We will compute the module definition file for the 7 skeleton of
the extended square (r=2) of S^1. (This is just the suspension of
the truncated projective space P_1^6, so it is simple to verify that
it is correct.)
In MAGMA, do the following, where the file s1 contains
1
1
(meaning that it has 1 generator in degree 1.)
> N:=7;
> r:=2;
> file := "s1";
> load newDL;
Loading "newDL"
[
[]
]
6 generators
6 monomials
Mons sorted
ops on monomial 1
ops on monomial 2
ops on monomial 3
ops on monomial 4
ops on monomial 5
ops on monomial 6
> quit;
Total time: 1.520 seconds, Total memory usage: 15.00MB
The result has been written in the file D_2s1to7mod:
~/magma[39]: more D_2s1to7mod
6
2 3 4 5 6 7
0 1 1 1
1 2 1 3
2 1 1 3
2 2 1 4
2 3 1 5
4 1 1 5
It is easy to verify that these are the correct sSteenrod operations
for the truncated projective space P_2^6 suspended once.
Now let us compute the 10-skeleton of the extended square of
the module we just computed (the 7-skeleton of Susp P_2^6):
> N := 10;
> r := 2;
> file := "D_2s1to7mod";
> load newDL;
Loading "newDL"
[ <--- You may ignore this. The program is just
[], mumbling to itself.
[
[ 1 ]
],
[],
[
[ 3 ],
[ 2 ]
],
[
undef,
[ 3 ]
],
[
[ 5 ],
undef,
[ 3 ]
]
]
18 generators
27 monomials
Mons sorted
ops on monomial 1
ops on monomial 2
ops on monomial 3
ops on monomial 4
ops on monomial 5
ops on monomial 6
ops on monomial 7
ops on monomial 8
ops on monomial 9
ops on monomial 10
ops on monomial 11
ops on monomial 12
ops on monomial 13
ops on monomial 14
ops on monomial 15
ops on monomial 16
ops on monomial 17
ops on monomial 18
ops on monomial 19
ops on monomial 20
ops on monomial 21
ops on monomial 22
ops on monomial 23
ops on monomial 24
ops on monomial 25
ops on monomial 26
ops on monomial 27
The following will help you interpret the results. Mons contains an
F_2 basis for the module in the ordering used to write out the results.
In the explanation below, let i = iota_1 be the fundamental class of S^1
that we started with, so the module generators for its extended square
were Q^1(i), Q^2(i), ... Q^6(i) in degrees 2, ..., 7.
I have compressed the output below a bit: MAGMA spreads things
out a lot, for good reason, but I compressed the first few rows
to facilitate my explanation of them. Note MAGMA 1-indexes, while
my ext code 0-indexes, hence the shift i-1.
> [ : i in [1..#Mons]];
[
<0, [ [ 1 ], [ 1 ] ], 4>, gen #0 is Q^1(i) Q^1(i) in degree 4
<1, [ [ 1 ], [ 2 ] ], 5>, gen #1 is Q^1(i) Q^2(i) in degree 5
<2, [ [ 3, 1 ] ], 5>, gen #2 is Q^3(Q^1(i)) in degree 5
<3, [ [ 1 ], [ 3 ] ], 6>,
<4, [ [ 4, 1 ] ], 6>,
<5, [ [ 2 ], [ 2 ] ], 6>,
<6, [ [ 5, 1 ] ], 7>,
<7, [ [ 1 ], [ 4 ] ], 7>,
<8, [
[ 4, 2 ]
], 7>,
<9, [
[ 2 ],
[ 3 ]
], 7>,
<10, [
[ 1 ],
[ 5 ]
], 8>,
<11, [
[ 6, 1 ]
], 8>,
<12, [
[ 2 ],
[ 4 ]
], 8>,
<13, [
[ 5, 2 ]
], 8>,
<14, [
[ 3 ],
[ 3 ]
], 8>,
<15, [
[ 1 ],
[ 6 ]
], 9>,
<16, [
[ 3 ],
[ 4 ]
], 9>,
<17, [
[ 2 ],
[ 5 ]
], 9>,
<18, [
[ 7, 1 ]
], 9>,
<19, [
[ 5, 3 ]
], 9>,
<20, [
[ 6, 2 ]
], 9>,
<21, [
[ 2 ],
[ 6 ]
], 10>,
<22, [
[ 8, 1 ]
], 10>,
<23, [
[ 4 ],
[ 4 ]
], 10>,
<24, [
[ 7, 2 ]
], 10>,
<25, [
[ 3 ],
[ 5 ]
], 10>,
<26, [ [ 6, 3 ] ], 10> gen # 26 is Q^6(Q^3(i)) in degree 10.
]
>
Now the file it wrote:
~/magma[48]: more D_2D_2s1to7modto10mod
27
4 5 5 6 6 6 7 7 7 7 8 8 8 8 8 9 9 9 9 9 9 10 10 10 10 10 10
0 1 1 1
0 2 2 4 5
0 3 1 8
1 2 1 7
1 3 1 12
2 1 1 4
2 2 2 6 8
2 3 1 11
2 4 1 20
3 1 2 7 9
3 2 2 10 12
3 3 2 15 17
3 4 1 21
4 2 1 13
4 3 1 20
4 4 2 22 24
5 1 1 8
5 2 2 12 13
5 3 1 20
5 4 1 23
6 1 1 11
6 2 1 20
7 1 1 12
9 1 1 12
9 2 2 16 17
9 3 1 21
10 1 2 15 17
10 2 1 21
11 2 2 22 24
13 1 1 20
14 1 1 16
14 2 2 23 25
15 1 1 21
17 1 1 21
18 1 1 22
19 1 1 26