INTRODUCTION xxv
In order to construct that nullhomotopy, one needs to understand the homotopy
type of gl1(tmf )—this is done one prime at a time. The crucial observation is that
there is a map of spectra gl
1
(tmf )p ˆ tmf
p
ˆ, the ‘topological logarithm’, and a
homotopy pullback square
gl
1
(tmf )
log
tmf
LK(1)(tmf )
1−Up
LK(1)(tmf )
where Up is a topological refinement of Atkin’s operator on p-adic modular forms.
The fiber of the topological logarithm is particularly intriguing: Hopkins speculates
that it is related to exotic smooth structures on free loop spaces of spheres.
Chapters 11 and 12: The sheaf of E∞ ring spectra and The construc-
tion of tmf . We outline a roadmap for the construction of tmf , the connective
spectrum of topological modular forms. The major steps in the construction are
given in reverse order.
The spectrum tmf is the connective cover of the nonconnective spectrum
Tmf ,
tmf := τ≥0Tmf ,
and Tmf is the global sections of a sheaf of spectra,
Tmf :=
Otop(Mell
),
where Mell is the moduli stack of elliptic curves with possibly nodal sin-
gularities. This stack is the Deligne–Mumford compactification of the
moduli stack of smooth elliptic curves. Here,
Otop
is a sheaf on Mell
in the ´ etale topology. Also, TMF is the global sections of
Otop
over the
substack Mell of smooth elliptic curves,
TMF :=
Otop(Mell
).
The uppercase ‘T in Tmf signifies that the spectrum is no longer connective (but
it is also not periodic). The ‘top’ stands for topological, and
Otop
can be viewed as
a kind of structure sheaf for a spectral version of Mell .
We are left now to construct the sheaf of spectra
Otop.
The first step is to isolate
the problem at every prime p and at Q. That is, one constructs Op
top,
a sheaf of
spectra on the p-completion (Mell )p and then pushes this sheaf forward along the
inclusion map ιp : (Mell )p Mell . One then assembles these pushforwards to
obtain
Otop,
as follows.
The sheaf
Otop
is the limit in a diagram
Otop
p
ιp,∗Optop
ιQ,∗OQ
top
p
ιp,∗Optop
Q
for a given choice of map αarith : ιQ,∗OQ
top

p
ιp,∗Op top
Q
.
Previous Page Next Page