ELLIPTIC GENERA AND ELLIPTIC COHOMOLOGY 5

Rationally, MU

∗

⊗ Q is generated by the complex projective spaces

CPi

for i ≥ 1.

The book [Sto] is an excellent source of further information on cobordism.

Example 1.2 (Complex K-theory). Isomorphism classes of complex vector

bundles over a space X form an abelian monoid via the direct sum ⊕ operation. For-

mally adjoining inverses gives the associated Grothendieck group known as K(X)

or

K0(X);

elements in K(X) are formal differences of vector bundles up to iso-

morphism. The reduced group K0(X) is naturally isomorphic to [X, Z × BU], and

Bott periodicity gives a homotopy equivalence

Ω2(Z

× BU) Z × BU. Therefore,

we can extend Z × BU to an Ω-spectrum known as K, where

K2n = Z × BU,

K2n+1 = Ω(Z × BU) U.

This defines the multiplicative cohomology theory known as (complex) K-theory,

with ring structure induced by the tensor product of vector bundles. A straightfor-

ward evaluation shows that the coeﬃcients π∗K are

K2n(pt)

∼

=

π0(Z × BU) = Z,

K2n+1(pt)

∼

= π0(U) = 0.

Furthermore, Bott periodicity is manifested in K-theory by the Bott class β =

[ξ] − 1 ∈ K(S2)

∼

=

K−2(pt), where ξ → S2 is the Hopf bundle and 1 is the

isomorphism class of the trivial line bundle. The class β is invertible in K∗, and

multiplication by β and β−1 induces the periodicity in general rings K∗(X).

The periodicity in K-theory turns out to be a very convenient property, and it

motivates the following definition.

Definition 1.3. A multiplicative cohomology theory E is even periodic if

Ei(pt)

= 0 whenever i is odd and there exists β ∈

E−2(pt)

such that β is in-

vertible in

E∗(pt).

The existence of β−1 ∈ E2(pt) implies that for general X there are natural

isomorphisms

E∗+2(X)

·β

E∗(X)

·β−1

∼

=

.

given by multiplication with β and

β−1,

so E is periodic with period 2.

A number of cohomology theories, such as ordinary cohomology, are even (i.e.

Eodd(pt)

= 0) but not periodic. Given an arbitrary even cohomology theory, we

can create an even periodic theory A by defining

An(X)

:=

k∈Z

En+2k(X).

For example, if we perform this construction on ordinary cohomology with coeﬃ-

cients in a ring R, we obtain a theory known as periodic ordinary cohomology. The

coeﬃcients of MU in (1) show that MU also is even but not periodic. We define

periodic complex cobordism MP by

MP

n(X)

:=

k∈Z

MU

n+2k(X),