Cohomology of modules over the mod 2 Steenrod algebra
Robert R. Bruner
Wayne State University
Download the latest version of the program from my
publications, etc., page, under computer code.
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,
- S0, that is F_2 , s <= 39 , 0 <= t <= 140
Brief:
0 to 22 ,
22 to 44 ,
44 to 66 ,
66 to 88 ,
88 to 110 ,
GIFS
Complete files are illegible, hence omitted.
Briefer (h0 and h1 only):
0 to 22 ,
22 to 44 ,
44 to 66 ,
66 to 88 ,
88 to 110 ,
- P^3 smash R over A(2), s <= 35, 0 <= t <= 100
0 to 22 ,
22 to 44 ,
44 to 66 ,
66 to 88 ,
- RP^2 = Mod 2 Moore space , s <= 10, 0 <= t <= 50
Brief:
0 to 22 ,
22 to 44 ,
44 to 66 ,
Complete:
0 to 22 ,
22 to 44 ,
44 to 66 ,
GIFS
- CP^2 = Cofiber(eta) , s <= 20, 0 <= t <= 45
Brief:
0 to 22 ,
22 to 44
Complete:
0 to 22 ,
22 to 44
- RP^2 smash CP^2 , s <= 19 , 0 <= t <= 70
definition
Brief:
0 to 22 ,
22 to 44 ,
44 to 66 ,
66 to 88 ,
Complete:
0 to 22 ,
22 to 44 ,
44 to 66 ,
66 to 88 ,
-
RP^2 smash RP^2, s < 40 , 0 <= t <= 60
Brief:
0 to 22 ,
22 to 44 ,
44 to 66 ,
Detail of all h_i products.
-
RP^12, s < 40 , 0 <= t <= 60
Brief:
1 to 23 ,
23 to 45 ,
45 to 60 (incomplete) ,
Detail of all h_i products.
- A 4 cell complex with Sq(3)Sq(4) nonzero, s < 40 , 0 <= t <= 60
definition
Brief:
0 to 22 ,
22 to 44 ,
- Cofiber eta again , s <= 10 , 0 <= t <= 50
definition
Brief:
0 to 22 ,
22 to 44 ,
44 to 66 ,
Complete:
0 to 22 ,
22 to 44 ,
44 to 66 ,
- Misc1 , s <= 19 , 0 <= t <= 43
definition
Brief:
0 to 22 ,
22 to 44
Complete:
0 to 22 ,
22 to 44
- Misc2 , s <= 19 , 3 <= t <= 43
definition
Brief:
3 to 25 ,
25 to 47 ,
Complete:
3 to 25 ,
25 to 47 ,
- Misc3 , s <= 39 , 00 <= t <= 13
definition
Brief:
0 to 22 ,
Complete:
0 to 22 ,
- Misc4 , s <= 39 , 0 <= t <= 13
definition
Brief:
0 to 22 ,
Complete:
0 to 22 ,
- Stunted projective spaces P_n are RP(infinity)/RP(n-1)
- BZ/2, i.e. P_1 , s <= 39 , 1 <= t <= 75
Brief:
1 to 23 ,
23 to 45 ,
45 to 67 ,
67 to 89 ,
Complete:
1 to 23 ,
23 to 45 ,
45 to 67 ,
67 to 89 ,
- P_2 , s <= 20 , 2 <= t <= 30
Brief:
2 to 24 ,
24 to 46 ,
Complete:
2 to 24 ,
24 to 46 ,
- P_3 , s <= 20 , 3 <= t <= 31
Brief:
3 to 25 ,
25 to 47 ,
Complete:
3 to 25 ,
25 to 47 ,
- P_4 , s <= 20 , 4 <= t <= 30
Brief:
4 to 26 ,
26 to 48 ,
Complete:
4 to 26 ,
26 to 48 ,
- P_8 , s <= 39 , 8 <= t <= 20
Brief:
8 to 30 ,
Complete:
8 to 30 ,
- P_-1 , s <= 39 , -1 <= t <= 45
Brief:
-1 to 21 ,
-1 to 21 unzipped ,
21 to 43 ,
Complete:
-1 to 21 ,
