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 E_{A}^{∗}(·) is orientable if complex line bundles are well behaved. More precisely, we let ℂP (𝒰) denote the space of lines in a complete complex A-universe, and ϵ denote the trivial representation. We say that E_{A}^{∗}(·) is a complex stable ring theory if there are suspension isomorphisms

σ_{V} : Ẽ_{A}^{n}(X) ⟶^{≅} Ẽ_{A}^{n+|V |}(S^{V} ∧ X)

for all complex representations V, where S^{V} 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 Ẽ^{|V |}(S^{V}). We say the theory is complex orientable if in addition, there is a cohomology class x(ϵ) ∈ E_{A}^{∗}(ℂP (𝒰), ℂP (ϵ)) which restricts to a generator of

E_{A}^{∗}(ℂP (α ⊕ ϵ), ℂP (ϵ)) ≅ Ẽ_{A}^{∗}(S^{α^{−1}})

for all one dimensional representations α. Thus if the complex orientation x(ϵ) 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 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 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 [7] has proved that by construction MU_{A}^{∗}(·) is universal for cohomology theories with Thom classes. Combined with our main result this implies the following universality statement. Theorem 1.2. If E_{A}^{∗}(·) 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.

IK acknowledges support from an NSF National Young Investigator award.

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 E_{A}^{∗}(·) is a complex orientable theory then E_{∗}^{A}(MU) is a polynomial E_{A}^{∗}- algebra and

Ring_{A}(MU, E) = E_{∗}^{A}-alg(E_{∗}^{A}(MU), E_{∗}^{A}).

This is proved in Section 8. We shall be more specific about the polynomial generators in due course. The good behaviour of E_{∗}^{A}(MU) should be contrasted with the facts that E_{A}^{∗}(ℂP (𝒰)) is not a power series ring in general, and MU_{A}^{∗} 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 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.

We are grateful to the referee for his interesting comments, some of which are reproduced in Remark 5.8.

2. Equivariant Grassmannians.

Let Gr_{n}(V) be the complex Grassmannian of complex n-dimensional subspaces of V. For example Gr_{1}(V) = ℂP (V). Lemma 2.1. The A-space Gr_{n}(𝒰) is a classifying space for A-equivariant complex n-plane bundles:

Gr_{n}(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))^{×n} = Gr_{1}(𝒰)^{×n} ⟶ Gr_{n}(𝒰^{⊕n}) ≅ Gr_{n}(𝒰),

where the final homeomorphism arises from an isometric isomorphism 𝒰^{⊕n} ≅ 𝒰. This sum map induces maps

E_{∗}^{A}(ℂP (𝒰))^{⊗n} = E_{∗}^{A}((ℂP (𝒰))^{×n}) ⟶ E_{∗}^{A}(Gr_{n}(𝒰))

and

E_{A}^{∗}(Gr_{n}(𝒰)) ⟶ E_{A}^{∗}((ℂP (𝒰))^{×n}) = E_{A}^{∗}(ℂP (𝒰))^{ˆ⊗n},

where the completed tensor producte ˆ⊗ refers to the skeletal filtration topology. The Künneth theorems implicit in this statement are corollaries of Cole’s Splitting Theorem [3].

Since any permutation of the copies of 𝒰 in 𝒰^{⊕n} is homotopic to the identity through isometric isomorphisms, the induced maps factor through coinvariants and invariants for the symmetric group Σ_{n}.

Theorem 2.2. For any complex orientable cohomology theory E_{A}^{∗}(·), the direct sum of lines induces isomorphisms

E_{∗}^{A}(Gr_{n}(𝒰)) ≅ {E_{∗}^{A}(ℂP (𝒰))^{⊗n}}_{Σ_{n}}

and

E_{A}^{∗}(Gr_{n}(𝒰)) ≅ {E_{A}^{∗}(ℂP (𝒰))^{ˆ⊗n}}^{Σ_{n}}.

Remark 2.3. The homological result can be neatly stated if we take all n together. Indeed the spectrum ⋁_{n≥0} Gr_{n}(𝒰) is a ring under direct sum, and

E_{∗}^{A}(⋁_{n≥0} Gr_{n}(𝒰)) = Symm(E_{∗}^{A}(ℂP (𝒰))).

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 Grassmannians follows from that of ℂP (𝒰) made explicit in [4].

Indeed, Cole showed that E_{∗}^{A}(ℂP (𝒰)) is additively free over E_{A}^{∗} and E_{A}^{∗}(ℂP (𝒰)) is a product of suspensions of E_{A}^{∗}. Furthermore, he showed how an orientation x(ϵ) 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}), … of E_{A}^{∗}(ℂP (𝒰)), and we may let β_{0}(ℱ), β_{1}(ℱ), β_{2}(ℱ), … denote the dual basis of E_{∗}^{A}(ℂP (𝒰)). The notation for the homology generators reflects the fact that β_{i}(ℱ) depends on the initial segment V^{0} ⊂ V^{1} ⊂ ⋯ ⊂ V^{i} of the flag. Furthermore, a Künneth theorem holds for the homology or cohomology of products of ℂP (𝒰). Lemma 2.4. An E_{A}^{∗}-basis of the coinvariants {E_{∗}^{A}(ℂP (𝒰))^{⊗n}}_{Σ_{n}} is given by the images of all products β_{i_{1}}(ℱ) ⊗ β_{i_{2}}(ℱ) ⊗ ⋯ ⊗ β_{i_{n}}(ℱ) so that 0 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n}. A topological E_{A}^{∗}-basis of the invariants {E_{A}^{∗}(ℂP (𝒰))^{ˆ⊗n}}^{Σ_{n}} corresponds to the collection of sequences 0 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n}; the basis elements are the symmetric sums

Σ′y(V^{i_{σ(1)}}) ⊗ y(V^{i_{σ(2)}}) ⊗ ⋯ ⊗ y(V^{i_{σ(n)}})

where Σ′ denotes the sum over the orbit of (i_{1}, i_{2}, …, i_{n}). Proof: The result is clear once we remark that the Σ_{n} action arises from an action on the basis of the homology or cohomology of (ℂP (𝒰))^{×n}. In the case of cohomology, we pass to limits from the case of ℂP (V)^{×n}. □

Corollary 2.5. If E is a complex oriented cohomology theory then E ∧ Gr_{n}(𝒰) splits as a wedge of copies of E indexed by sequences 0 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n}:

E ∧ Gr_{n}(𝒰) ≃ ⋁_{i} Σ^{2|i|}E,

where |i| = i_{1} + i_{2} + ⋯ + i_{n}. The splitting depends on the orientation. Proof: In the usual way, from the basis of E_{∗}^{A}(Gr_{n}(𝒰)), we may construct a map

s : ⋁_{i} Σ^{2|i|}E ⟶ E ∧ Gr_{n}(𝒰)

of A-spectra, using the product on E.

