Steenrod algebra learning seminar

Milnor A(1)
The Milnor subalgebra A(1)

Fix a prime p. The Steenrod algebra A = Ap is the algebra of mod p cohomology operations; it is the mainspring of homotopy theory. The goal of this seminar is to understand the work of Miller, Palmieri, Wilkerson and others on the ring-theoretic properties of Ext A * k k and the work of Hopkins and Palmieri on “stable homotopy over the Steenrod algebra.” Future topics may include computations with the May spectral sequence, generic representation theory and unstable modules, and Lannes' T -functor.

Schedule of talks

6 November 14:30 MC108 Sam Isaacson Introduction: What are cohomology operations?
13 November 14:30 MC108 Dan Christensen The dual of the Steenrod algebra
19 November 11:30 MC107 Dan Christensen The dual of the Steenrod algebra: part 2
26 November 11:30 MC107 Enxin Wu Freeness of modules over the Steenrod algebra
3 December 11:30 MC107 Enxin Wu Freeness of modules over the Steenrod algebra: part 2
10 December 11:30 MC107 Enxin Wu Freeness: part 3
17 December 11:30 MC107 David Barnes Sub-Hopf algebras of the Steenrod algebra
15 January 10:30 MC108 Sam Isaacson Vanishing lines in Ext
22 January 10:30 MC108 Enxin Wu Self-maps and periodicity for modules over the Steenrod algebra
29 January 13:30 MC108 Enxin Wu Margolis' killing construction
5 February 13:30 MC106 Dan Christensen The cohomology of Hopf algebras
26 February 13:30 MC106 Sam Isaacson The algebraic Whitehead conjecture

Bibliography

Introduction to the Steenrod algebra and its dual

There are two ways to define the Steenrod algebra: as the ring of all stable additive mod p cohomology operations, or as the free algebra generated by the Bockstein and power operations modulo the Ádem relations. That these agree is a theorem of Serre (at the prime 2) and Cartan (at odd primes). Mosher and Tangora give an account at the prime 2; compare the work of Rothenberg and Steenrod.

Classically, Steenrod operations act on the mod p cohomology of a space. “Steenrod-like” operations also arise in the mod p homology of an infinite loop space, the cohomology of a cocommutative connected Hopf algebra over a finite field, etc. May's paper gives a unified algebraic construction of all of these sorts of operations and various Serre/Kudo transgression theorems. The power operations acting on various Ext groups are tremendously useful in computing spectral sequence differentials—we'll make use of this machinery later.

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  4. May, J. Peter. A general algebraic approach to Steenrod operations. The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod's Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970) pp. 153–231 Lecture Notes in Mathematics, Vol. 168 Springer, Berlin, 1970 MR0281196 (43 #6915)
  5. Milnor, John. The Steenrod algebra and its dual. Ann. of Math. (2) 67 1958 150–171. MR0099653 (20 #6092)
  6. Mosher, Robert E. and Tangora, Martin C. Cohomology operations and applications in homotopy theory. Harper & Row, Publishers, New York-London 1968 x+214 pp. MR0226634 (37 #2223)
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  8. Serre, Jean-Pierre. Sur les groupes d'Eilenberg-MacLane. (French) C. R. Acad. Sci. Paris 234, (1952). 1243–1245. MR0046047 (13,675c)
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  12. Steenrod, N. E. Cohomology operations. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. Annals of Mathematics Studies, No. 50 Princeton University Press, Princeton, N.J. 1962 vii+139 pp MR0145525 (26 #3056)
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Modules over sub-Hopf algebras of the Steenrod algebra, the Quillen-Venkov argument

Adams and Margolis describe a beautiful criterion to detect when a module over a sub-Hopf algebra of the mod 2 Steenrod algebra is free, namely that all relevant Margolis homology groups vanish; see Moore and Peterson for the odd primary version. Anderson and Davis refined this result at the prime 2 to show that the vanishing of certain Margolis homology groups implies the existence of a vanishing line in Ext. Miller and Wilkerson proved an analogous result in the odd primary case using a generalization of the “Quillen-Venkov argument.”

