18.915 | Fall 2014 | Graduate

Graduate Topology Seminar: Kan Seminar

Reading List

In this section, Professor Haynes Miller discusses the reading list for the course and explains how it creates a narrative arc for understanding algebraic topology.

Narrative Arc

The exact set and sequence of papers read in the Kan seminar varies from one year to the next. The sequence in this particular year seemed to work out very well. The narrative arc of the course begins with Serre’s work from the early 1950’s: Fibrations, spectral sequences, and the cohomology of Eilenberg-Maclane spaces. As a prologue, we read about the construction and basic properties of Steenrod operations. These lead to a construction of the Stiefel-Whitney classes (from Milnor and Stasheff). We get started with simplicial sets using Milnor’s paper on geometric realization. This is followed by Thom’s work on the Steenrod problem and transversality. The oriented bordism ring and the Hirzebruch signature theorem is next (from Milnor and Stasheff). We are then in a position to understand Milnor’s paper on exotic 7-spheres.

Then K-theory comes along (from Atiyah’s book). (Sometimes we do a proof of Bott periodicity, but this is often part of a course in global analysis so I don’t mind omitting it.) We can then do Adams’s and Atiyah’s proof of Hopf invariant one, and (if there’s an ambitious student) Adams’s resolution of the vector field problem.

Handwritten list of research papers in the field of algebraic topology.

A portion of the original reading list (PDF), in Dan Kan’s handwriting. (Image courtesy of Michael Kan.)

Quillen’s work on equivariant cohomology is up next. Wall’s finiteness obstruction paper is very difficult for students, and may be inadvisable.

Stable homotopy theory follows, first Boardman’s construction of CW spectra following Adams, then May’s treatment of operads and the geometry of iterated loop spaces. Enough expertise with classifying spaces is in place for Quillen’s higher algebraic K-theory paper to make sense now. Nishida’s paper on nilpotence is always very hard and quite mysterious, somehow too far ahead of its time.

Atiyah and Hirzebruch’s paper on topological obstructions to the integral Hodge conjecture was a student suggestion. I think it worked well: An argument very much like Thom’s, but integrating K-theory and the topological Riemann-Roch theorem.

Now comes Quillen’s theory of model categories, followed immediately by Bousfield’s use of this formalism in the construction of localization functors and Sullivan’s rational homotopy theory (following Griffiths and Morgan).

After a while, the students will often start to propose their own papers. If the paper they propose seems reasonable, and connects well with the sequence of the course, then I welcome its inclusion. That’s exactly what I want to have happen.

Conversations about the Readings

I invite students to have individual conversations with me about the papers they are reading. Sometimes students feel frustrated with the papers because they have a difficult time understanding the writing. I listen patiently. I usually cannot help them very much with that aspect of the work, but the conversation gives me an opportunity to talk with them about the mathematics included in the papers. Students are ready to hear my thoughts about the mathematics in a way that they wouldn’t be if they weren’t struggling with reading the papers. Their struggles prime them for engagement and learning. I advertise these conversations as being 30-minute meetings, but in reality, we talk as long as students need to in order to make progress with their work.

Serre, J. -P. “Cohomologie modulo 2 des complexes d’Eilenberg-Maclane.” Commentarii Mathematici Helvetici 27, no. 1 (1953): 198–232.

Thom, R. “Quelques propriétés globales des variétés différentiables.” Commentarii Mathematici Helvetici 28 (1954): 17–86.

Milnor, J. W. “On the Construction FK.” In Algebraic Topology: A Student’s Guide (London Mathematical Society Lecture Note). Vol. 4. Cambridge University Press, 1972, pp. 119–36. ISBN: 9780521080767.

———. “On Manifolds Homeomorphic to the 7–sphere.” Annals of Mathematics 64, no. 2 (1956): 399–405.

Milnor, J. W., and J. D. Stasheff. Characteristic Classes (Annals of Mathematics Studies). Vol. 76_._ Princeton University Press, 1974. ISBN: 9780691081229. [Preview with Google Books]

Milnor, J. W. “The Geometric Realization of a Semi–simplicial Complex.” Annals of Mathematics 65, no. 2 (1957): 357–62.

Moore, J. C. “Semi-simplicial complexes and Postnikov systems.” In Symposium Internacional de Topologia Algebraica. La Universidad Nacional Autonoma de Mexico y la UNESCO, 1958, pp. 232–47.

Bott, R. “The stable Homotopy Groups of the Classical Groups.” Annals of Mathematics 70, no. 2 (1959): 313–37.

Adams, J. F. “On the Non-existence of Elements of Hopf Invariant One.” Annals of Mathematics 72, no. 1 (1960): 20–104.

