14 CORBETT REDDEN
When M admits a string structure (i.e. M is a spin manifold with spin characteristic
(M) = 0 ∈
Z)), Witten formally defined the Dirac operator on LM
and showed its S1-equivariant index equals ϕW (M), up to a normalization involving
η. If M is spin, then the q-series ϕJ (M) and ϕW (M) both have integer coeﬃcients.
If M is string, ϕW (M) is the q-expansion of a modular form over all of SL2(Z).
Note that the integrality properties can be proven by considering ϕJ and ϕW as a
power series where each coeﬃcient is the index of a twisted Dirac operator on M.
For more detailed information on elliptic genera, an excellent reference is the text
4. Elliptic Cohomology
There is an important description of K-theory via Conner-Floyd. As described
in Section 2, the formal group law for K-theory is given by a map MP
Z or MU
Z β, β−1 , depending on our desired grading conven-
tion. From this map of coeﬃcients encoding the formal group law, one can in fact
recover all of K-theory.
Theorem 4.1. (Conner-Floyd) For any finite cell complex X,
In general, Quillen’s theorem shows that a formal group law F over a graded
ring R is induced by a map MU
→ R. Can we construct a complex-oriented
cohomology theory E with formal group law F over E∗
R? Imitating the Conner-
Floyd description of K-theory, we can define
While E is a functor satisfying the homotopy, excision, and additivity axioms of
a cohomology theory, the “long exact sequence of a pair” will not necessarily be
exact. This is due to the fact that exact sequences are not in general exact after
tensoring with an arbitrary ring. If R is flat over MU
then E will satisfy the long
exact sequence of a pair and will be a cohomology theory.
The condition that R is flat over MU
is very strong and not usually satisfied.
However, the Landweber exact functor theorem states that R only needs to satisfy a
much weaker set of conditions. This criterion, described in more detail in Chapter 5,
states one only needs to check that multiplication by certain elements vi is injective
on certain quotients R/Ii. In the case of the elliptic formal group law, the elements
v1 and v2 can be given explicitly in terms of and δ, and the quotients R/In are
trivial for n 2. Therefore, one can explicitly check Landweber’s criterion and
conclude the following.
Theorem 4.2. (Landweber, Ravenel, Stong) There is a homology theory Ell
Ell∗(X) = MU ∗(X) ⊗MU
, δ, ,
whose associated cohomology theory is complex oriented with formal group law given
by the Euler formal group law. For finite CW complexes X,
, δ, ,
|δ| = −4, | | = −8.