6 CORBETT REDDEN

Letting |β| = −2, we could equivalently define MP

n

(X) ⊂ MU

∗

(X) β,

β−1

as

formal series which are homogeneous of degree n.

Definition 1.4. In E-cohomology, a Thom class for the vector bundle V → X

(with dimR V = n) is a class UV ∈

En(Th(V

)) such that for each x ∈ X there

exists ϕx :

Rn

→ Vx so that UV → 1 under the following composition:

En(Th(V )) En(Th(Vx))

∼

=

ϕx∗

En(Sn)

∼

=

E0(pt)

UV

✤

1

Thom classes give rise to Thom isomorphisms

E∗(X)

·UV

→

E∗+n(Th(V

)).

The existence of Thom isomorphisms allows one to construct pushforward maps

in cohomology theories, which in turn gives important invariants generalizing the

Euler class. Ordinary cohomology with Z/2 coeﬃcients admits Thom classes for

all vector bundles, but only oriented bundles have Thom classes in

H∗(−;

Z). In

general, we would like to functorially define Thom classes compatible with the ⊕

operation. Such a choice for vector bundles with lifts of the structure group to Gk is

called a G-orientation of the cohomology theory E, and E is said to be G-orientable

if there exists such an orientation. A specific orientation will be given by universal

Thom classes in En(MGn) and is equivalent (at least up to homotopy) to a map

of ring spectra MG → E. We will mostly be concerned with complex orientable

theories, and summarizing the above discussion gives the following definition.

Definition 1.5. A complex orientation of E is a natural, multiplicative, col-

lection of Thom classes UV ∈ E2n(Th(V )) for all complex vector bundles V → X,

where dimC V = n. More explicitly, these classes must satisfy

• f

∗(UV

) = Uf

∗V

for f : Y → X,

• UV1⊕V2 = UV1 · UV2 ,

• For any x ∈ X, the class UV maps to 1 under the

composition1

E2n(Th(V

)) →

E2n(Th(Vx))

∼

=

→

E2n(S2n)

∼

=

→

E0(pt).

Given a complex orientation, we can define Chern classes in the cohomology

theory E. Because the zero-section

CP∞

→ ξ induces a homotopy equivalence

CP∞

∼

→ Th(ξ), the universal Thom class for line bundles is naturally a class c1 ∈

E2(CP∞),

and it plays the role of the universal first Chern class. If one computes

E∗(CP∞),

the existence of c1 implies the Atiyah–Hirzebruch spectral sequence must

collapse at the E2 page. This implies the first part of the following theorem.

Theorem 1.6. A complex orientation of E determines an isomorphism

E∗(CP∞)

∼

=

E∗(pt)

c1 ,

and such an isomorphism is equivalent to a complex orientation. Furthermore, any

even periodic theory is complex orientable.

1The

complex structure induces an orientation on Vx, hence there is a canonical homotopy

class of map ϕx :

R2n

→ Vx.