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 oen and Vezzosi. Lurie interpreted
the stack Mell with its sheaf
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
is isomorphic to
the E-cohomology ring of a point adjoin a formal
power series generator in degree 2. The tensor product of line bundles defines a

which in turn defines a comultiplication on
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
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