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.
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