Chapter 1
Elliptic genera and elliptic cohomology
Corbett Redden
The goal of this overview is to introduce concepts which underlie elliptic coho-
mology and reappear in the construction of tmf . We begin by defining complex-
oriented cohomology theories and looking at the two special cases of complex cobor-
dism and K-theory. We then see that a complex orientation of a cohomology theory
naturally leads to a formal group law. Furthermore, Quillen’s theorem states that
the universal complex-oriented theory (complex cobordism) encodes the universal
formal group law. This implies that complex genera, or homomorphisms from the
complex cobordism ring to a ring R, are equivalent to formal group laws over R.
The group structure on an elliptic curve naturally leads to the notion of an elliptic
genus. Finally, we use the Landweber exact functor theorem to produce an elliptic
cohomology theory whose formal group law is given by the universal elliptic genus.
Elliptic cohomology was introduced by Landweber, Ravenel, and Stong in the
mid-1980’s as a cohomological refinement of elliptic genera. The notion of elliptic
genera had previously been invented by Ochanine to address conjectured rigidity
and vanishing theorems for certain genera on manifolds admitting non-trivial group
actions. Witten played an important role in this process by using intuition from
string theory to form many of these conjectures. He subsequently interpreted the
elliptic genus as the signature of the free loop space of a spin manifold, beginning a
long and interesting interaction between theoretical physics and algebraic topology
that is still active today. While we don’t have the space to adequately tell this story,
there are already several excellent references: the introductory article in [Lan] gives
the history of elliptic genera and elliptic cohomology, [Seg] explains how they should
be related to more geometric objects, and [Hop] summarizes important properties
of tmf . Finally, both [Lur] and [Goe] give a detailed survey of elliptic cohomology
and tmf from the more modern perspective of derived algebraic geometry.
1. Complex-oriented cohomology theories
A generalized cohomology theory E is a functor from (some subcategory of)
topological spaces to the category of abelian groups. This functor must satisfy all
the Eilenberg–Steenrod axioms except for the dimension axiom, which states the
cohomology of a point is only non-trivial in degree 0. Any cohomology theory is
represented by a spectrum which we also call E, and from a spectrum the reduced
3
http://dx.doi.org/10.1090/surv/201/01
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