- P_-2 , s <= 19 , -2 <= t <= 15
Brief:
-2 to 20 ,
Complete:
-2 to 20 ,
- P_-3 , s <= 19 , -3 <= t <= 4
Brief:
-3 to 19 ,
Complete:
-3 to 19 ,
- P_-4 , s <= 19 , -4 <= t <= 20
Brief:
-4 to 18 ,
18 to 40 ,
Complete:
-4 to 18 ,
18 to 40 ,
- P_-5 , s <= 19 , -5 <= t <= 6
Brief:
-5 to 17 ,
Complete:
-5 to 17 ,
- P_-6 , s <= 19 , -6 <= t <= 20
Brief:
-6 to 16 ,
16 to 38 ,
Complete:
-6 to 16 ,
16 to 38 ,
- P_-7 , s <= 19 , -7 <= t <= 8
Brief:
-7 to 15 ,
Complete:
-7 to 15 ,
- P_-8 , s <= 19 , -8 <= t <= 20
Brief:
-8 to 14 ,
14 to 36 ,
Complete:
-8 to 14 ,
14 to 36 ,
- P_-9 , s <= 19 , -9 <= t <= 65
Brief:
-9 to 13 ,
13 to 35 ,
35 to 57 ,
57 to 79 ,
Complete:
-9 to 13 ,
13 to 35 ,
35 to 57 ,
57 to 79 ,
- P_-10 , s <= 19 , -10 <= t <= 20
Brief:
-10 to 12 ,
12 to 34 ,
Complete:
-10 to 12 ,
12 to 34 ,
- P_-11 , s <= 19 , -11 <= t <= 12
Brief:
-11 to 11 ,
11 to 33 ,
Complete:
-11 to 11 ,
11 to 33 ,
- P_-12 , s <= 19 , -12 <= t <= 20
Brief:
-12 to 10 ,
10 to 32 ,
Complete:
-12 to 10 ,
10 to 32 ,
- P_-13 , s <= 19 , -13 <= t <= 20
Brief:
-13 to 9 ,
9 to 31 ,
Complete:
-13 to 9 ,
9 to 31 ,
- P_-14 , s <= 19 , -14 <= t <= 20
Brief:
-14 to 8 ,
8 to 30 ,
Complete:
-14 to 8 ,
8 to 30 ,
- P_-15 , s <= 19 , -15 <= t <= 20
Brief:
-15 to 7 ,
7 to 29 ,
Complete:
-15 to 7 ,
7 to 29 ,
- P_-16 , s <= 19 , -16 <= t <= 20
Brief:
-16 to 6 ,
6 to 28 ,
Complete:
-16 to 6 ,
6 to 28 ,
- P_-17 , s <= 19 , -17 <= t <= 60
Brief:
-17 to 5 ,
5 to 27 ,
27 to 49 ,
49 to 71 ,
Complete:
-17 to 5 ,
5 to 27 ,
27 to 49 ,
49 to 71 ,
- P_-18 , s <= 19 , -18 <= t <= 20
Brief:
-18 to 4 ,
4 to 26 ,
Complete:
-18 to 4 ,
4 to 26 ,
- P_-19 , s <= 19 , -19 <= t <= 20
Brief:
-19 to 3 ,
3 to 25 ,
Complete:
-19 to 3 ,
3 to 25 ,
- P_-20 , s <= 19 , -20 <= t <= 22
Brief:
-20 to 2 ,
2 to 24 ,
Complete:
-20 to 2 ,
2 to 24 ,
- P_-21 , s <= 19 , -21 <= t <= 30
Brief:
-21 to 1 ,
1 to 23 ,
23 to 45 ,
Complete:
-21 to 1 ,
1 to 23 ,
23 to 45 ,
- P_-22 , s <= 19 , -22 <= t <= 24
Brief:
-22 to 0 ,
0 to 22 ,
22 to 44 ,
Complete:
-22 to 0 ,
0 to 22 ,
22 to 44 ,
- P_-23 , s <= 19 , -23 <= t <= 40
Brief:
-23 to -1 ,
-1 to 21 ,
21 to 43 ,
Complete:
-23 to -1 ,
-1 to 21 ,
21 to 43 ,
- P_-24 , s <= 19 , -24 <= t <= 26
Brief:
-24 to -2 ,
-2 to 20 ,
20 to 42 ,
Complete:
-24 to -2 ,
-2 to 20 ,
20 to 42 ,
- P_-25 , s <= 19 , -25 <= t <= 26
Brief:
-25 to -3 ,
-3 to 19 ,
19 to 41 ,
Complete:
-25 to -3 ,
-3 to 19 ,
19 to 41 ,
- P_-26 , s <= 19 , -26 <= t <= 28
Brief:
-26 to -4 ,
-4 to 18 ,
18 to 40 ,
