14 CORBETT REDDEN

When M admits a string structure (i.e. M is a spin manifold with spin characteristic

class

p1

2

(M) = 0 ∈

H4(M;

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

[HBJ].

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

0(pt)

→

K0(pt)

∼

=

Z or MU

∗

→ K∗

∼

=

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,

K∗(X)

∼

=

MP

∗(X)

⊗MP0 Z

∼

=

MU

∗(X)

⊗MU

∗

Z β,

β−1

.

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

E∗(X)

:= MU

∗

(X) ⊗MU

∗

R.

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

∗

Z[

1

2

, δ, ,

Δ−1]

whose associated cohomology theory is complex oriented with formal group law given

by the Euler formal group law. For finite CW complexes X,

Ell∗(X)

= MU

∗

(X) ⊗MU

∗

Z[

1

2

, δ, ,

Δ−1].

In

Ell∗,

|δ| = −4, | | = −8.