) x Hom (fln-1; ffk+1)) -! S(Hom (fln; ffk+1)) ' G*
*rn-1(U - V k)
on unit sphere bundles.
*
* 9
Proof: The inclusion corresponding to the retraction r is the composite
Grn-1(U - V k) ' S(Hom (fl1; ffk+1)) x D(Hom (fln-1; ffk+1)) -i!
S(Hom (fl1; ffk+1) x Hom (fln-1; f*
*fk+1)) = X(k):
It is easily checked that with the equivalence S(Hom (fln; ffk+1)) ' Grn*
*-1(U - V k)_specified_
in 4.3, we have ri = 1. *
* |__|
To guarantee a map of spectral sequences E(2)**;*-! E(1)**;*converging*
* to the map
EA*(CP (U) x Grn-1(U)) -! EA*(Grn(U));
the spectral sequence for EA*(CP (U) x Grn-1(U)) is constructed from the*
* sequence of maps
1) Hom(fln;V 2)
(CP (U)xGrn-1(U))0 -! (CP (U)xGrn-1(U))Hom(fln;V-! (CP (U)xGrn-1(U)) -!*
* . . .
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 filtration. Note that over*
* CP (U) x Grn-1(U)
the bundle Hom (fln; V k) is Hom (fl1; V k) x Hom (fln-1; V k), and hence
k) Hom(fl ;V k) Hom(fl *
*;V k)
(CP (U) x Grn-1(U))Hom(fln;V' CP (U) 1 ^ Grn-1(U) n-1*
* :
We therefore define
(CP (U) x Grn-1(U))[k]:= CP (U) x (Grn-1(U)[k]) [ (CP (U)[k]) x G*
*rn-1(U);
so as to ensure a cofibre sequence
*
* k)
(CP (U)xGrn-1(U))[k-1]-! CP (U)xGrn-1(U) -! (CP (U-V k)xGrn-1(U-V k))Hom*
*(fln;V:
Applying EA*(.) to the resulting filtration we obtain a right half-pla*
*ne homological spectral
sequence with
E(2)12p;q= EA2p+q((CP (U)xGrn-1(U))[p]; (CP (U)xGrn-1(U))[p-1]) =) EA2p+q(CP *
*(U)xGrn-1(U)):
By induction this spectral sequence is under complete control.
Proposition 5.6. If E*A(.) is a complex oriented cohomology theory then *
*the spectral se-
quence E(2)**;*above collapses at E(2)1*;*.
Proof: This follows from the inductive hypothesis, since E ^ Grn-1(U)+ s*
*plits as a wedge of
suspensions of E indexed by the Schubert cells of Grn-1(U)+. Smashing th*
*is with CP (U)+,
we again get splitting from Cole's Splitting Theorem [3]. The filtration*
* of the new spectral
sequence is compatible with that of the splitting in view of the smash p*
*roduct decomposition_
of the Thom spaces. The collapse of the spectral sequence follows. *
* |__|
To see that the map E(2)*;*1-! E(1)*;*1is surjective, recall that
k)
Grn(U)[k]=Grn(U)[k-1]' Grn-1(U - V k)Hom(fln;V
and
*
* k)
(CP (U) x Grn-1(U))[k]=(CP (U - V k) x Grn-1(U - V k))[k-1]' X(k)H*
*om(fln;V:
10 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
It therefore suffices to show that for each k, the map
k) k Hom(fln;V k)
X(k)Hom(fln;V-! Grn-1(U - V )
of subquotients is surjective in EA*(.). This follows for the unThomified 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 CP (U - V k) x Grn-1(U - V k) the bundle fln*
* splits as
the product fl1 x fln-1. Thus Hom (fln; V k) splits as a sum of bundles of dime*
*nsion n - 1
for each of which we have a Thom isomorphism by induction. Surjectivity now fol*
*lows from
that of the unThomified map for the universe U - V k.