Complete:
-26 to -4 ,
-4 to 18 ,
18 to 40 ,
- P_-27 , s <= 19 , -27 <= t <= 28
Brief:
-27 to -5 ,
-5 to 17 ,
17 to 39 ,
Complete:
-27 to -5 ,
-5 to 17 ,
17 to 39 ,
- P_-28 , s <= 19 , -28 <= t <= 30
Brief:
-28 to -6 ,
-6 to 16 ,
16 to 38 ,
Complete:
-28 to -6 ,
-6 to 16 ,
16 to 38 ,
- P_-29 , s <= 19 , -29 <= t <= 50
Brief:
-29 to -7 ,
-7 to 15 ,
15 to 37 ,
37 to 59 ,
Complete:
-29 to -7 ,
-7 to 15 ,
15 to 37 ,
37 to 59 ,
- P_-30 , s <= 19 , -30 <= t <= 32
Brief:
-30 to -8 ,
-8 to 14 ,
14 to 36 ,
Complete:
-30 to -8 ,
-8 to 14 ,
14 to 36 ,
- P_-31 , s <= 19 , -31 <= t <= 32
Brief:
-31 to -9 ,
-9 to 13 ,
13 to 35 ,
Complete:
-31 to -9 ,
-9 to 13 ,
13 to 35 ,
- P_-32 , s <= 19 , -32 <= t <= 34
Brief:
-32 to -10 ,
-10 to 12 ,
12 to 34 ,
Complete:
-32 to -10 ,
-10 to 12 ,
12 to 34 ,
- P_-33 , s <= 19 , -33 <= t <= 55
Brief:
-33 to -11 ,
-11 to 11 ,
11 to 33 ,
33 to 55 ,
Complete:
-33 to -11 ,
-11 to 11 ,
11 to 33 ,
33 to 55 ,
- P_-34 , s <= 19 , -34 <= t <= 36
Brief:
-34 to -12 ,
-12 to 10 ,
10 to 32 ,
32 to 54 ,
Complete:
-34 to -12 ,
-12 to 10 ,
10 to 32 ,
32 to 54 ,
- P_-35 , s <= 19 , -35 <= t <= 37
Brief:
-35 to -13 ,
-13 to 9 ,
9 to 31 ,
31 to 53 ,
Complete:
-35 to -13 ,
-13 to 9 ,
9 to 31 ,
31 to 53 ,
- P_-36 , s <= 19 , -36 <= t <= 38
Brief:
-36 to -14 ,
-14 to 8 ,
8 to 30 ,
30 to 52 ,
Complete:
-36 to -14 ,
-14 to 8 ,
8 to 30 ,
30 to 52 ,
- P_-37 , s <= 19 , -37 <= t <= 48
Brief:
-37 to -15 ,
-15 to 7 ,
7 to 29 ,
29 to 51 ,
Complete:
-37 to -15 ,
-15 to 7 ,
7 to 29 ,
29 to 51 ,
- P_-38 , s <= 19 , -38 <= t <= 40
Brief:
-38 to -16 ,
-16 to 6 ,
6 to 28 ,
28 to 50 ,
Complete:
-38 to -16 ,
-16 to 6 ,
6 to 28 ,
28 to 50 ,
- P_-39 , s <= 19 , -39 <= t <= 43
Brief:
-39 to -17 ,
-17 to 5 ,
5 to 27 ,
27 to 49 ,
Complete:
-39 to -17 ,
-17 to 5 ,
5 to 27 ,
27 to 49 ,
- P_-40 , s <= 19 , -40 <= t <= 42
Brief:
-40 to -18 ,
-18 to 4 ,
4 to 26 ,
26 to 48 ,
Complete:
-40 to -18 ,
-18 to 4 ,
4 to 26 ,
26 to 48 ,
- P_-41 , s <= 19 , -41 <= t <= 44
Brief:
-41 to -19 ,
-19 to 3 ,
3 to 25 ,
25 to 47 ,
Complete:
-41 to -19 ,
-19 to 3 ,
3 to 25 ,
25 to 47 ,
- P_-42 , s <= 19 , -42 <= t <= 44
Brief:
-42 to -20 ,
-20 to 2 ,
2 to 24 ,
24 to 46 ,
Complete:
-42 to -20 ,
-20 to 2 ,
2 to 24 ,
24 to 46 ,
- P_-43 , s <= 19 , -43 <= t <= 45
Brief:
-43 to -21 ,
-21 to 1 ,
1 to 23 ,
23 to 45 ,
Complete:
-43 to -21 ,
-21 to 1 ,
1 to 23 ,
23 to 45 ,
- P_-44 , s <= 19 , -44 <= t <= 46
Brief:
-44 to -22 ,
-22 to 0 ,
0 to 22 ,
22 to 44 ,
