Papers, talks and computer code of Robert R. Bruner

• Database of the Cohomology of the Steenrod Algebra
• Computer code
• Talks
• Publications
• Preprints
• Notes

Cohomology charts

Visualization of A(2)

Database: The cohomology of the Steenrod Algebra

• (Sept 2021, joint with John Rognes)(Revised and extended, Feb 2022) Database: DOI:10.11582/2022.00015
  ArXiv copy of the document included in the database explaining it content: The Cohomology of the Steenrod Algebra
  The dataset contains a minimal resolution of the mod 2 Steenrod algebra in the range 0 le s le 128, 0 le t le 200, together with chain maps for each cocycle in that range and for the squaring operation Sq^0 in the cohomology of the Steenrod algebra. The included document CohomA2.pdf explains the contents and usage of the dataset in detail.

Computer Code

• ext.1.9.5 (1 September 2022)
  This release
  • fixes an important bug, which fortunately does not affect anyone who uses it as distributed,
  • fixes a couple of minor bugs, and
  • makes a number of other improvements.
  • I have also deleted obsolete documentation and provided up-to-date documentation which gives full details about the use of the software.
• ext.1.9.3
  (6 August 2018) Fixes problem with checkmap which prevented it from working. This had been fixed in 1.8.7 but crept back in to 1.9.2. Adds 'induceup' which takes an A(2) resolution and tensors it up to A. Fixes an error in sortDef which made it fail. Makes 'dolifts' uniform: there had been an old version in A/F2 which did less than the generic version does. Now all directories point to the generic version. Gives better instructions for the use of the option to resolve a finitely presented module. Fixes a typo (extra ",0" in Tikz charts) and improves spacing and margins in chart. Adds script 'missing' which looks for missing generators over Ext_A.
• ext.1.9.2
  (7 July 2016) Makes obsolete the other ext.1.9.x versions. The chart function now numbers items. Also added 'missing_gens' to speed up the search for sufficient cocycles to generate over the cohomology of the algebra (A or A(2)). Finally, fixes numerous minor annoyances that older compilers ignored by newer compilers are cranky about. (Read NEW to learn about these.)
• ext.1.9.1
  (24 October 2015) Changed the ordering of generators to run from right to left rather left to right, to get fewer crossing lines. Either this or 1.9.0's left to right produces better h0-towers than the old display program. Also, finished removing detritus no longer needed (setup.[ch]) now that the old display programs are gone.
• ext.1.9.0
  (18 October 2015) replaces the old chart code ('display') by a new one, 'chart', which produces Tikz code in a Tex file. Run your favorite TeX compiler on its output to get a pdf, and/or incorporate the figure directly into your own TeX file. It also has a newer version of vsumm for A(2)-modules called vsummA2 which produces a TeX file using longtable with h0, h1 and h2 multiples, and a 'Notes' column in which you can record product information and/or differentials that you compute.
  This is quick and dirty version to make the chart function available to people. If you have an ext.1.8.7 version and want to keep all the caculations that you have already done, extract the programs 1. summarize.c, 2.chart.c, 3. A/Install, 4. A/Clean, 5. A2/Install, and 6. A2/Clean from ext.1.9.0, and put them in place in your existing version.
  Soon, I hope to replace all the ugly old charts on my Cohomology charts web page with these pretty new ones.
• ext.1.8.7
  (14 April 2014) The latest stable version of the ext calculator, fixes a memory allocation error which prevented checkmap from functioning. Also corrected a typo in the instructions for cocycle.
• ext.1.8.6
  Fixes an irritant experienced on systems which have aliased "rm" to "rm -i".
• ext.1.8.5
  The previous version of the ext calculator, in case there are problems with ext.1.8.6.
  
  
• FP_Modules
  (This link is now OBSOLETE -- the FP modules code is now part of Sage, as of version 9.6 (May 2022))
  SAGE code to compute with finitely presented modules over the mod p Steenrod algebra, any p. Written by Mike Catanzaro. You can define finitely presented modules and homomorphisms between them. You can compute kernels, images, cokernels, direct sums, resolutions, and modules defined by elements of Ext^1.
  