Finally, we may remark that the analysis identifies a basis. We may visualiz*
*e the basis
of the homology (CP (U))xn as the points of Nn, with the complex dimension of t*
*he cell
corresponding to i = (i1; i2; : :;:in) being |i| = i1 + i2 + . .+.in. The homol*
*ogy 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 CP (U) = CP (U) x * corre*
*sponds to
the points (i1; 0; : :;:0), and that for Grn-1(U) = * x 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 homolo*
*gy in even
degrees, with basis corresponding to the points (i1; i2; : :;:in) either (i) wi*
*th i1 = 0 and
0 i2 i3 . . .in or (ii) with i1 arbitrary, i2 = 0 and 0 i3 i4 . . .in. Si*
*nce 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 effect of the Thom isomorph*
*ism is to
replace standard homology generators relative to the flag F=V kby those for F: *
*the precise
meaning becomes clear from the special case CP (U).
k)
Lemma 5.7. Naming the cohomology E*A(CP (U-V k)Hom(fl1;V) by the equivalence *
*CP (U-
k) k k
V k)Hom(fl1;V' CP (U)=CP (V ), the element y(V ) is a Thom class. The resulti*
*ng Thom
k) * k A k Hom(fl *
*;V k)
isomorphisms E*A(CP (U-V k)Hom(fl1;V) ~=EA(CP (U-V ) and E* (CP (U-V ) 1*
* ) ~=
EA*(CP (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
fil(F=V k) - fik+l(F)
*
* __
in homology for l 0. *
* |__|
Now we return to find a basis for EA*(Grn(U)) itself. By induction we know t*
*hat the
elements fii1(F) fii2(F) fii3(F) . . .fiin(F) with i1 arbitrary and 0 i2 i*
*3 . . .in
give a basis of EA*(CP (U) x Grn-1(U)). The proof of this shows that the filtra*
*tion 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 col-
lapse at E1, and since the E1 terms are free over E*A, it follows that the basi*
*s elements of
*
*11
EA*(CP (U) x Grn-1(U)) giving rise to the basis elements (i)p for some p (namel*
*y 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 theo*
*ry has
Thom classes for arbitrary bundles. It is the purpose of the present section t*
*o make this
explicit: we prove Theorem 1.1.
Following Okonek [6] we say that E*A(.) has Thom classes if for any complex v*
*ector bundle
over a space X, there is an element o() 2 E*A(X ) so that this system of elemen*
*ts is
1. natural for bundle maps,
2. product preserving in the sense that o( x j) = o() ^ o(j) and
3. normalized so that if X = A=B and the fibre of over B is the representati*
*on W of B
then o() = oeV(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 o*
*() was
a unit multiple of oeW (1). However, we show that any complex oriented cohomolo*
*gy theory
has Thom classes in the present stricter sense.
To prove that a system of Thom classes gives Thom isomorphisms
~= *
E*A(X) -! "EA(X )
(essentially by cup product with o()) or
"EA*(X ) -~=!EA*(X)
(essentially by cap product with o()) we proceed from the fact that oeV is an i*
*somorphism.
We may then work our way up the cellular filtration of X using the normalizatio*
*n condition.
Alternatively, if has a complement j so that j = V we may use the product pr*
*eserving
property.
Before proceeding we make a reduction.
Lemma 6.2. A cohomology theory E*A(.) has a system of Thom classes if and onl*
*y if it has
universal Thom classes,
on 2 "E*A(Grn(U)fln)
for each n, so that
1. the system is product preserving in the sense that under the direct sum ma*
*p Grm (U) x
Grn(U) -! Grm+n (U), the class om+n pulls back to om ^ on
2. Over the point W of Grn(U) with isotropy B A the class on restricts to oe*
*W (1);
We may then define the Thom class for an n-plane bundle by pulling back on unde*
*r the __
Thomification of its classifying map. *
* |__|
Theorem 6.3. If E is a complex oriented theory with orientation in cohomologi*
*cal degree 2
and complex stable structure determined by the orientation, there is an associa*
*ted system of
Thom classes.
12 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Proof: We shall construct classes on as specified in 6.2. Assume that the flag *
*begins with
V 1= ffl. By pullback along conjugation, we see that
* [0]
Grn(U)fln~=Grn(U)fln' Grn(U)=Grn(U) ;
and we have identified the cohomology of this space exactly. From the orientati*
*on we con-
struct the cohomology class y(ffl) 2 E*A(CP (U)), and hence by 2.2, the element*
* y(ffl)n 2
E*A(Grn(U)). By construction this restricts to zero in E*A(Grn(U)[0]), and thus*
* we obtain
on = y(ffl)n 2 E*A(Grn(U); Grn(U)[0]) ~="E*A(Grn(U)fln):
The element on is identified with a map
on : Grn(U)fln-! E:
Lemma 6.4. The classes on are compatible under product in the sense that
om^on
Grm (U)flm^ Grn(U)fln________//E ^ E
| ||
| |
fflffl| fflffl|
Grm+n (U)flm+n____om+n______//E
commutes up to homotopy.
