ELLIPTIC GENERA AND ELLIPTIC COHOMOLOGY 5
⊗ Q is generated by the complex projective spaces
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)
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
× 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
π0(Z × BU) = Z,
= 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
= 0 whenever i is odd and there exists β ∈
such that β is in-
The existence of β−1 ∈ E2(pt) implies that for general X there are natural
given by multiplication with β and
so E is periodic with period 2.
A number of cohomology theories, such as ordinary cohomology, are even (i.e.
= 0) but not periodic. Given an arbitrary even cohomology theory, we
can create an even periodic theory A by defining
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