4 CORBETT REDDEN

homology and cohomology groups of a finite CW complex X are given by

En(X) = lim

k→∞

πn+k(X ∧ Ek),

En(X)

= lim

k→∞

[ΣkX,

En+k].

The coeﬃcient groups are abbreviated by

E∗

=

E∗(pt)

and E∗ = E∗(pt), and

they are naturally related by π∗E = E∗

∼

=

E−∗.

We restrict to theories with a

graded commutative ring structure

Ei(X)

×

Ej(X)

→

Ei+j(X)

analogous to the

cup product in ordinary cohomology. They are known as multiplicative cohomology

theories and are represented by ring spectra.

Example 1.1 (Cobordism). A smooth closed (compact with no boundary)

manifold M is said to be null-bordant if there exists a compact manifold W whose

boundary is M. A singular manifold (M, f) in X, where f : M → X is a continu-

ous map, is null-bordant if there exists a singular manifold (W, F ) with boundary

(M, f). The n-th unoriented bordism group of X, denoted by Ωn

O(X),

is the set of

smooth closed singular n-manifolds in X modulo null-bordism; the group structure

is given by the disjoint union of manifolds.

Let {Gk} be a sequence of topological groups with representations {Gk

→kρ

O(k)} which are compatible with the inclusion maps. We define a G-structure on

M as a stable lift of the structure groups to Gk for the stable normal bundle νM .

Suppose a manifold W with ∂W = M has a G-structure on νW that extends to

the G-structure on νM . This is considered a null-bordism of M as a G-manifold.

The abelian group Ωn

G(X)

is then defined as before; it is the set of smooth closed

singular n-manifolds on X with G-structure on ν, modulo null-bordism. Up to

homotopy, G-structures on the stable tangent bundle and stable normal bundle are

equivalent; we later use this fact in geometric constructions.

The functors Ω∗

G

are examples of generalized homology theories, and the

Pontryagin–Thom construction shows they are represented by the Thom spectra

MG = {MGk} = {Th(ρkξk)}.

∗

Here, ξk → BO(k) is the universal k-dimensional

vector bundle (ξk = EO(k) ×O(k)

Rk),

and for any vector bundle V → X the Thom

space Th(V ) is defined as the unit disc bundle modulo the unit sphere bundle

D(V )/S(V ). Particularly common examples of G-bordism include oriented bor-

dism, spin bordism, and complex bordism, corresponding to the groups SO(k),

Spin(k), and U(k), respectively. Bordism classes in these examples have an orien-

tation, spin structure, or complex structure on the manifold’s stable normal bundle

(or stable tangent bundle).

The spectrum MG defines a generalized cohomology theory known as G-

cobordism. It is also a multiplicative cohomology theory (assuming there are maps

Gk1 × Gk2 → Gk1+k2 compatible with the orthogonal representations). The coeﬃ-

cient ring of MG is simply the bordism ring of manifolds with stable G-structure,

MG−∗(pt)

∼

= MG∗(pt) = Ω∗

G,

and the product structure is induced by the product of manifolds. Of particular in-

terest to us will be oriented cobordism and complex cobordism. The first coeﬃcient

calculation is due to Thom, and the second is from Thom, Milnor, and Novikov:

MSO∗

⊗ Q

∼

=

Q[[CP2], [CP4],...]

(1)

MU

∗

∼

=

Z[a1,a2,...]; |ai| = −2i.