  1. Adams, J. F. A finiteness theorem in homological algebra. Proc. Cambridge Philos. Soc. 57 1961 31–36. MR0122852 (23 #A184)
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  3. Adams, J. F. and Margolis, H. R. Modules over the Steenrod algebra. Topology 10 (1971), 271–282. MR0294450 (45 #3520)
  4. Adams, J. F. and Margolis, H. R. Sub-Hopf-algebras of the Steenrod algebra. Proc. Cambridge Philos. Soc. 76 (1974), 45–52. MR0341487
  5. Anderson, Donald W. and Davis, Donald M. A vanishing theorem in homological algebra. Comment. Math. Helv. 48 (1973), 318–327. MR0334207 (48 #12526)
  6. Margolis, H. R. Spectra and the Steenrod algebra. Modules over the Steenrod algebra and the stable homotopy category. North-Holland Mathematical Library, 29. North-Holland Publishing Co., Amsterdam, 1983. xix+489 pp. ISBN: 0-444-86516-0 MR0738973 (86j:55001)
  7. Miller, Haynes R. A localization theorem in homological algebra. Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 1, 73–84. MR0494105 (58 #13036)
  8. Miller, Haynes and Wilkerson, Clarence. Vanishing lines for modules over the Steenrod algebra. J. Pure Appl. Algebra 22 (1981), no. 3, 293–307. MR0629336 (82m:55024)
  9. Moore, John C. and Peterson, Franklin P. Nearly Frobenius algebras, Poincaré algebras and their modules. J. Pure Appl. Algebra 3 (1973), 83–93. MR0335572 (49 #353)
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  11. Quillen, D. and Venkov, B. B. Cohomology of finite groups and elementary abelian subgroups. Topology 11 (1972), 317–318. MR0294506 (45 #3576)

Cohomology of sub-Hopf algebras of the Steenrod algebra, properties of Ext

Lin, Palmieri, and Wilkerson compute some ring-theoretic properties of Ext of the Steenrod algebra using methods inspired largely by Quillen's work on ring-theoretic properties of Borel cohomology. In particular, Palmieri computes Ext of the Steenrod algebra “up to F-isomorphism,” i.e., modulo nilpotent elements.

  1. Lin, Wen Hsiung. Cohomology of sub-Hopf-algebras of the Steenrod algebra. J. Pure Appl. Algebra 10 (1977/78), no. 2, 101–113. MR0454975 (56 #13217)
  2. Lin, Wen Hsiung. Cohomology of sub-Hopf-algebras of the Steenrod algebra. II. J. Pure Appl. Algebra 11 (1977/78), no. 1-3, 105–110. MR0506680 (58 #22249)
  3. Palmieri, John H. A note on the cohomology of finite-dimensional cocommutative Hopf algebras. J. Algebra 188 (1997), no. 1, 203–215. MR1432354 (98a:16043)
  4. Palmieri, John H. Quillen stratification for the Steenrod algebra. Ann. of Math. (2) 149 (1999), no. 2, 421–449. MR1689334 (2000g:55026)
  5. Quillen, Daniel. The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR0298694 (45 #7743)
  6. Wilkerson, Clarence. The cohomology algebras of finite-dimensional Hopf algebras. Trans. Amer. Math. Soc. 264 (1981), no. 1, 137–150. MR0597872 (82e:16019)

The stable category of comodules over the dual of the Steenrod algebra, algebraic chromatic theory

Hovey, Palmieri, and Strickland give an axiomization of stable homotopy theory in the book Axiomatic stable homotopy theory. Palmieri uses this machinery in his 2001 Memoirs book to describe algebraic analogues of the Devinatz-Hopkins-Smith nilpotence and periodicity theorems. Some of this work dates back to the early 1990's.

  1. Hopkins, Michael J. and Palmieri, John H. A nilpotence theorem for modules over the mod 2 Steenrod algebra. Topology 32 (1993), no. 4, 751–756. MR1241871 (94f:55014)
  2. Hopkins, Michael J. and Smith, Jeffrey H. Nilpotence and stable homotopy theory. II. Ann. of Math. (2) 148 (1998), no. 1, 1–49. MR1652975 (99h:55009)
  3. Hovey, Mark; Palmieri, John H.; and Strickland, Neil P. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114 pp. MR1388895 (98a:55017)
  4. Mahowald, Mark and Shick, Paul. Periodic phenomena in the classical Adams spectral sequence. Trans. Amer. Math. Soc. 300 (1987), no. 1, 191–206. MR0871672 (88e:55019)
  5. Palmieri, John H. Self-maps of modules over the Steenrod algebra. J. Pure Appl. Algebra 79 (1992), no. 3, 281–291. MR1167577 (93d:55023)
  6. Palmieri, John H. Nilpotence for modules over the mod 2 Steenrod algebra. I, II. Duke Math. J. 82 (1996), no. 1, 195–208, 209–226. MR1387226 (97c:55028)
  7. Palmieri, John H. Stable homotopy over the Steenrod algebra. Mem. Amer. Math. Soc. 151 (2001), no. 716, xiv+172 pp. MR1821838 (2002a:55019)
  8. Palmieri, John H. and Sadofsky, Hal. Self-maps of spectra, a theorem of J. Smith, and Margolis' killing construction. Math. Z. 215 (1994), no. 3, 477–490. MR1262528 (94k:55016)

Ext computations, the lambda algebra, the May spectral sequence

The May spectral sequence is the most powerful method for computing Ext of the Steenrod algebra (the algebraic input of the Adams spectral sequence). Also, hidden inside the cobar complex of the Steenrod algebra is a sub-dga called the lambda algebra. It not only efficiently computes Ext, but contains a sequence of subcomplexes each computing the E1 page of an unstable Adams spectral sequence. Its construction is the subject of Priddy's papers.