———. “Vector Fields on Spheres.” Annals of Mathematics 75, no. 3 (1962): 603–32.

Brown, E. H., Jr.  “Cohomology Theories.” Annals of Mathematics 75, no. 3 (1962): 467–84.

Kervaire, M. A., and J. W. Milnor. “Groups of Homotopy Spheres: I.” Annals of Mathematics 77, no. 3 (1963): 504–37.

Atiyah, M. F. K–theory (Advanced Book Classics). 2nd ed. Notes by D. W. Anderson. Addison Wesley Longman Publishing Company, 1989. ISBN: 9780201093940.

Wall, C. T. C. “Finiteness Conditions for CW–complexes.” Annals of Mathematics 81, no. 1 (1965): 56–69.

Adams, J. F. “On the Groups J (X)–IV.” Topology 5, no. 1 (1966): 21–71.

Adams, J. F., and M. F. Atiyah. “K–theory and the Hopf Invariant.” The Quarterly Journal of Mathematics 17, no. 1 (1966): 31–38.

Quillen, D. G. Homotopical Algebra (Lecture Notes in Mathematics). Vol. 43. Springer, 1967. ISBN: 9783540039143.

Segal, G. “Classifying Spaces and Spectral Sequences.” Publications Math´ematiques de I’lnstitut Hautes des E’tudes Scientifique 34, no. 1 (1968): 105–12.

Atiyah, M. F., and G. Segal. “Equivariant K-theory and Completion.” Journal of Differential Geometry 3, no. 1–2 (1969): 1–18.

Adams, J. F. “Quillen’s Work on Formal Groups and Complex Cobordism.” In Stable Homotopy and Generalised Homology (Chicago Lectures in Mathematics). University of Chicago Press, 1995. ISBN: 9780226005249. [reprint of the 1974 original]

Quillen, D. G. “The Spectrum of an Equivariant Cohomology Ring: I.” Annals of Mathematics 94, no. 3 (1971): 549–72.

May, J. P. The Geometry of Iterated Loop Spaces (Lectures Notes in Mathematics). Vol. 271. Springer-Verlag, 1972.

Quillen, D. G. “On the Cohomology and K-theory of the General Linear Groups Over a Finite Field.” Annals of Mathematics 96, no. 3 (1972): 552–86.

———. “Algebraic K–theory I.” In Higher K-Theories (Lecture Notes in Mathematics). Vol. 341. Springer, 1973, pp. 85–147. ISBN: 9783540064343.

Nishida, G. “The Nilpotency of Elements of the Stable Homotopy Groups of Spheres.” Journal of the Mathematical Society of Japan 25, no. 4 (1973): 707–32.

Adams, J. F. Stable Homotopy and Generalised Homology (Chicago Lectures in Mathematics). University of Chicago Press, 1995. ISBN: 9780226005249. [reprint of the 1974 original]

Segal, G. “Categories and Cohomology Theories.” Topology 13, no. 3 (1974): 293–312.

Bousfield, A. K. “The Localization of Spaces with Respect to Homology.” Topology 14, no. 2 (1975): 133–50.

Griffiths, P. A., and J. W. Morgan. Rational Homotopy Theory and Differential Forms. Birkh ̈auser, 1981. ISBN: 9783764330415.

Adams, J. F. “Prerequisites (on equivariant stable homotopy theory) for Carlsson’s lecture.” In Algebraic Topology (Lecture Notes in Mathematics). Vol. 1051. Springer-Verlag, 1984, pp. 483–532. ISBN: 9780387129020.

Waldhausen, F. “Algebraic K-theory of Spaces.” In Algebraic and Geometric Topology (Lecture Notes in Mathematics). Vol. 1126. Springer, 1985, pp. 318–419. ISBN: 9783540152354.

Most of the papers read in the seminar are available online, through the MIT library’s VERA database. You need an MIT certificate to use it. Many journals are also available directly through MathSciNet.

For mainly German documents visit Goettinger Digitalisierungs-Zentrum and follow links to Mathematical Literature.

For much earlier work, try la bibliotheque Gallica-Math. The most useful link from there is to NUMDAM, an archive of seminars and other mathematical documents. I especially commend to you the Seminaire Henri Cartan:

Year 1950–51: Cohomologie des groupes, suite spectrale, faisceaux

Years 1953–55: Algebre d’Eilenberg-Maclane et homotopie

Year 1958–59: Invariant de Hopf et operations cohomologiques secondaires

Year 1959–60: Periodicite des groupes d’homotopie stable des groupes classiques, d’apres Bott

For more recent work, the standard preprint server is the Front for the Mathematics ArXiv.

Course Info

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Fall 2014
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