By construction it induces an isomorphism in π_{∗}^{A}(·). Now suppose B ⊆ A, and consider the B-equivariant situation. The restriction map E_{A}^{∗}(ℂP (𝒰)) ⟶ E_{B}^{∗}(ℂP (𝒰)) takes an Aorientation to a B-orientation, and the resulting basis corresponding to a complete flag to the basis corresponding to the same flag regarded B-equivariantly. It therefore follows that the restriction E_{A}^{∗}(Gr_{n}(𝒰)) ⟶ E_{B}^{∗}(Gr_{n}(𝒰)) takes the sequence basis to another basis. Thus s induces an isomorphism of π_{∗}^{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 specific orientation in very concrete terms. Corollary 2.6. If E is a complex oriented cohomology theory then

E_{∗}^{A}(Gr_{n}(𝒰)) = E_{∗}^{A}{β(V^{i_{1}}) ⊗ β(V^{i_{2}}) ⊗ ⋯ ⊗ β(V^{i_{n}}) | 0 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n}}

and

\begin{aligned}E_{A}^{∗}(Gr_{n}(𝒰)) & = & E_{∗}^{A}-mod(E_{∗}^{A}(Gr_{n}(𝒰)), E_{∗}^{A}) \\ & = & E_{A}^{∗}\{\{Σ′y(V^{i_{σ(1)}}) ⊗ y(V^{i_{σ(2)}}) ⊗ ⋯ ⊗ y(V^{i_{σ(n)}}) | 0 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n}\}\}. □ \end{aligned}

Finally we may consider the extra structure on the system

{E_{∗}^{A}(Gr_{n}(𝒰))}_{n≥0} or {E_{A}^{∗}(Gr_{n}(𝒰))}_{n≥0}.

The relevant extra structure is given by the conjugation map

\overline{·} : Gr_{n}(𝒰) ⟶ Gr_{n}(\overline{𝒰}) ≅ Gr_{n}(𝒰),

the action maps

⊗α : Gr_{n}(𝒰) ⟶ Gr_{n}(𝒰 ⊗ α) ≅ Gr_{n}(𝒰)

for α ∈ A^{∗}, the direct sum maps

⊕ : Gr_{m}(𝒰) × Gr_{n}(𝒰) ⟶ Gr_{m+n}(𝒰 ⊕ 𝒰) ≅ Gr_{m+n}(𝒰)

and the tensor product maps

⊗ : Gr_{m}(𝒰) × Gr_{n}(𝒰) ⟶ Gr_{mn}(𝒰 ⊗ 𝒰) ≅ Gr_{mn}(𝒰).

The structure constants for all these maps in homology and cohomology can be deduced from those ⊗α for n = 1 and ⊗ for m = n = 1. Furthermore the coproduct on E_{∗}^{A}(Gr_{n}(𝒰)) and the product on E_{A}^{∗}(Gr_{n}(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 Gr_{n}(V) into Schubert cells, as described for example in [6]. We choose a complete A-invariant flag

0 = V^{0} ⊂ V^{1} ⊂ V^{2} ⊂ V^{3} ⊂ ⋯ ⊂ V^{m} = V,

and let α_{i} = V^{i}/V^{i−1} as usual. It is well known that Gr_{n}(V) admits a non-equivariant CW-structure in which the cells are indexed by sequences of integers

1 ≤ σ_{1} < σ_{2} < ⋯ < σ_{n} ≤ m;

the sequence σ = (σ_{1}, σ_{2}, …, σ_{n}) is called a Schubert symbol. It is convenient to use σ to select a complete flag

0 = V (σ)^{0} ⊂ V (σ)^{1} ⊂ V (σ)^{2} ⊂ V (σ)^{3} ⊂ ⋯ ⊂ V (σ)^{n}

of length n where

V (σ)^{i} = α_{σ_{1}} ⊕ α_{σ_{2}} ⊕ ⋯ ⊕ α_{σ_{i}}.

The cell e(σ) corresponding to the Schubert symbol σ consists of all n-planes X with

dim(X ∩ V^{σ_{i}}) = i and dim(X ∩ V^{σ_{i}−1}) = i − 1

for i = 1, 2, …, n. Such an n-plane X admits a basis x_{1}, x_{2}, …, x_{n} with x_{i} ∈ V^{σ_{i}} and nonzero in V^{σ_{i}}/V^{σ_{i}−1}. This is usually represented as an m × n-matrix with rows x_{1}, x_{2}, …, x_{n}, and columns indexed by α_{1}, α_{2}, …, α_{m}. Dividing x_{i} by its α_{σ_{i}} coordinate, we may assume the last entry in each row is 1, and then subtracting a suitable multiple of x_{i} 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(σ) is an A-subspace. Accordingly, these same Schubert cells e(σ) 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 α_{σ_{i}}^{−1} ⊗ (V^{σ_{i}}/V (σ)^{i}), so that the cell e(σ) is of Rep(A)-dimension

⨁_{i=1}^{n} α_{σ_{i}}^{−1} ⊗ (V^{σ_{i}}/V (σ)^{i}).