  
• Chern
  MAGMA code to compute Chern classes. Set G to the group you want and "load Chern;". It will print the character table and then report the Chern clases of each irreducible, both as a character and as a linear combination of the irreducibles.
  Sample output: Dihedral of order 16 and Alternating on 5 letters.
  
  
• Dyer-Lashof
  MAGMA code to compute the mod 2 cohomology of the r^th extended power of a spectrum. Input is a module over the mod 2 Steenrod algebra in the 'module definition format'. Set
  • N to be the upper limit of the calculation; we will compute the N-skeleton in other words,
  • r to indicate the r^th extended power, and
  • file to the name of the file containing the description of the input module.
  It will write the output in a file named D_rfiletoNmod (substitute r, file and N that you set) which is in the module defintion format so that it can be fed directly into the newmodule command of the ext package. The variable Mons will be useful in interpreting that file. See some sample runs to see the calculation of the module definition for for the 7 skeleton of D_2(S^1) and for the 10 skeleton of D_2 of this 6 cell complex. Here is the Ext chart for the 40 (rather than 10) skeleton of the D_2(D_2^7(S^1)) produced by ext.1.8.3 while I was typing this up. (1 min.) From it you can see, for example, that pi_4,5,6,7 are each Z/2, with pi_4,5,6 mapping monomorphically under the Hurewicz map and with nu acting nontrivially on all three of these.

Talks

• The Finiteness Conjecture
       at the Workshop on the Kervaire invariant and stable homotopy theory, 28 April 2011.
  
  
• Characteristic Classes in Connective K-Theory ( Part 1 and Part 2 )
       University of Muenster, Oberseminar Topologie, 20 June 2011.
  
  
• Commutative Ring Spectra and Spectral Sequences
       at the Conference on Structured Ring Spectra, in Hamburg, 1-5 August 2011.
  
  
• The Finiteness Conjecture
       Topology Seminar, Universitetet i Bergen, 9 August 2011. (Quite similar to Edinburgh talk.)
  
  
• Characteristic Classes in Connective K-Theory ( General Theory and Compact Lie Groups )
       Topology Seminar, Universitetet i Bergen, 10 August 2011. (Substantially reorganized since the Muenster talk. Still too long.)
  
  
• A(2)-modules and the Adams spectral sequence
       Equivariant, Chromatic and Motivic Homotopy Theory, Northwestern University, Evanston, Illinois, 25 March 2013.
  Sage code and output referred to in the talk.
  
  
• A(2)-Modules and their Cohomology
       Topology Symposium, Universitetet i Bergen, 1 June 2017.
  
  
• A(2)-Modules and their Cohomology
       Topology Ecuador 2017, Universidad San Francisco de Quito, San Cristobal, August 18, 2017.
  
  
• The mod 2 Adams Spectral Sequence for tmf_*
       Isaac Newton Institute, Cambridge, England, 11 September 2018.
  
  
• The Root Invariant
       A Jolly Pleasant Conference, NTNU (Norges teknisk-naturvitenskapelige universitet), Trondheim, Norway, 29 July 2019.
  
  
• ext and its uses
       eCHT seminar on machine computation in homotopy theory, 30 September 2021.
  
  

Publications

1. (with John Greenlees and John Rognes) The local cohomology spectral sequence for topological modular forms
   Mathematische Zeitschrift, online 13 June 2022, (arXiv:2107.02272).
   
   
2. (with John Rognes) The Adams spectral sequence for the image-of-J spectrum,
   Transactions of the AMS 375, pp. 5803-5827, online 23 May 2022, ( arXiv:2105.02601).
   
   
3. (with John Rognes, 2021) Dataset: The Cohomology of the Steenrod Algebra, DOI:10.11582/2021.00077.
   arXiv copy of the document in the dataset describing its contents: The Cohomology of the Steenrod Algebra
   
   
4. (with John Rognes) The Adams spectral sequence for topological modular forms, Mathematical Surveys and Monographs, 253.
   American Mathematical Society, Providence, RI, [2021]. ©2021. xix+690 pp. ISBN: 978-1-4704-5674-0
   Charts and supplemental material from the book
   Errata
   
   
5. (with David Benson) A Counterexample for lightning flash modules over E(e1,e2)
   Archiv der Mathematik, 23 Feb. 2016, DOI 10.1007/s00013-016-0880-8.
   