*
* __
Proof: This follows from the fact that y(ffl)m y(ffl)n = y(ffl)(m+n) . *
* |__|
Lemma 6.5. The class on is a Thom class for fln.
Proof: An A-fixed point of Grn(U) is a representation V = fi1 fi2 . . .fin, a*
*nd such a
point lies in the image of the direct sum map from (CP (U))xn. By naturality it*
* therefore
suffices to deal with the case n = 1. However, by definition of an orientation*
* o1 = y(ffl)
restricts to a generator of E*A(CP (ff ffl); CP (ffl)). Since we have used the*
* orientation to give
the complex stable structure, the generator is oeff-1(1) as required.
For the fibre over a non-fixed point we use the fact that orientations behave*
* well_under
restriction to subgroups. *
* |__|
7. Universality of complex cobordism.
Let us turn now to the spectrum level statements, and the connection with map*
*s MU -!
E: we prove Theorem 1.2.
First we deduce that the existence of a ring map MU -! E gives E a complex or*
*ientation.
This is not quite obvious because of the distinction between a map of spectra a*
*nd 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*
* ori-
entable, 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 domai*
*n is MU
or the fact that we use its canonical orientation. However our results show tha*
*t the general
case follows from this one.
*
*13
*
Proof: Let x(ffl) 2 M"U (CP (U)) denote an orientation of MU. By hypothesis the*
* restric-
* ff-1 *
tion (ff-1) of x(ffl) to M"U (S ) is a generator as an MUA-module. From the s*
*uspension
* ff-1 *-ff-1 -1
isomorphism M"U (S ) ~=M"U , and (ff ) is a unit in the RO(G)-graded coe*
*fficient
ring. Hence *((ff-1)) is also a unit in the RO(G)-graded sense, and therefore *
**(x(ffl))
-1 *
* __
restricts to a generator of the integer graded module "E*A(Sff ) as required. *
* |__|
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)fln:
If U V with U of dimension m, the structure map
SV -U^ MU(U) = Grm (U U)(V -U)flm-! Grn(V U)fln
is given by Thomifying the map adding V - U. Thus
MU ' holim -V Grn(V U)fln;
! V
and the cohomology of MU may be calculated from that of Thom spaces of Grassman*
*nians
by the Milnor exact sequence.
It is useful to be more explicit about the homology.
Lemma 7.3. Provided the complex stable structure is defined by the orientatio*
*n, the struc-
ture map V -UGrn(U U)flm-! Grn(V U)fln, takes the element
oeV -Uom (fii1(F) fii2(F) . . .fiim(F))
to __
on(fi0(F) . . .fi0(F) fii1(F)) fii2(F) . . .fiim(F)): |__|
Proposition 7.4. The classes on assemble to give a unique map
o : MU -! E
of ring spectra.
Proof: Since the structure maps are surjective in cohomology,
E*A(MU) = lim E*A(-V Grn(U)fln):
V
The existence of the map o therefore follows from compatibility of the elements*
* on under
suspension.
Lemma 7.5. The classes on are compatible under suspension in the sense that
V -Uom
V -UGrm (U U)flm _________//_E
| ||
| |
fflffl| on fflffl|
Grn(V U)fln ____________//_E
commutes up to homotopy.
14 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Proof: First note that the space of isometric isomorphisms U -! U is contractib*
*le, so the
particular identification V U ~=U is not important. Now, since the structure m*
*aps in_MU_
arise from bundles, the lemma follows from 6.4. *
* |__|
Since the smash product MU ^ MU -! MU is induced by the Thomification of the *
*map
classifying direct sum of bundles, the compatibility of the elements on under p*
*roducts shows_
that the map o 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 a*
*nd ring maps
MU - ! E which makes no explicit reference to the orientation. The main new re*
*sult is
Theorem 1.3.