4. Thom complexes and the Schubert filtration.

There is a convenient filtration associated to the Schubert cells, which we shall need to use. We let Gr_{n}(𝒰)^{[k]} be the subcomplex of Gr_{n}(𝒰) corresponding to Schubert cells with σ_{1} ≤ k + 1; the indexing is chosen since k + 1 is the lowest complex dimension of a cell not in Gr_{n}(𝒰)^{[k]}. The resulting filtration

Gr_{n}(𝒰)^{[0]} ⊂ Gr_{n}(𝒰)^{[1]} ⊂ Gr_{n}(𝒰)^{[2]} ⊂ ⋯ ⊂ Gr_{n}(𝒰)

will be called the Schubert filtration. Of course the Schubert filtration depends on the complete flag ℱ; when we use the universe 𝒰 − V^{k}, we use the cell structure associated to the complete flag ℱ/V^{k}.

To obtain sufficient naturality we need to interpret the filtration in terms of Thom complexes. For this we let γ_{n} denote the tautological n-plane bundle over Gr_{n}(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

Gr_{n}(𝒰)/Gr_{n}(𝒰)^{[k−1]} ≃ (Gr_{n}(𝒰 − V^{k}))^{Hom(γ_{n},V^{k})}.

This may be chosen natural as k varies in the sense that there is a homotopy commutative diagram

\xymatrix{Gr_{n}(𝒰)/Gr_{n}(𝒰)^{[k−1]} \ar[r]^{≃} \ar[dd] & (Gr_{n}(𝒰 − V^{k}))^{Hom(γ_{n},V^{k})} \ar@{-->}[dr]^{j} \\  &  & Gr_{n}(𝒰 − V^{k}))^{Hom(γ_{n},V^{k+1})} \\ Gr_{n}(𝒰)/Gr_{n}(𝒰)^{[k]} \ar[r]^{≃} & (Gr_{n}(𝒰 − V^{k+1}))^{Hom(γ_{n},V^{k+1})} \ar@{-->}[ur]_{≃}}

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 finite dimensional subspace V ⊂ 𝒰, the inclusion 𝒰 − V ⊆ 𝒰 induces the stabilization equivalence

Gr_{n}(𝒰 − V) ≃ Gr_{n}(𝒰).

Proof: The proof is by induction on k, using the following two equivalences. Lemma 4.3. There is an equvialence

Gr_{n}(𝒰 − V^{k})/Gr_{n}(𝒰 − V^{k})^{[0]} ≃ Gr_{n}(𝒰 − V^{k+1})^{Hom(γ_{n},α_{k+1})},

Remark 4.4. In view of the fact that Gr_{n}(𝒰 − V^{k})^{[0]} ≅ Gr_{n−1}(𝒰 − V^{k+1}) the lemma may be viewed as giving a Gysin cofibre sequence

Gr_{n−1}(𝒰 − V^{k+1}) ⟶^{α_{k+1}⊕} Gr_{n}(𝒰 − V^{k}) ⟶ Gr_{n}(𝒰 − V^{k+1})^{Hom(γ_{n},α_{k+1})}. □

Lemma 4.5. There is a homeomorphism

Gr_{n}(𝒰)^{[k]}/Gr_{n}(𝒰)^{[k−1]} ≅ (Gr_{n}(𝒰 − V^{k})^{[0]})^{Hom(γ_{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 Gr_{n}(𝒰)/Gr_{n}(𝒰)^{[k−1]} has been identified as in the statement of the theorem. We then calculate

Gr_{n}(𝒰)/Gr_{n}(𝒰)^{[k]} = (Gr_{n}(𝒰)/Gr_{n}(𝒰)^{[k−1]}) / (Gr_{n}(𝒰)^{[k]}/Gr_{n}(𝒰)^{[k−1]})
≃ {Gr_{n}(𝒰 − V^{k})/Gr_{n}(𝒰 − V^{k})^{[0]}}^{Hom(γ_{n},V^{k})}
≃ {Gr_{n}(𝒰 − V^{k+1})^{Hom(γ_{n},α_{k+1})}}^{Hom(γ_{n},V^{k})}
≅ Gr_{n}(𝒰 − V^{k+1})^{Hom(γ_{n},V^{k+1})},

where the first equivalence uses the inductive hypothesis and 4.5, and the second 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 the unit sphere bundle. It therefore suffices to construct a homotopy commutative square

S(Hom(γ_{n}, α_{k+1})) ⟶ Gr_{n}(𝒰 − V^{k+1})
≃↓  ↓≃
Gr_{n}(𝒰 − V^{k})^{[0]} ⟶ Gr_{n}(𝒰 − V^{k}).

The right hand equivalence is the stabilization equivalence induced by the inclusion of complete universes. The right hand vertical S(Hom(γ_{n}, α_{k+1})) ⟶ Gr_{n}(𝒰 − V^{k})^{[0]} is defined by (z, X) ⟼ α_{k+1} ⊕ ker(z), where z is a unit vector in Hom(X, α_{k+1}). To see 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 : α_{k+1} ⊕ X ⟶ α_{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 compactification of the vector bundle. We may then define a map

Gr_{n}(𝒰)^{[k]}/Gr_{n}(𝒰)^{[k−1]} ⟶ (Gr_{n}(𝒰 − V^{k})^{[0]})^{Hom(γ_{n},V^{k})}

by taking a representative n-plane X, expressed in standard form as X = (X_{≤k}|X_{>k}), to the vector X_{≤k} 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 continuous at the basepoint. □

This completes the proof of 4.1. □

5. The spectral sequence of universal Thom complexes.

We now construct a spectral sequence for calculating E_{∗}^{A}(Gr_{n}(𝒰)) using the Schubert cell filtration

Gr_{n}(𝒰)^{[0]} ⊂ Gr_{n}(𝒰)^{[1]} ⊂ Gr_{n}(𝒰)^{[2]} ⊂ ⋯ ⊂ Gr_{n}(𝒰)

introduced in Section 4. We think of the superscript k in Gr_{n}(𝒰)^{[k]} as a complex dimension, and applying E_{∗}^{A}(·) to the filtration we obtain a right half-plane homological spectral sequence E(1)_{∗,∗}^{∗} concentrated in even filtration degrees with

E(1)_{2p,q}^{1} = E_{2p+q}^{A}(Gr_{n}(𝒰)^{[p]}, Gr_{n}(𝒰)^{[p−1]}) ⟹ E_{2p+q}^{A}(Gr_{n}(𝒰)).