   
6. Idempotents, Localizations and Picard Groups of A(1)−modules
   in An Alpine Expedition through Algebraic Topology, Contemporary Mathematics, vol. 617, Amer. Math. Soc., Providence, RI, 2014, pp. 81-108.
   (http://dx.doi.org/10.1090/conm/617/12346) (Formerly On Ossa's Theorem and Local Picard Groups, arXiv:1211.0213.)
   
   
7. (with Marcel Bokstedt, Sverre Lunoe-Nielsen, and John Rognes) On cyclic fixed points of spectra
   Math. Zeit. 11 July 2013, (DOI) 10.1007/s00209-013-1187-0
   arXiv:1211.0213.
   
   
8. (with John Greenlees) Connective real K-theory of finite groups,
   Mathematical Surveys and Monographs 169, Amer. Math. Soc., Providence, RI 2010.
   
   
9. (with John Rognes) Differentials in the homological homotopy fixed point spectral sequence
   Algebr. Geom. Topol. 5, 653--690 (electronic) 2005.
   
   
10. (with Le Minh Ha and Nguyen H. V. Hung) On the behavior of the algebraic transfer, Trans. Amer. Math. Soc. 357, 473--487 2005.
    
    
11. (with John Greenlees) The Connective K-theory of Finite Groups
    prepublication version of Memoirs AMS V. 165 No. 785, Sept 2003.
    
    
12. (with Donald M. Davis and Mark Mahowald) Nonimmersions of real projective spaces implied by tmf
    Recent progress in homotopy theory (Baltimore, MD, 2000)
    Contemp. Math. 293, 45--68, Amer. Math. Soc., Providence, RI 2002.
    
    
13. Extended powers of manifolds and the Adams spectral sequence ,
    Homotopy methods in algebraic topology (Boulder, CO, 1999),
    Contemp. Math. 271, 41--51, Amer. Math. Soc., Providence, RI 2001.
    
    
14. (ed. with J. P. C. Greenlees and Nicholas Kuhn) Homotopy methods in algebraic topology
    Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held at the University of Colorado, Boulder, CO, June 20–24, 1999.
    Contemporary Mathematics 271, Amer. Math. Soc., Providence, RI 2001.
    
    
15. Ossa's Theorem and Adams covers
    Proc. Amer. Math. Soc. 127 (1999), no. 8, 2443--2447.
    
    
16. Some root invariants and Steenrod operations in Ext_A(F2,F2)
    Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997),
    Contemp. Math. 220, 27--33, Amer. Math. Soc., Providence, RI, 1998.
    
    
17. Some Remarks on the Root Invariant
    Stable and Unstable Homotopy (Toronto, ON 1996), Fields Inst. Commun. 19 (1998) 31--37.
    Correction: at the bottom of page 3, the element mu is in pi_9, not pi_8.
    
    
18. A Yoneda Description of the Steenrod Operations
    Proc. Symp. Pure Math. 63 (1998), problem session.
    
    
19. (with Dilip Bayen) Real connective K-theory and the quaternion group
    Trans. AMS 348 (1996), 2201-2216.
    
    
20. (with F. R. Cohen and C. A. McGibbon) On stable homotopy equivalences
    Oxford Quarterly J. of Math., (2) 46 (1995), 11-20.
    
    
21. (with J. P. C. Greenlees) The Bredon-Loffler conjecture
    Exper. Math. 4 (1995), 289-297.
    
    
22. (with Lawrence Brenton) On recursive solutions of a unit fraction equation
    J. Austral. Math. Soc. Ser. A 57 (1994), no. 3, 341--356.
    