First we need to calculate the homology of MU. We use the notation of Sectio*
*n 7. Of
course we have
n n A -V n n fln
EA*(MU) = lim EA*(-V MU(V )) = lim E* ( Grn(V U) ):
! n ! n
Thus we may construct elements using the maps
~= A -V n fl A -V n n A
EA*(Grn(U)) -! E* ( Grn(U) n) = E* ( MU(V )) -! E* (MU):
We identified the effect of the maps in the direct system in 7.3. It is natural*
* to suppress the
terms fi(V 0).
Definition 8.1. We write bi(F) for the image of fii+1(F) in EA*(MU). It follows*
* that the
image of fii1+1(F) fii2+1(F) . . .fiin+1(F) is bi1(F)bi2(F) . .b.in(F). In th*
*e nonequivariant
context, it is standard to write fii+1for fii+1(F) and bifor bi(F).
Note that Whitney sum of bundles gives a product MU ^ MU -! MU, and hence
EA*(MU) is an E*A-algebra. The simplicity of the following theorem is somewhat *
*surprising.
Theorem 8.2. If E*A(.) is a complex oriented theory, the homology of MU is po*
*lynomial
over E*A:
EA*(MU) = E*A[b1(F); b2(F); b3(F); . .].:
Furthermore there is a K"unneth theorem in the sense that E*A(MU^s) = E*A(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 rel*
*ation __
(oe o) \ (fi fl) = (oe \ fi) (o \ fl). *
* |__|
It is now straightforward to deduce the global statements we require. The add*
*itive part is
no problem since the homology of MU is free over E*A, and we have a K"unneth is*
*omorphism.
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 us*
*eful conse-
quence.
Corollary 8.5. If x(ffl) is a complex orientation of E*A(.), then there is a na*
*tural bijective
correspondence between orientations of E*A(.) in cohomological degree 2 which g*
*ive the same
complex stable structure as x(ffl) and
Ring[MU; E]A = EA*-alg(EA*(MU); EA*):
More precisely, suppose x0(ffl) is an orientation in cohomological degree 2 giv*
*ing the same
complex stable structure as x(ffl). If V 1= ffl and the associated parameter is
y0(ffl) = iiy(V i)
then 0 = 0 and 1 = 1. The associated algebra homomorphism
EA*(MU) = E*A[b1(F); b2(F); b3(F); : :]:-! E*A
is then determined by
bi(F) 7-! i+1
for i 1.
Proof: In general, if x0(ffl) is an orientation of E, then the associated map M*
*U - ! E
restricts to x0(ffl)n as a map
-2n((CP (U))xn)fln-! -2n(Grn(U))fln-! E:
Thus the algebra homomorphism associated to x0(ffl) may be calculated on an ele*
*ment z
arising from -2nMU(V ) as x0(ffl)n (z).
Now suppose x0(ffl) is expressed in terms of the basis associated to the orie*
*ntation x(ffl).
The coefficient 0 = 0 since y0(ffl) restricts trivially to CP (ffl). The coeffi*
*cient 1 is obtained
-1
by restricting to (CP (V 2); CP (V 1)) ~=Sff2; since the complex stable structu*
*res defined by
x(ffl) and x0(ffl) agree, the coefficient is 1. It is easy to see the images of*
* bi(F) for i 1 are as
1 1 *
* __
specified since they arise from -V MU(V ). *
* |__|
Remark 8.6. (i) For a complex stable structure arising from an orientation, t*
*he element
oeff(1) for any chosen ff determines the rest, because of the action of A*. Thi*
*s explains why
consideration of ff-12alone was enough to show 1 = 1.
(ii) The statement of the corollary is consistent with the conjecture that MUA**
*MU classifies
strict isomorphisms of A-equivariant formal group laws.
16 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
References
[1]J.F.Adams "Stable homotopy and generalized homology." Chicago University Pr*
*ess (1974).
[2]M.Cole "Complex oriented RO(G)-graded equivariant cohomology theories and t*
*heir formal group laws."
Thesis, University of Chicago (1996)
[3]M.Cole "Complex oriented equivariant cohomology theories and equivariant co*
*mplex projective spaces"
(In preparation)
[4]Michael Cole, J.P.C.Greenlees and I.Kriz "Equivariant formal group laws." S*
*ubmitted for publication
(1997) pp 29
[5]J.W.Milnor and J.Stasheff "Characteristic classes." Princeton University Pr*
*ess (1974)
[6]C.Okonek "Der Conner-Floyd-Isomorphismus f"ur Abelsche Gruppen." Math. Z. 1*
*79 (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