Cohomology of modules over the mod 2 Steenrod algebra
Robert R. Bruner
Wayne State University

Stunted projective spaces

(for computing the root invariant)

P-1 has lower cell in dimension -1, et cetera. Upper cells are 100 and the resolution is computed through internal degree 60 or 70.

The charts all start in the minus 30 stem so that the sphere being built on the -1 stem is in the same place in each chart.

I have assembled these into a single pdf document. If you view it in single page mode, then you can scroll through it and watch the sphere being built on the -1 cell. The blizzard of other classes surrounding the classes which are building up the -1 sphere come and go, and make it more apparent how miraculous Lin's theorem really is.

Here are the individual files, if you want them.

P-1 , P-2 , P-3 , P-4 , P-5 , P-6 , P-7 , P-8 ,
P-9 , P-10 , P-11 , P-12 , P-13 , P-14 , P-15 , P-16 ,
P-17 , P-18 , P-10 , P-20 , P-21 , P-22 , P-23 , P-24 ,
P-25 , P-26 , P-27 , P-28 , P-29 .

Some Demo videos

Download the latest version of the program from my publications, etc., page, under computer code. Currently ext.1.9.5.

Dreadfully old charts

These charts are dreadfully old, and you can compute ones for yourself through 60 degrees above the bottom cell in 1 minute or two for yourself. Of course, these charts are perfectly alright: one of the wonderful things about mathematics is that it stays true.

Ext(M,F_2) for a number of modules M over the mod 2 Steenrod algebra. Here F_2 is short for GF(2). Two variants are presented. The second set shows multiplications by all h_i's. The first shows only h0, h1, and h2. The second is generally readable only under magnification.

NEW!
At the end will be found a few charts of Ext(\Susp^n F_2,F_2) in the category of unstable modules over the Steenrod algebra. These are the E_2 term of the unstable Adams spectral sequence for the homotopy of S^n. More to come.

For a few modules, GIFs are presented as well as Postcript files. The GIFs are more convenient, but the .ps files have better resolution, so view the GIFs first, and use the Postscript for detail.

Several of the modules are described by an ascii file in the format needed by the programs, which I will refer to as the module definition format .

For each module M, I note the internal degrees t and homological degree s through which the calculation is complete. Homological degree is displayed vertically and total degree t-s is displayed horizontally as customary.

If you would like to see a module not shown here, or would like to see the calculation for a particular module extended, email me. If you want to propose a module, it would help if you send it to me in the module definition format . Calculations over other GF(2)-algebras can also be done. Chain maps between many of the resolutions displayed here have also been computed, though at present I have no simple way to display them.

N.B. Some modules are described by giving a spectrum with that module as its cohomology. In such cases, I often omit (de-)suspensions. They should be obvious from the internal degrees.

Back home.

Visit Christian Nassau's web site for some modern approaches to these calculations, odd primary results, programs you can download and other interesting items,