The homotopy of tmf can be computed directly using the Adams spectral
sequence. Alternatively, one can use the elliptic spectral sequence to compute the
homotopy of Tmf . The Adams spectral sequence has the form
(Fp, Fp) ⇒ π∗(tmf )ˆ,p
where Ap tmf := homtmf
(HFp,HFp) is a tmf -analog of the Steenrod algebra.
At the prime 2, the map A2
→ A ≡ A2 to the classical Steenrod algebra is
injective, and the tmf -module Adams spectral sequence can be identified with the
classical Adams spectral sequence
), F2) = ExtA(A/ /A(2), F2) = ExtA(2)(F2, F2) ⇒ π∗(tmf
The elliptic spectral sequence has the form
⇒ πt−s(Tmf ). The
is concentrated in even degrees and is the t/2-th power of a line
bundle ω; the spectral sequence thus has the form
⇒ π2p−q(Tmf ).
The manuscripts. The book concludes with three of the original, previously
unpublished, manuscripts on tmf : “Elliptic curves and stable homotopy I” (1996)
by Hopkins and Miller, “From elliptic curves to homotopy theory” (1998) by Hop-
kins and Mahowald, and “K(1)-local E∞ ring spectra” (1998) by Hopkins. The
first focuses primarily on the construction of the sheaf of (associative) ring spectra
on the moduli stack of elliptic curves, the second on the computation of the homo-
topy of the resulting spectrum of sections around the supersingular elliptic curve at
the prime 2, and the third on a direct cellular construction of the K(1)-localization
of tmf . These documents have been left, for the most part, in their original draft
form; they retain the attendant roughness and sometimes substantive loose ends,
but also the dense, heady insight of their original composition. The preceding chap-
ters of this book can be viewed as a communal exposition, more than fifteen years
on, of aspects of these and other primary sources about tmf .
4. Reader’s guide
This is not a textbook. Though the contents spans all the way from classical
aspects of elliptic cohomology to the construction of tmf , there are substantive
gaps of both exposition and content, and an attempt to use this book for a lecture,
seminar, or reading course will require thoughtful supplementation.
Reading straight through the book would require, among much else, some fa-
miliarity and comfort with commutative ring spectra, stacks, and spectral sequences.
Many of the chapters, though, presume knowledge of none of these topics; instead
of thinking of them as prerequisites, we suggest one simply starts reading, and as
appropriate or necessary selects from among the following as companion sources:
Commutative ring spectra:
• May, J. Peter. E∞ ring spaces and E∞ ring spectra. With contributions by
Frank Quinn, Nigel Ray, and Jorgen Tornehave. Lecture Notes in Mathemat-
ics, Vol. 577. Springer-Verlag, Berlin-New York, 1977.