The spectral sequence is indexed so that a Rep(A)-s-cell in filtration 2p contributes to E_{2p,2s}^{1}. 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 concentrated in even filtration degrees:

E(1)_{1}^{2p,q} = E_{A}^{2p+q}(Gr_{n}(𝒰)^{[p]}, Gr_{n}(𝒰)^{[p−1]}) ⟹ E_{A}^{2p+q}(Gr_{n}(𝒰)).

Remark 5.1. In view of the equivalence

Gr_{n}(𝒰)/Gr_{n}(𝒰)^{[k−1]} ≃ Gr_{n}(𝒰 − V^{k})^{Hom(γ_{n},V^{k})}

obtained from 4.1 by stabilization, the same spectral sequences may be obtained from the sequence of maps

Gr_{n}(𝒰)^{0} ⟶ Gr_{n}(𝒰)^{Hom(γ_{n},V^{1})} ⟶ Gr_{n}(𝒰)^{Hom(γ_{n},V^{2})} ⟶ Gr_{n}(𝒰)^{Hom(γ_{n},V^{3})} ⟶ ….

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 E_{∗}^{A}(Gr_{n}(𝒰)) is free over E_{A}^{∗} on the basis of all products β_{i_{1}}(ℱ) ⊗ β_{i_{2}}(ℱ) ⊗ ⋯ ⊗ β_{i_{n}}(ℱ) so that 0 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n} as described in 2.4. 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

ℂP (𝒰) × Gr_{n−1}(𝒰) = Gr_{1}(𝒰) × Gr_{n−1}(𝒰) ⟶^{⊕} Gr_{n}(𝒰 ⊕ 𝒰) ≅ Gr_{n}(𝒰).

The proof proceeds by constructing a compatible spectral sequence E(2)_{∗,∗}^{∗} for calculating E_{∗}^{A}(ℂP (𝒰) × Gr_{n−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. Lemma 5.3. There is a cofibre sequence

X(k)_{+} ⟶ (ℂP (𝒰−V^{k})×Gr_{n−1}(𝒰−V^{k}))_{+} ⟶ (ℂP (𝒰−V^{k+1})×Gr_{n−1}(𝒰−V^{k+1}))^{Hom(γ_{n},α_{k+1})}

where X(k) is a pushout

\xymatrix{Gr_{n−2}(𝒰 − V^{k+1}) \ar[r]^{⊕α_{k+1}} \ar[d]_{\{c_{α_{k+1}},1\}} & Gr_{n−1}(𝒰 − V^{k}) \ar[d]^{i} \\ ℂP (𝒰 − V^{k}) × Gr_{n−2}(𝒰 − V^{k+1}) \ar[r] & X(k),}

where the top horizontal is induced by adding α_{k+1} to the (n − 1)-plane and universe as in 4.4, and the left hand vertical uses the inclusion of the A-fixed line α_{k+1} in ℂP (𝒰 − V^{k}) in the first factor. Proof: Over ℂP (𝒰 − V^{k+1}) × Gr_{n−1}(𝒰 − V^{k+1}) the bundle Hom(γ_{n}, α_{k+1}) is the product Hom(γ_{1}, α_{k+1})×Hom(γ_{n−1}, α_{k+1}). Thus the unit disc bundle is the product D(Hom(γ_{1}, α_{k+1}))× D(Hom(γ_{n−1}, α_{k+1})) and the Thom space is the smash product

ℂP (𝒰 − V^{k+1})^{Hom(γ_{1},α_{k+1})} ∧ Gr_{n−1}(𝒰 − V^{k+1})^{Hom(γ_{n−1},α_{k+1})}.

This identifies the unit sphere bundle S(Hom(γ_{1}, α_{k+1}) × Hom(γ_{n−1}, α_{k+1})) as the pushout

\xymatrix{S(Hom(γ_{1}, α_{k+1})) × S(Hom(γ_{n−1}, α_{k+1})) \ar[r] \ar[d] & S(Hom(γ_{1}, α_{k+1})) × D(Hom(γ_{n−1}, α_{k+1})) \ar[d]^{i} \\ D(Hom(γ_{1}, α_{k+1})) × S(Hom(γ_{n−1}, α_{k+1})) \ar[r] & S(Hom(γ_{1}, α_{k+1}) × Hom(γ_{n−1}, α_{k+1})).}

The description of X(k) follows. □

Corollary 5.4. There is a cofibre sequence

X(k)^{Hom(γ_{n},V^{k})} ⟶

(ℂP (𝒰−V^{k})×Gr_{n−1}(𝒰−V^{k}))^{Hom(γ_{n},V^{k})} ⟶ (ℂP (𝒰−V^{k+1})×Gr_{n−1}(𝒰−V^{k+1})^{Hom(γ_{n},V^{k+1})}. □

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-decomposition (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}) × Gr_{n−1}(𝒰 − V^{k+1}) ⟶^{⊕} Gr_{n}(𝒰 − V^{k+1})

induces a retraction

r : X(k) = S(Hom(γ_{1}, α_{k+1}) × Hom(γ_{n−1}, α_{k+1})) ⟶ S(Hom(γ_{n}, α_{k+1})) ≃ Gr_{n−1}(𝒰 − V^{k})

on unit sphere bundles. Proof: The inclusion corresponding to the retraction r is the composite

Gr_{n−1}(𝒰 − V^{k}) ≃ S(Hom(γ_{1}, α_{k+1})) × D(Hom(γ_{n−1}, α_{k+1})) ⟶^{i}

S(Hom(γ_{1}, α_{k+1}) × Hom(γ_{n−1}, α_{k+1})) = X(k).

