• Rezk, Charles. Notes on the Hopkins–Miller theorem. Homotopy theory via
algebraic geometry and group representations (Evanston, IL, 1997), 313–366,
Contemp. Math., 220, Amer. Math. Soc., Providence, RI, 1998.
• Schwede, Stefan. Book project about symmetric spectra. Book preprint. Avail-
• Complex oriented cohomology theories and the language of stacks. Course
notes for 18.917, taught by Mike Hopkins (1999), available at
• Naumann, Niko. The stack of formal groups in stable homotopy theory. Adv.
Math. 215 (2007), no. 2, 569–600.
• The stacks project. Open source textbook, available at
• Vistoli, Angelo. Grothendieck topologies, fibered categories and descent theory.
Fundamental algebraic geometry, 1–104, Math. Surveys Monogr., 123, Amer.
Math. Soc., Providence, RI, 2005.
• Hatcher, Allen. Spectral sequences in algebraic topology. Book preprint. Avail-
• McCleary, John. A user’s guide to spectral sequences. Second edition. Cam-
bridge Studies in Advanced Mathematics, 58. Cambridge University Press,
Cambridge, 2001. xvi+561 pp. ISBN: 0-521-56759-9
• Weibel, Charles A. An introduction to homological algebra. Cambridge Studies
in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994.
xiv+450 pp. ISBN: 0-521-43500-5; 0-521-55987-1
The contents of this book span four levels: the first five chapters (elliptic co-
homology, elliptic curves, the moduli stack, the exact functor theorem, sheaves in
homotopy theory) are more elementary, classical, and expository and we hope will
be tractable for all readers and instructive or at least entertaining for all but the
experts; the next three chapters (the Hasse square, the local structure of the moduli
stack, obstruction theory) are somewhat more sophisticated in both content and
tone, and especially for novice and intermediate readers will require more deter-
mination, patience, and willingness to repeatedly pause and read other references
before proceeding; the last five chapters (from spectra to stacks, string orientation,
the sheaf of ring spectra, the construction, the homotopy groups) are distinctly
yet more advanced, with Mike Hopkins’ reflective account of and perspective on
the subject, followed by an extensive technical treatment of the construction and
homotopy of tmf ; finally the three classic manuscripts (Hopkins–Miller, Hopkins–
Mahowald, Hopkins) illuminate the original viewpoint on tmf —a careful reading
of them will require serious dedication even from experts.
In addition to the references listed above, we encourage the reader to consult
the following sources about tmf more broadly: