Introduction

Contents

1. Elliptic cohomology

2. A brief history of tmf

3. Overview

4. Reader’s guide

1. Elliptic cohomology

A ring-valued cohomology theory E is complex orientable if there is an ‘orien-

tation class’ x ∈

E2(CP∞)

whose restriction along the inclusion S2

∼

=

CP1 →

CP∞

is the element 1 in E0S0

∼

=

E2CP1. The existence of such an orientation class

implies, by the collapse of the Atiyah–Hirzebruch spectral sequence, that

E∗(CP∞)

∼

=

E∗[[x]].

The class x is a universal characteristic class for line bundles in E-cohomology; it

is the E-theoretic analogue of the first Chern class. The space

CP∞

represents the

functor

X → {isomorphism classes of line bundles on X},

and the tensor product of line bundles induces a multiplication map

CP∞ ×CP∞

→

CP∞.

Applying

E∗

produces a ring map

E∗[[x]]

∼

=

E∗(CP∞)

→

E∗(CP∞

×

CP∞)

∼

=

E∗[[x1,x2]];

the image of x under this map is a formula for the E-theoretic first Chern class

of a tensor product of line bundles in terms of the first Chern classes of the two

factors. That ring map

E∗[[x]]

→

E∗[[x1,x2]]

is a (1-dimensional, commutative)

formal group law—that is, a commutative group structure on the formal completion

ˆ

A 1 at the origin of the aﬃne line A1 over the ring E∗.

A formal group often arises as the completion of a group scheme at its identity

element; the dimension of the formal group is the dimension of the original group

scheme. There are three kinds of 1-dimensional group schemes:

(1) the additive group Ga = A1 with multiplication determined by the map

Z[x] → Z[x1,x2] sending x to x1 + x2,

(2) the multiplicative group Gm = A1\{0} with multiplication determined by

the map

Z[x±1]

→ Z[x1

±1

, x2

±1

] sending x to x1x2, and

(3) elliptic curves (of which there are many isomorphism classes).

xi