It is easily checked that with the equivalence S(Hom(γ_{n}, α_{k+1})) ≃ Gr_{n−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

E_{∗}^{A}(ℂP (𝒰) × Gr_{n−1}(𝒰)) ⟶ E_{∗}^{A}(Gr_{n}(𝒰)),

the spectral sequence for E_{∗}^{A}(ℂP (𝒰) × Gr_{n−1}(𝒰)) is constructed from the sequence of maps

(ℂP (𝒰)×Gr_{n−1}(𝒰))^{0} ⟶ (ℂP (𝒰)×Gr_{n−1}(𝒰))^{Hom(γ_{n},V^{1})} ⟶ (ℂP (𝒰)×Gr_{n−1}(𝒰))^{Hom(γ_{n},V^{2})} ⟶ ⋯

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) × Gr_{n−1}(𝒰) the bundle Hom(γ_{n}, V^{k}) is Hom(γ_{1}, V^{k}) × Hom(γ_{n−1}, V^{k}), and hence

(ℂP (𝒰) × Gr_{n−1}(𝒰))^{Hom(γ_{n},V^{k})} ≃ ℂP (𝒰)^{Hom(γ_{1},V^{k})} ∧ Gr_{n−1}(𝒰)^{Hom(γ_{n−1},V^{k})}.

We therefore define

(ℂP (𝒰) × Gr_{n−1}(𝒰))^{[k]} := ℂP (𝒰) × (Gr_{n−1}(𝒰)^{[k]}) ∪ (ℂP (𝒰)^{[k]}) × Gr_{n−1}(𝒰),

so as to ensure a cofibre sequence

(ℂP (𝒰)×Gr_{n−1}(𝒰))^{[k−1]} ⟶ ℂP (𝒰)×Gr_{n−1}(𝒰) ⟶ (ℂP (𝒰−V^{k})×Gr_{n−1}(𝒰−V^{k}))^{Hom(γ_{n},V^{k})}.

Applying E_{∗}^{A}(·) to the resulting filtration we obtain a right half-plane homological spectral sequence with

E(2)_{2p,q}^{1} = E_{2p+q}^{A}((ℂP (𝒰)×Gr_{n−1}(𝒰))^{[p]}, (ℂP (𝒰)×Gr_{n−1}(𝒰))^{[p−1]}) ⟹ E_{2p+q}^{A}(ℂP (𝒰)×Gr_{n−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 sequence E(2)_{∗,∗}^{∗} above collapses at E(2)_{∗,∗}^{1}.

Proof: This follows from the inductive hypothesis, since E ∧ Gr_{n−1}(𝒰)_{+} splits as a wedge of suspensions of E indexed by the Schubert cells of Gr_{n−1}(𝒰)_{+}. Smashing this 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 product decomposition of the Thom spaces. The collapse of the spectral sequence follows. □

To see that the map E(2)_{1}^{∗,∗} ⟶ E(1)_{1}^{∗,∗} is surjective, recall that

Gr_{n}(𝒰)^{[k]}/Gr_{n}(𝒰)^{[k−1]} ≃ Gr_{n−1}(𝒰 − V^{k})^{Hom(γ_{n},V^{k})}

and

(ℂP (𝒰) × Gr_{n−1}(𝒰))^{[k]}/(ℂP (𝒰 − V^{k}) × Gr_{n−1}(𝒰 − V^{k}))^{[k−1]} ≃ X(k)^{Hom(γ_{n},V^{k})}.

It therefore suffices to show that for each k, the map

X(k)^{Hom(γ_{n},V^{k})} ⟶ Gr_{n−1}(𝒰 − V^{k})^{Hom(γ_{n},V^{k})}

of subquotients is surjective in E_{∗}^{A}(·). This follows for the unThomified map, since by 5.5 the map r : X(k) ⟶ Gr_{n−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}) × Gr_{n−1}(𝒰 − V^{k}) the bundle γ_{n} splits as the product γ_{1} × γ_{n−1}. Thus Hom(γ_{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 unThomified map for the universe 𝒰 − V^{k}.

Finally, we may remark that the analysis identifies a basis. We may visualize the basis of the homology (ℂP (𝒰))^{×n} as the points of ℕ^{n}, with the complex dimension of the cell corresponding to i = (i_{1}, i_{2}, …, i_{n}) being |i| = i_{1} + i_{2} + ⋯ + i_{n}. The homology of all other relevant spaces have bases given by the images of these basis elements. These correspond to subsets of ℕ^{n} as follows. The basis for the space ℂP (𝒰) = ℂP (𝒰) × ∗ corresponds to the points (i_{1}, 0, …, 0), and that for Gr_{n−1}(𝒰) = ∗ × Gr_{n−1}(𝒰) corresponds to the points (0, i_{2}, …, i_{n}) with 0 ≤ i_{2} ≤ i_{3} ≤ ⋯ ≤ i_{n}. The basis for the subspace Gr_{n−2}(𝒰) ⊆ Gr_{n−1}(𝒰) corresponds to the points (0, 0, i_{3}, …, i_{n}) with 0 ≤ i_{3} ≤ i_{4} ≤ ⋯ ≤ i_{n}.

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 (i_{1}, i_{2}, …, i_{n}) either (i) with i_{1} = 0 and 0 ≤ i_{2} ≤ i_{3} ≤ ⋯ ≤ i_{n} or (ii) with i_{1} arbitrary, i_{2} = 0 and 0 ≤ i_{3} ≤ i_{4} ≤ ⋯ ≤ i_{n}. Since the map i occurs in the pushout description of X(k) we see that r : X(k) ⟶ Gr_{n−1}(𝒰 − V^{k}) maps the subspace on the basis elements elements corresponding to 0 ≤ i_{2} ≤ i_{3} ≤ ⋯ ≤ i_{n} isomorphically to the homology of Gr_{n−1}(𝒰−V^{k}). The effect of the Thom isomorphism is to replace standard homology generators relative to the flag ℱ/V^{k} by those for ℱ: the precise meaning becomes clear from the special case ℂP (𝒰).

