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.
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