44 to 66 ,
Complete:
-44 to -22 ,
-22 to 0 ,
0 to 22 ,
22 to 44 ,
44 to 66 ,
- P_-45 , s <= 19 , -45 <= t <= 45
Brief:
-45 to -23 ,
-23 to -1 ,
-1 to 21 ,
21 to 43 ,
43 to 65 ,
Complete:
-45 to -23 ,
-23 to -1 ,
-1 to 21 ,
21 to 43 ,
43 to 65 ,
- P_-46 , s <= 19 , -46 <= t <= 45
Brief:
-46 to -24 ,
-24 to -2 ,
-2 to 20 ,
20 to 42 ,
42 to 64 ,
Complete:
-46 to -24 ,
-24 to -2 ,
-2 to 20 ,
20 to 42 ,
42 to 64 ,
- P_-47 , s <= 19 , -47 <= t <= 47
Brief:
-47 to -25 ,
-25 to -3 ,
-3 to 19 ,
19 to 41 ,
41 to 63 ,
Complete:
-47 to -25 ,
-25 to -3 ,
-3 to 19 ,
19 to 41 ,
41 to 63 ,
- P_-48 , s <= 19 , -48 <= t <= 45
Brief:
-48 to -26 ,
-26 to -4 ,
-4 to 18 ,
18 to 40 ,
40 to 62 ,
Complete:
-48 to -26 ,
-26 to -4 ,
-4 to 18 ,
18 to 40 ,
40 to 62 ,
- P_-49 , s <= 19 , -49 <= t <= 51
Brief:
-49 to -27 ,
-27 to -5 ,
-5 to 17 ,
17 to 39 ,
39 to 61 ,
Complete:
-49 to -27 ,
-27 to -5 ,
-5 to 17 ,
17 to 39 ,
39 to 61 ,
- P_-50 , s <= 19 , -50 <= t <= 45
Brief:
-50 to -28 ,
-28 to -6 ,
-6 to 16 ,
16 to 38 ,
38 to 60 ,
Complete:
-50 to -28 ,
-28 to -6 ,
-6 to 16 ,
16 to 38 ,
38 to 60 ,
- P_-51 , s <= 19 , -51 <= t <= 44
Brief:
-51 to -29 ,
-29 to -7 ,
-7 to 15 ,
15 to 37 ,
37 to 59 ,
Complete:
-51 to -29 ,
-29 to -7 ,
-7 to 15 ,
15 to 37 ,
37 to 59 ,
- P_-52 , s <= 19 , -52 <= t <= 45
Brief:
-52 to -30 ,
-30 to -8 ,
-8 to 14 ,
14 to 36 ,
36 to 58 ,
Complete:
-52 to -30 ,
-30 to -8 ,
-8 to 14 ,
14 to 36 ,
36 to 58 ,
- P_-53 , s <= 19 , -53 <= t <= 43
Brief:
-53 to -31 ,
-31 to -9 ,
-9 to 13 ,
13 to 35 ,
35 to 57 ,
Complete:
-53 to -31 ,
-31 to -9 ,
-9 to 13 ,
13 to 35 ,
35 to 57 ,
- P_-54 , s <= 19 , -54 <= t <= 45
Brief:
-54 to -32 ,
-32 to -10 ,
-10 to 12 ,
12 to 34 ,
34 to 56 ,
Complete:
-54 to -32 ,
-32 to -10 ,
-10 to 12 ,
12 to 34 ,
34 to 56 ,
- P_-55 , s <= 19 , -55 <= t <= 43
Brief:
-55 to -33 ,
-33 to -11 ,
-11 to 11 ,
11 to 33 ,
33 to 55 ,
Complete:
-55 to -33 ,
-33 to -11 ,
-11 to 11 ,
11 to 33 ,
33 to 55 ,
- P_-56 , s <= 19 , -56 <= t <= 45
Brief:
-56 to -34 ,
-34 to -12 ,
-12 to 10 ,
10 to 32 ,
32 to 54 ,
Complete:
-56 to -34 ,
-34 to -12 ,
-12 to 10 ,
10 to 32 ,
32 to 54 ,
- P_-60 , s <= 19 , -60 <= t <= 42
Brief:
-60 to -38 ,
-38 to -16 ,
-16 to 6 ,
6 to 28 ,
28 to 50 ,
Complete:
-60 to -38 ,
-38 to -16 ,
-16 to 6 ,
6 to 28 ,
28 to 50 ,
- P_-65 , s <= 19 , -65 <= t <= 43
Brief:
-65 to -43 ,
-43 to -21 ,
-21 to 1 ,
1 to 23 ,
23 to 45 ,
Complete:
-65 to -43 ,
-43 to -21 ,
-21 to 1 ,
1 to 23 ,
23 to 45 ,
- Unstable Exts