Lemma 5.7. Naming the cohomology E_{A}^{∗}(ℂP (𝒰−V^{k})^{Hom(γ_{1},V^{k})}) by the equivalence ℂP (𝒰− V^{k})^{Hom(γ_{1},V^{k})} ≃ ℂP (𝒰)/ℂP (V^{k}), the element y(V^{k}) is a Thom class. The resulting Thom isomorphisms E_{A}^{∗}(ℂP (𝒰−V^{k})^{Hom(γ_{1},V^{k})}) ≅ E_{A}^{∗}(ℂP (𝒰−V^{k}) and E_{∗}^{A}(ℂP (𝒰−V^{k})^{Hom(γ_{1},V^{k})}) ≅ E_{∗}^{A}(ℂP (𝒰 − V^{k}) are given by

y(V^{k+l}/V^{k}) ⟼ y(V^{k})y(V^{k+l}/V^{k}) = y(V^{k+l})

in cohomology and

β_{l}(ℱ/V^{k}) ⟵ β_{k+l}(ℱ)

in homology for l ≥ 0. □

Now we return to find a basis for E_{∗}^{A}(Gr_{n}(𝒰)) itself. By induction we know that the elements β_{i_{1}}(ℱ) ⊗ β_{i_{2}}(ℱ) ⊗ β_{i_{3}}(ℱ) ⊗ ⋯ ⊗ β_{i_{n}}(ℱ) with i_{1} arbitrary and 0 ≤ i_{2} ≤ i_{3} ≤ ⋯ ≤ i_{n} give a basis of E_{∗}^{A}(ℂP (𝒰) × Gr_{n−1}(𝒰)). The proof of this shows that the filtration is such that the basis of E(2)_{2p,∗}^{1} is given by the images of the elements either

(i)_{p} with p = i_{1} ≤ i_{2} ≤ i_{3} ≤ ⋯ ≤ i_{n} or (ii)_{p} with i_{1} ≥ p and p = i_{2} ≤ i_{3} ≤ ⋯ ≤ i_{n}.

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)_{2p,∗}^{2}. Since the spectral sequences collapse at E^{1}, and since the E^{1} terms are free over E_{A}^{∗}, it follows that the basis elements of E_{∗}^{A}(ℂP (𝒰) × Gr_{n−1}(𝒰)) giving rise to the basis elements (i)_{p} for some p (namely those with i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n}) map to a basis of E_{∗}^{A}(Gr_{n}(𝒰)) as required. □

Remark 5.8. The referee suggests the following attractive repackaging of the results of this section. For any representation W, we can define a spectrum

R_{W} := ⋁_{k≥0} Gr_{k}(𝒰)^{Hom(γ,W)}.

Direct sum makes this into a ring spectrum and indeed there is a strictly commutative ring spectrum in the category of [5] weakly equivalent to R_{W}: one takes the spectrum associated to the pre-spectrum whose V th space is ⋁_{k} Σ^{V} Gr_{k}(V)^{Hom(γ,W)}.

The construction is natural in the sense that if V ⊆ W, the inclusion of Thom spaces gives a ring map R_{V} ⟶ R_{W}. The particular case V = 0 gives a cofibre sequence

J_{W} ⟶ R_{0} ⟶ R_{W} .

Theorem 4.1 then states

J_{V^{k}} ≃ ⋁_{n} Gr_{n}(𝒰)^{[k−1]}.

A one dimensional representation α gives a map

i_{α} : S^{Hom(α,W)} ⟶ Gr_{1}(𝒰)^{Hom(γ,W)} ⊆ R_{W}

By Lemma 4.5 there is a cofibre sequence

S^{Hom(α,W)} ∧ R_{W} ⟶^{i_{α}} R_{W} ⟶ R_{W ⊕α},

so that in particular

J_{W ⊕α}/J_{W} ≃ fibre(R_{W} ⟶ R_{W ⊕α}) ≃ S^{Hom(α,W)} ∧ R_{W} .

Now for a complex orientable theory we may package the results of this section as saying

E_{∗}^{A}(R_{0}) = Symm(E_{∗}^{A}(ℂP (𝒰))),

and more generally

E_{∗}^{A}(R_{W}) = Symm(E_{∗}^{A}(ℂP (𝒰), ℂP (W))). □

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 [7] we say that E_{A}^{∗}(·) has Thom classes if for any complex vector bundle over a space X, there is an element τ(ξ) ∈ E_{A}^{∗}(X^{ξ}) so that this system of elements is

(1) natural for bundle maps,

(2) product preserving in the sense that τ(ξ × η) = τ(ξ) ∧ τ(η) and

(3) normalized so that if X = A/B and the fibre of ξ over B is the representation W of

B then τ(ξ) = σ_{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 τ(ξ) was a unit multiple of σ_{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

E_{A}^{∗}(X) ⟶^{≅} Ẽ_{A}^{∗}(X^{ξ})

(essentially by cup product with τ(ξ)) or

Ẽ_{∗}^{A}(X^{ξ}) ⟶^{≅} E_{∗}^{A}(X)

(essentially by cap product with τ(ξ)) we proceed from the fact that σ_{V} is an isomorphism. We may then work our way up the cellular filtration of X using the normalization condition. Alternatively, if ξ has a complement η so that ξ ⊕ η = V we may use the product preserving property.

Before proceeding we make a reduction. Lemma 6.2. A cohomology theory E_{A}^{∗}(·) has a system of Thom classes if and only if it has universal Thom classes,

τ_{n} ∈ Ẽ_{A}^{∗}(Gr_{n}(𝒰)^{γ_{n}})

for each n, so that

(1) the system is product preserving in the sense that under the direct sum map Gr_{m}(𝒰)× Gr_{n}(𝒰) ⟶ Gr_{m+n}(𝒰), the class τ_{m+n} pulls back to τ_{m} ∧ τ_{n}

(2) Over the point W of Gr_{n}(𝒰) with isotropy B ⊆ A the class τ_{n} restricts to σ_{W} (1); We may then define the Thom class for an n-plane bundle by pulling back τ_{n} under the Thomification 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.

