INTRODUCTION xxiii

Chapter 8: Goerss–Hopkins obstruction theory. Goerss–Hopkins ob-

struction theory is a technical apparatus for approaching questions such as the fol-

lowing: for a ring spectrum E and a commutative E∗-algebra A in E∗E-comodules,

is there an E∞-ring spectrum X such that E∗X is equivalent to A? What is the

homotopy type of the space of all such E∞-ring spectra X?

That space is called the realization space of A and is denoted BR(A). There

is an obstruction theory for specifying points of BR(A), and the obstructions

live in certain Andr´ e–Quillen cohomology groups of A. More precisely, there is

a Postnikov-type tower

. . . → BRn(A) → BRn−1(A) → . . . → BR0(A)

with inverse limit BR(A) whose layers are controlled by the Andr´ e–Quillen co-

homology groups of A, as follows. If we let Hn+2(A;ΩnA) be the Andr´e–Quillen

cohomology space (the Eilenberg–MacLane space for the Andr´ e–Quillen cohomol-

ogy group) of the algebra A with coeﬃcients in the nth desuspension of A, then

Hn+2(A;ΩnA) is acted on by the automorphism group of the pair (A, ΩnA) and

we can form, by the Borel construction, a space Hn+2(A;ΩnA) over the classifying

space of Aut(A, ΩnA). This is a bundle of pointed spaces and the base points pro-

vide a section BAut(A,

ΩnA)

→

Hn+2(A;ΩnA).

The spaces BRn(A) then fit into

homotopy pullback squares

BRn(A) BAut(A,

ΩnA)

BRn−1(A) Hn+2(A;ΩnA).

Chapter 9: From spectra to stacks. We have focussed on constructing

spectra using stacks, but one can also go the other way, associating stacks to spectra.

Given a commutative ring spectrum X, let MX be the stack associated to the Hopf

algebroid

(MU∗X, MU∗MU ⊗MU∗ MU∗X).

If X is complex orientable, then MX is the scheme Spec(π∗X)—the stackiness of

MX therefore measures the failure of complex orientability of X. The canonical

Hopf algebroid map (MU∗,MU∗MU) → (MU∗X, MU∗MU ⊗MU∗ MU∗X) induces

a map of stacks from MX to MFG,

(1)

the moduli stack of formal groups with first

order coordinate. Moreover, under good circumstances, the stack associated to a

smash product of two ring spectra is the fiber product over

MFG:(1)

MX∧Y

∼

= MX ×

MF

(1)

G

MY .

It will be instructive to apply the above isomorphism to the case when Y is

tmf , and X is one of the spectra in a filtration

S0

= X(1) → X(2) → · · · X(n) → · · · → MU

of the complex cobordism spectrum. By definition, X(n) is the Thom spectrum

associated to the subspace ΩSU(n) of ΩSU BU; the spectrum X(n) is an E2-

ring spectrum because ΩSU(n) is a double loop space. Recall that for a complex

orientable theory R, multiplicative maps MU → R correspond to coordinates on

the formal group of R. There is a similar story with X(n) in place of MU, where the

formal groups are now only defined modulo terms of degree n+1, and multiplicative