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
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