Proof: We shall construct classes τ_{n} as specified in 6.2. Assume that the flag begins with V^{1} = ϵ. By pullback along conjugation, we see that

Gr_{n}(𝒰)^{γ_{n}} ≅ Gr_{n}(𝒰)^{γ_{n}^{∗}} ≃ Gr_{n}(𝒰)/Gr_{n}(𝒰)^{[0]},

and we have identified the cohomology of this space exactly. From the orientation we construct the cohomology class y(ϵ) ∈ E_{A}^{∗}(ℂP (𝒰)), and hence by 2.2, the element y(ϵ)^{⊗n} ∈ E_{A}^{∗}(Gr_{n}(𝒰)). By construction this restricts to zero in E_{A}^{∗}(Gr_{n}(𝒰)^{[0]}), and thus we obtain

τ_{n} = y(ϵ)^{⊗n} ∈ E_{A}^{∗}(Gr_{n}(𝒰), Gr_{n}(𝒰)^{[0]}) ≅ Ẽ_{A}^{∗}(Gr_{n}(𝒰)^{γ_{n}}).

The element τ_{n} is identified with a map

τ_{n} : Gr_{n}(𝒰)^{γ_{n}} ⟶ E.

Lemma 6.4. The classes τ_{n} are compatible under product in the sense that

\xymatrix{Gr_{m}(𝒰)^{γ_{m}} ∧ Gr_{n}(𝒰)^{γ_{n}} \ar[r]^{τ_{m}∧τ_{n}} \ar[d] & E ∧ E \ar[d] \\ Gr_{m+n}(𝒰)^{γ_{m+n}} \ar[r]_{τ_{m+n}} & E}

commutes up to homotopy. Proof: This follows from the fact that y(ϵ)^{⊗m} ⊗ y(ϵ)^{⊗n} = y(ϵ)^{⊗(m+n)}. □

Lemma 6.5. The class τ_{n} is a Thom class for γ_{n}. Proof: An A-fixed point of Gr_{n}(𝒰) is a representation V = β_{1} ⊕ β_{2} ⊕ ⋯ ⊕ β_{n}, and such a point lies in the image of the direct sum map from (ℂP (𝒰))^{×n}. By naturality it therefore suffices to deal with the case n = 1. However, by definition of an orientation τ_{1} = y(ϵ) restricts to a generator of E_{A}^{∗}(ℂP (α ⊕ ϵ), ℂP (ϵ)). Since we have used the orientation to give the complex stable structure, the generator is σ_{α^{−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 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 θ : 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.

Proof: Let x(ϵ) ∈ M̃U^{∗}(ℂP (𝒰)) denote an orientation of MU. By hypothesis the restriction λ(α^{−1}) of x(ϵ) to M̃U^{∗}(S^{α^{−1}}) is a generator as an MU_{A}^{∗}-module. From the suspension isomorphism M̃U^{∗}(S^{α^{−1}}) ≅ M̃U^{∗−α^{−1}}, and λ(α^{−1}) is a unit in the RO(G)-graded coefficient ring. Hence θ_{∗}(λ(α^{−1})) is also a unit in the RO(G)-graded sense, and therefore θ_{∗}(x(ϵ)) restricts to a generator of the integer graded module Ẽ_{A}^{∗}(S^{α^{−1}}) 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) = Gr_{n}(V ⊕ 𝒰)^{γ_{n}}.

If U ⊆ V with U of dimension m, the structure map

S^{V −U} ∧ MU(U) = Gr_{m}(U ⊕ 𝒰)^{(V −U)⊕γ_{m}} ⟶ Gr_{n}(V ⊕ 𝒰)^{γ_{n}}

is given by Thomifying the map adding V − U. Thus

MU ≃ holim_{→}_{V} Σ^{−V} Gr_{n}(V ⊕ 𝒰)^{γ_{n}},

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 defined by the orientation, the structure map Σ^{V −U}Gr_{n}(U ⊕ 𝒰)^{γ_{m}} ⟶ Gr_{n}(V ⊕ 𝒰)^{γ_{n}}, takes the element

σ_{V −U}τ_{m}(β_{i_{1}}(ℱ) ⊗ β_{i_{2}}(ℱ) ⊗ ⋯ ⊗ β_{i_{m}}(ℱ))

to

τ_{n}(β_{0}(ℱ) ⊗ ⋯ ⊗ β_{0}(ℱ) ⊗ β_{i_{1}}(ℱ)) ⊗ β_{i_{2}}(ℱ) ⊗ ⋯ ⊗ β_{i_{m}}(ℱ)). □

Proposition 7.4. The classes τ_{n} assemble to give a unique map

τ : MU ⟶ E

of ring spectra. Proof: Since the structure maps are surjective in cohomology,

E_{A}^{∗}(MU) = lim_{←}_{V} E_{A}^{∗}(Σ^{−V} Gr_{n}(𝒰)^{γ_{n}}).

The existence of the map τ therefore follows from compatibility of the elements τ_{n} under suspension. Lemma 7.5. The classes τ_{n} are compatible under suspension in the sense that

\xymatrix{Σ^{V −U}Gr_{m}(U ⊕ 𝒰)^{γ_{m}} \ar[r]^{Σ^{V −U} τ_{m}} \ar[d] & E \ar[d] \\ Gr_{n}(V ⊕ 𝒰)^{γ_{n}} \ar[r]^{τ_{n}} & E}

commutes up to homotopy.

Proof: First note that the space of isometric isomorphisms 𝒰 ⟶ 𝒰 is contractible, so the particular identification 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 Thomification of the map classifying direct sum of bundles, the compatibility of the elements τ_{n} under products shows that the map τ 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

E_{∗}^{A}(MU) = lim_{→}_{n} E_{∗}^{A}(Σ^{−V^{n}}MU(V^{n})) = lim_{→}_{n} E_{∗}^{A}(Σ^{−V^{n}}Gr_{n}(V^{n} ⊕ 𝒰)^{γ_{n}}).

