Papers, talks and computer code of
Robert R. Bruner
(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.)
(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.
(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.
(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.
Fixes an irritant experienced on systems which have aliased "rm" to "rm -i".
The previous version of the ext calculator, in case there are problems with ext.1.8.6.
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.
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.
Dihedral of order 16 and
Alternating on 5 letters.
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
'module definition format'.
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.
- 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,
- file to the name of the file containing the description of the input module.
The Finiteness Conjecture
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
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,
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.
A Counterexample for lightning flash modules over E(e1,e2)
Archiv der Mathematik, 23 Feb. 2016, DOI 10.1007/s00013-016-0880-8.
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.
(Formerly On Ossa's Theorem and Local Picard Groups,
On cyclic fixed points of spectra
(with Marcel Bokstedt, Sverre Lunoe-Nielsen, and John Rognes)
Math. Zeit. 11 July 2013, (DOI) 10.1007/s00209-013-1187-0
Connective real K-theory of finite groups (with John Greenlees),
Mathematical Surveys and Monographs 169, Amer.
Math. Soc., Providence, RI 2010.
Differentials in the homological homotopy fixed point spectral sequence
(with John Rognes),
Algebr. Geom. Topol. 5, 653--690 (electronic) 2005.
On the behavior of the algebraic transfer,
(with Le Minh Ha and Nguyen H. V. Hung)
Trans. Amer. Math. Soc. 357, 473--487 2005.
The Connective K-theory of Finite Groups
(with John Greenlees),
prepublication version of Memoirs AMS V. 165 No. 785, Sept 2003.
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.
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.
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.
Ossa's Theorem and Adams covers
Proc. Amer. Math. Soc. 127 (1999),
no. 8, 2443--2447.
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.
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.
A Yoneda Description of the Steenrod Operations
Proc. Symp. Pure Math. 63 (1998), problem session.
Real connective K-theory and the quaternion group
(with Dilip Bayen)
Trans. AMS 348 (1996), 2201-2216.
On stable homotopy equivalences
(with F. R. Cohen and C. A. McGibbon)
Oxford Quarterly J. of Math., (2) 46 (1995), 11-20.
The Bredon-Loffler conjecture
(with J. P. C. Greenlees)
Exper. Math. 4 (1995), 289-297.
On recursive solutions of a unit fraction equation
(with Lawrence Brenton)
J. Austral. Math. Soc. Ser. A 57 (1994), no. 3,
Ext in the nineties
Contemp. Math., 146 (1993), 71-90.
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.
An example in the cohomology of augmented algebras
J. Pure Appl. Algebra 55 (1988), no. 1-2,
H_infinity ring spectra and their applications
(with J. P. May, J. E. McClure and M. Steinberger)
Lecture Notes in Mathematics, 1176. Springer-Verlag,
A new differential in the Adams spectral sequence
Topology 23 (1984), no. 3, 271--276.
Two Generalizations of the Adams Spectral Sequence
Canadian Mathematical Society Conference Proceedings, V. 2, part 1 (1982) pp. 275--287.
An infinite family in pi*(S^0) derived from Mahowald's eta-j family
Proc. Amer. Math. Soc. 82 (1981), no.
Algebraic and geometric connecting homomorphisms in the Adams spectral sequence
Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, pp. 131--133.
Notes in Math., 658, Springer, Berlin, 1978.
Locally compact groups without distinct isomorphic closed subgroups (with D. L. Armacost)
Proc. Amer. Math. Soc. 40
- Radicals and Torsion Theories in Locally
undergraduate thesis, Amherst College, Dec 1972.
Ossa's Theorem via the Kunneth formula
(with Khairia Mira, Laura Stanley and Victor Snaith)
On the Postnikov towers for real and complex connective K-theory
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.
The Cohomology of ku
An account of Adams' calculation of the mod 2 cohomology
of complex connective K-theory.
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.
An Adams Spectral Sequence primer
A draft of an introduction to the classical Adams spectral sequence.
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.
How to solve a quartic
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.
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.
A Relation in the Steenrod Algebra
The Adem relation needed to reduce all squaring operations to those given by a power of 2.
Some squaring operations in Ext
Calculated by machine, using every tool available.
The semi-dihedral algebra in algebraic topology
Why the 8 dimensional semi-dihedral algebra is of interest to algebraic topologists.
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).