    
23. Ext in the nineties
    Contemp. Math., 146 (1993), 71-90.
    
    
24. Calculation of large Ext modules
    Computers in geometry and topology (Chicago, IL, 1986),
    Lecture Notes in Pure and Appl. Math., 114 79--104, Marcel Dekker, New York, 1989.
    
    
25. An example in the cohomology of augmented algebras
    J. Pure Appl. Algebra 55 (1988), no. 1-2, 81--84.
    
    
26. (with J. P. May, J. E. McClure and M. Steinberger) H_infinity ring spectra and their applications (my corrections added in red)
    Lecture Notes in Mathematics, 1176. Springer-Verlag, Berlin, 1986. (Peter's copy)
    
    
27. A new differential in the Adams spectral sequence
    Topology 23 (1984), no. 3, 271--276.
    
    
28. Two Generalizations of the Adams Spectral Sequence
    Canadian Mathematical Society Conference Proceedings, V. 2, part 1 (1982) pp. 275--287.
    
    
29. An infinite family in pi*(S^0) derived from Mahowald's eta-j family
    Proc. Amer. Math. Soc. 82 (1981), no. 4, 637--639.
    
    
30. Algebraic and geometric connecting homomorphisms in the Adams spectral sequence
    Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, pp. 131--133.
    Lecture Notes in Math., 658, Springer, Berlin, 1978.
    
    
31. Locally compact groups without distinct isomorphic closed subgroups (with D. L. Armacost)
    Proc. Amer. Math. Soc. 40 (1973), 260--264.
    
    
32. Radicals and Torsion Theories in Locally Compact Groups
    undergraduate thesis, Amherst College, Dec 1972.
    
    

Preprints

33. Steenrod operations and A-module extensions (with Christian Nassau and Sean Tilson)
    arXiv:1909:03117
    
    
34. Ossa's Theorem via the Kunneth formula (with Khairia Mira, Laura Stanley and Victor Snaith)
    arXiv:1008.0166.
    
    
35. On the Postnikov towers for real and complex connective K-theory
    arXiv:1208.2232. (local copy)
    
    

Notes

36. The Complex Numbers are Algebraically Closed
    A very simple proof that a finite dimensional division algebra over C must be one dimensional.
    
    
37. The Untwisting Isomorphism
    The untwisting isomorphism for modules over a Hopf algebra.
    
    
38. The tangent bundle of projective space
    Uses equivariance to compute the tangent bundles of real and complex projective spaces as an alternative to the argument given in Milnor and Stasheff.
    It is more direct and explicit.
    
    
39. The Cohomology of ku
    An account of Adams' calculation of the mod 2 cohomology of complex connective K-theory.
    
    
40. The Cohomology of the mod 2 Steenrod Algebra
    Results of the penultimate run, complete with all products, out to t=141, s=40. The exposition is a very rough draft, but the results should be correct.
    
    
41. An Adams Spectral Sequence primer
    A draft of an introduction to the classical Adams spectral sequence.
    
    
42. The connective complex K-theory of an elementary abelian p-group
    A note written to answer a question asked about the rank 3 case at an odd prime. Contains some general remarks about all ranks.
    
    
43. How to solve a quartic
    
    
44. Asymmetry and efficiency in Toda brackets
    I show that computing Toda brackets via Yoneda composites requires less data than might be expected. This is useful in doing actual computations.
    
    
45. Cup 1 and symmetric Toda brackets
    Derivation of the formula for a symmetric 3-fold Toda bracket in terms of cup-1 operations, valid when cup-1 satisfies the Hirsch formula.
    
    
46. A Relation in the Steenrod Algebra
    An Adem relation allowing one to reduce all squaring operations to those given by a power of 2. Dreadfully slow in practive.
    
    
47. Some squaring operations in Ext
    Calculated by machine, using every tool available.
    
    
48. The semi-dihedral algebra in algebraic topology
    Why the 8 dimensional semi-dihedral algebra is of interest to algebraic topologists.
    
    
49. Tate Cohomology of the anto-involution of the Steenrod algebra
    Calculated by MAGMA, in the effort to gather further data on the questions remaining after the paper by Crossley and Whitehouse (Proc AMS 2000).