INTRODUCTION xvii

Hopkins and Miller conceived of the problem as lifting this presheaf of ho-

mology theories to a sheaf of spectra. In the 80s and early 90s, Dwyer, Kan,

Smith, and Stover had developed an obstruction theory for rigidifying a diagram

in a homotopy category (here a diagram of elliptic homology theories) to an honest

diagram (here a sheaf of spectra). Hopkins and Miller adapted the Dwyer–Kan–

Stover theory to treat the seemingly more diﬃcult problem of rigidifying a diagram

of multiplicative cohomology theories to a diagram of A∞-ring spectra. The re-

sulting multiplicative obstruction groups vanished, except at the prime 2—Hopkins

addressed that last case by a direct construction in the category of K(1)-local E∞-

ring spectra. Altogether the resulting sheaf of spectra provided a universal elliptic

cohomology theory, the spectrum TMF of global sections (and its connective ver-

sion tmf ). Subsequently, Goerss and Hopkins upgraded the A∞ obstruction theory

to an obstruction theory for E∞-ring spectra, leading to the definitive theorem of

Goerss–Hopkins–Miller: the presheaf of elliptic homology theories on the compact-

ified moduli stack of elliptic curves lifts to a sheaf of E∞-ring spectra.

Meanwhile, Ando–Hopkins–Strickland (2001) established a systematic connec-

tion between elliptic cohomology and elliptic genera by constructing, for every

elliptic cohomology theory E, an E-orientation for almost complex manifolds with

certain vanishing characteristic classes. This collection was expected to assemble

into a single unified multiplicative tmf -orientation. Subsequently Laures (2004)

built a K(1)-local E∞-map MO 8 → tmf and then finally Ando–Hopkins–Rezk

produced the expected integral map of E∞-ring spectra MO 8 → tmf that recov-

ers Witten’s genus at the level of homotopy groups.

Later, an interpretation of tmf was given by Lurie (2009) using the theory of

spectral algebraic geometry, based on work of T¨ oen and Vezzosi. Lurie interpreted

the stack Mell with its sheaf

Otop

as a stack not over commutative rings but over

E∞-ring spectra. Using Goerss–Hopkins–Miller obstruction theory and a spectral

form of Artin’s representability theorem, he identified that stack as classifying ori-

ented elliptic curves over E∞-ring spectra. Unlike the previous construction of tmf

and of the sheaf Otop, this description specifies the sheaf and therefore the spectrum

tmf up to a contractible space of choices.

3. Overview

Part I

Chapter 1: Elliptic genera and elliptic cohomology. One-dimensional

formal group laws entered algebraic topology though complex orientations, in an-

swering the question of which generalized cohomology theories E carry a theory of

Chern classes for complex vector bundles. In any such theory, the E-cohomology

of

CP∞

is isomorphic to

E∗[[c1]],

the E-cohomology ring of a point adjoin a formal

power series generator in degree 2. The tensor product of line bundles defines a

map

CP∞

×

CP∞

→

CP∞,

which in turn defines a comultiplication on

E∗[[c1]],

i.e., a formal group law. Ordinary homology is an example of such a theory; the

associated formal group is the additive formal group, since the first Chern class

of the tensor product of line bundles is the sum of the respective Chern classes,

c1(L ⊗ L ) = c1(L) + c1(L ). Complex K-theory is another example of such a

theory; the associated formal group is the multiplicative formal group.

Complex cobordism also admits a theory of Chern classes, hence a formal group.

Quillen’s theorem is that this is the universal formal group. In other words, the