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
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