  1. Adams, J. F. On the structure and applications of the Steenrod algebra. Comment. Math. Helv. 32 1958 180–214. MR0096219 (20 #2711)
  2. Adams, J. F. On the non-existence of elements of Hopf invariant one. Ann. of Math. (2) 72 1960 20–104. MR0141119 (25 #4530)
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  4. Curtis, Edward B. Simplicial homotopy theory. Advances in Math. 6 1971 107–209 (1971). MR0279808 (43 #5529)
  5. Kochman, S. O. Bordism, stable homotopy and Adams spectral sequences. Fields Institute Monographs, 7. American Mathematical Society, Providence, RI, 1996. xiv+272 pp. ISBN: 0-8218-0600-9 MR1407034 (97i:55017)
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  7. May, J. Peter. Matric Massey products. J. Algebra 12 1969 533–568. MR0238929 (39 #289)
  8. Miller, Haynes R. On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space. J. Pure Appl. Algebra 20 (1981), no. 3, 287–312. MR0604321 (82f:55029)
  9. Milnor, John W. and Moore, John C. On the structure of Hopf algebras. Ann. of Math. (2) 81 1965 211–264. MR0174052 (30 #4259)
  10. Nakamura, Osamu. On the squaring operations in the May spectral sequence. Mem. Fac. Sci. Kyushu Univ. Ser. A 26 (1972), no. 2, 293–308. MR0413103 (54 #1224)
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  14. Tangora, Martin C. On the cohomology of the Steenrod algebra. Math. Z. 116 1970 18–64. MR0266205 (42 #1112)
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  16. Wang, John S. P. On the cohomology of the mod 2 Steenrod algebra and the non-existence of elements of Hopf invariant one. Illinois J. Math. 11 1967 480–490. MR0214065 (35 #4917)

Unstable modules over the Steenrod algebra, generic representation theory, Lannes' T-functor

As we mentioned above, the mod p cohomology of a space is an algebra over the mod p Steenrod algebra. But it also satisfies a dimension axiom: the power operations are zero on sufficiently low-dimensional cohomology classes, for example. The category of all “space-like,”—or more properly, “unstable”—modules over the Steenrod algebra is an essential tool in the proof of the Sullivan conjecture. It is related to the modular representation theory of the general linear group. This is really a topic for its own semester.

  1. Dwyer, William G. and Henn, Hans-Werner. Homotopy theoretic methods in group cohomology. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2001. x+98 pp. ISBN: 3-7643-6605-2 MR1926776 (2003h:20093)
  2. Henn, Hans-Werner; Lannes, Jean; and Schwartz, Lionel. The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects. Amer. J. Math. 115 (1993), no. 5, 1053–1106. MR1246184 (94i:55024)
  3. Kuhn, Nicholas J. Generic representations of the finite general linear groups and the Steenrod algebra. I. Amer. J. Math. 116 (1994), no. 2, 327–360. MR1269607 (95c:55022)
  4. Kuhn, Nicholas J. Generic representations of the finite general linear groups and the Steenrod algebra. II. K-Theory 8 (1994), no. 4, 395–428. MR1300547 (95k:55038)
  5. Kuhn, Nicholas J. Generic representations of the finite general linear groups and the Steenrod algebra. III. K-Theory 9 (1995), no. 3, 273–303. MR1344142 (97c:55026)
  6. Lannes, Jean. Sur les espaces fonctionnels dont la source est le classifiant d'un p-groupe abélien élémentaire. (French) [Function spaces whose source is the classifying space of an elementary abelian p-group] With an appendix by Michel Zisman. Inst. Hautes Études Sci. Publ. Math. No. 75 (1992), 135–244. MR1179079 (93j:55019)
  7. Lurie, Jacob. 18.917 Topics in Algebraic Topology: The Sullivan conjecture. Massachusetts Institute of Technology, 2007. MIT OpenCourseWare
  8. Schwartz, Lionel. Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1994. x+229 pp. ISBN: 0-226-74202-4; 0-226-74203-2 MR1282727 (95d:55017)

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