Papers, talks and computer code of Robert R. Bruner

• Computer code
• Talks
• Publications
• Preprints
• Notes

Cohomology charts

Visualization of A(2)

Computer Code

• Ext.1.8.5
  The latest stable version of the ext calculator.
  
  
• FP_Modules
  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.
  

Publications

1. Connective real K-theory of finite groups (with John Greenlees),
   Mathematical Surveys and Monographs 169, Amer. Math. Soc., Providence, RI 2010.
   
   
2. Differentials in the homological homotopy fixed point spectral sequence (with John Rognes),
   Algebr. Geom. Topol. 5, 653--690 (electronic) 2005.
   
   
3. On the behavior of the algebraic transfer, (with Le Minh Ha and Nguyen H. V. Hung)
   Trans. Amer. Math. Soc. 357, 473--487 2005.
   
   
4. The Connective K-theory of Finite Groups (with John Greenlees),
   prepublication version of Memoirs AMS V. 165 No. 785, Sept 2003.
   
   
5. Nonimmersions of real projective spaces implied by tmf (with Donald M. Davis and Mark Mahowald),
   Recent progress in homotopy theory (Baltimore, MD, 2000)
   Contemp. Math. 293, 45--68, Amer. Math. Soc., Providence, RI 2002.
   
   
6. 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.
   
   
7. Homotopy methods in algebraic topology (ed. with J. P. C. Greenlees and Nicholas Kuhn),
   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.
   
   
8. Ossa's Theorem and Adams covers
   Proc. Amer. Math. Soc. 127 (1999), no. 8, 2443--2447.
   
   
9. 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.
   
   
10. Some Remarks on the Root Invariant
    Stable and Unstable Homotopy (Toronto, ON 1996), Fields Inst. Commun. 19 (1998) 31--37.
    
    
11. A Yoneda Description of the Steenrod Operations
    Proc. Symp. Pure Math. 63 (1998), problem session.
    
    
12. Real connective K-theory and the quaternion group (with Dilip Bayen)
    Trans. AMS 348 (1996), 2201-2216.
    
    
13. On stable homotopy equivalences (with F. R. Cohen and C. A. McGibbon)
    Oxford Quarterly J. of Math., (2) 46 (1995), 11-20.
    
    
14. The Bredon-Loffler conjecture (with J. P. C. Greenlees)
    Exper. Math. 4 (1995), 289-297.
    
    
15. On recursive solutions of a unit fraction equation (with Lawrence Brenton)
    J. Austral. Math. Soc. Ser. A 57 (1994), no. 3, 341--356.
    
    
16. Ext in the nineties
    Contemp. Math., 146 (1993), 71-90.
    
    
17. 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.
    
    
18. An example in the cohomology of augmented algebras
    J. Pure Appl. Algebra 55 (1988), no. 1-2, 81--84.
    
    
19. H_infinity ring spectra and their applications (with J. P. May, J. E. McClure and M. Steinberger)
    Lecture Notes in Mathematics, 1176. Springer-Verlag, Berlin, 1986.
    
    
20. A new differential in the Adams spectral sequence
    Topology 23 (1984), no. 3, 271--276.
    
    
21. Two Generalizations of the Adams Spectral Sequence
    Canadian Mathematical Society Conference Proceedings, V. 2, part 1 (1982) pp. 275--287.
    
    
22. An infinite family in pi*(S^0) derived from Mahowald's eta-j family
    Proc. Amer. Math. Soc. 82 (1981), no. 4, 637--639.
    
    
23. 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.
    
    
24. Locally compact groups without distinct isomorphic closed subgroups (with D. L. Armacost)
    Proc. Amer. Math. Soc. 40 (1973), 260--264.
    
    
25. Radicals and Torsion Theories in Locally Compact Groups
    undergraduate thesis, Amherst College, Dec 1972.
    
    

Preprints

26. On cyclic fixed points of spectra (with Marcel Bokstedt, Sverre Lunoe-Nielsen, and John Rognes)
    arXiv:1211.0213.
    
    
27. Ossa's Theorem via the Kunneth formula (with Khairia Mira, Laura Stanley and Victor Snaith)
    arXiv:1008.0166.
    
    
28. On the Postnikov towers for real and complex connective K-theory
    arXiv:1208.2232.
    
    
29. On Ossa's Theorem and Local Picard Groups
    arXiv:1211.0213.
    
    

Notes

30. 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.
    
    
31. An Adams Spectral Sequence primer
    A draft of an introduction to the classical Adams spectral sequence.
    
    
32. 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.
    
    
33. How to solve a quartic
    
    
34. 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.
    
    
35. 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.
    
    
36. A Relation in the Steenrod Algebra
    The Adem relation needed to reduce all squaring operations to those given by a power of 2.
    
    
37. Some squaring operations in Ext
    Calculated by machine, using every tool available.
    
    
38. The semi-dihedral algebra in algebraic topology
    Why the dimensional semi-dihedral algebra is of interest to algebraic topologists.
    
    
39. 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).