Thus we may construct elements using the maps

E_{∗}^{A}(Gr_{n}(𝒰)) ⟶^{≅} E_{∗}^{A}(Σ^{−V^{n}}Gr_{n}(𝒰)^{γ_{n}}) = E_{∗}^{A}(Σ^{−V^{n}}MU(V^{n})) ⟶ E_{∗}^{A}(MU).

We identified the effect of the maps in the direct system in 7.3. It is natural to suppress the terms β(V^{0}). Definition 8.1. We write b_{i}(ℱ) for the image of β_{i+1}(ℱ) in E_{∗}^{A}(MU). It follows that the image of β_{i_{1}+1}(ℱ) ⊗ β_{i_{2}+1}(ℱ) ⊗ ⋯ ⊗ β_{i_{n}+1}(ℱ) is b_{i_{1}}(ℱ)b_{i_{2}}(ℱ) ⋯ b_{i_{n}}(ℱ). In the nonequivariant context, it is standard to write β_{i+1} for β_{i+1}(ℱ) and b_{i} for b_{i}(ℱ).

Note that Whitney sum of bundles gives a product MU ∧ MU ⟶ MU, and hence E_{∗}^{A}(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 polynomial over E_{A}^{∗}:

E_{∗}^{A}(MU) = E_{A}^{∗}[b_{1}(ℱ), b_{2}(ℱ), b_{3}(ℱ), ⋯].

Furthermore there is a Künneth 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 relation (σ ⊗ τ) ∩ (β ⊗ γ) = ±(σ ∩ β) ⊗ (τ ∩ γ). □

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 E_{A}^{∗}, and we have a Künneth isomorphism. Corollary 8.3. If E is complex orientable, passage to E-homology gives

[MU^{∧s}, E]_{∗}^{A} = E_{∗}^{A}-mod(E_{∗}^{A}(MU^{∧s}), E_{∗}^{A}). □

We really want to understand the set Ring_{A}[MU, E] of homotopy classes of ring maps MU ⟶ E. Since E_{∗}^{A}(MU) is a free E_{∗}^{A}-algebra, this too is immediate.

Corollary 8.4. If E is complex orientable, passage to E-homology gives

Ring[MU, E]^{A} = E_{∗}^{A}-alg(E_{∗}^{A}(MU), E_{∗}^{A}). □

Combining this with 7.4 and being careful about normalization, we obtain a useful consequence. Corollary 8.5. If x(ϵ) is a complex orientation of E_{A}^{∗}(·), then there is a natural bijective correspondence between orientations of E_{A}^{∗}(·) in cohomological degree 2 which give the same complex stable structure as x(ϵ) and

Ring[MU, E]^{A} = E_{∗}^{A}-alg(E_{∗}^{A}(MU), E_{∗}^{A}).

More precisely, suppose x′(ϵ) is an orientation in cohomological degree 2 giving the same complex stable structure as x(ϵ). If V^{1} = ϵ and the associated parameter is

y′(ϵ) = Σ_{i}λ_{i}y(V^{i})

then λ_{0} = 0 and λ_{1} = 1. The associated algebra homomorphism

E_{∗}^{A}(MU) = E_{A}^{∗}[b_{1}(ℱ), b_{2}(ℱ), b_{3}(ℱ), …] ⟶ E_{A}^{∗}

is then determined by

b_{i}(ℱ) ⟼ λ_{i+1}

for i ≥ 1. Proof: In general, if x′(ϵ) is an orientation of E, then the associated map MU ⟶ E restricts to x′(ϵ)^{⊗n} as a map

Σ^{−2n}((ℂP (𝒰))^{×n})^{γ_{n}} ⟶ Σ^{−2n}(Gr_{n}(𝒰))^{γ_{n}} ⟶ E.

Thus the algebra homomorphism associated to x′(ϵ) may be calculated on an element z arising from Σ^{−2n}MU(V) as x′(ϵ)^{⊗n}(z).

Now suppose x′(ϵ) is expressed in terms of the basis associated to the orientation x(ϵ). The coefficient λ_{0} = 0 since y′(ϵ) restricts trivially to ℂP (ϵ). The coefficient λ_{1} is obtained by restricting to (ℂP (V^{2}), ℂP (V^{1})) ≅ S^{α_{2}^{−1}}; since the complex stable structures defined by x(ϵ) and x′(ϵ) agree, the coefficient is 1. It is easy to see the images of b_{i}(ℱ) for i ≥ 1 are as specified since they arise from Σ^{−V^{1}}MU(V^{1}). □

Remark 8.6. (i) For a complex stable structure arising from an orientation, the element σ_{α}(1) for any chosen α determines the rest, because of the action of A^{∗}. This explains why consideration of α_{2}^{−1} alone was enough to show λ_{1} = 1. (ii) The statement of the corollary is consistent with the conjecture that MU_{∗}^{A}MU classifies strict isomorphisms of A-equivariant formal group laws.

References

[1] J.F.Adams “Stable homotopy and generalized homology.” Chicago University Press (1974).

[2] M.Cole “Complex oriented RO(G)-graded equivariant cohomology theories and their formal group laws.” Thesis, University of Chicago (1996)

[3] M.Cole “Complex oriented equivariant cohomology theories and equivariant complex projective spaces” (In preparation)

[4] Michael Cole, J.P.C.Greenlees and I.Kriz “Equivariant formal group laws.” Proc. London Math. Soc 81 (2000) 355-386.

[5] A.D.Elmendorf, I.Kriz, M.A. Mandell and J.P.May “Rings, modules and algebras in stable homotopy theory.” American Mathematical Society Surveys and Monographs, American Mathematical Society, (1997)

[6] J.W.Milnor and J.Stasheff “Characteristic classes.” Princeton University Press (1974)

[7] C.Okonek “Der Conner-Floyd-Isomorphismus für 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