the cohomology of so(n) - cornell universitypi.math.cornell.edu/~hatcher/so/so(n).pdfthe cohomology...
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The Cohomology of SO(n)
Computer-generated pictures created by M.A.Agosto and J.J. Perez
Commentary by Allen Hatcher
The special orthogonal group SO(n) is high on the list of important topological spaces,
yet its homology and cohomology exhibit some surprising subtleties. The complications
arise from the presence of torsion in the integer homology and cohomology, but fortunately
the torsion consists just of elements of order 2. Both the integer cohomology ring modulo
torsion and the mod 2 cohomology ring have structures that are easy to describe (see
Section 3D of my book):
(1) H∗(SO(n); Z) modulo torsion is the exterior algebra on generators a3, a7, · · · , a4k−1
for n = 2k + 1 and a3, a7, · · · , a4k−1, a′
2k+1 for n = 2k + 2. Here subscripts denote degrees,
so ai ∈ Hi and a′
2k+1 ∈ H2k+1.
(2) H∗(SO(n); Z2) is the polynomial algebra on generators bi of odd degree i < n, trun-
cated by the relations bpi
i = 0 where pi is the smallest power of 2 such that bpi
i has degree
≥ n.
The subtleties arise when one tries to describe the actual integral cohomology ring itself.
In principle this follows from a calculation of mod 2 Bockstein homomorphisms, which is
not difficult and is described in Example 3E.7 of my book. The cases of SO(5) and SO(7)
are worked out in detail there. Here’s what the Bocksteins look like for SO(7):
The numbers across the top of the figure denote degrees. Each dot in the ith column
represents a basis element for Hi(SO(7); Z2) viewed as a vector space over Z2, with the
label on the dot telling which class the dot represents. For example the dot labeled 234 is
the product b2b3b4, where the relations b2i = b2
i allow the bi’s with even subscripts to be
expressed in terms of those with odd subscripts, the generators in statement (2) above. The
line segments in the diagram indicate the nonzero Bocksteins. These are homomorphisms
β : Hi(SO(7); Z2) → Hi+1(SO(7); Z2) satisying β2 = 0. The nontorsion in H∗(SO(7); Z)
corresponds to Ker β/Im β, while the torsion elements correspond to Im β. For example,
the nontorsion element a3 corresponds to b3 + b1b2 (these are Z2 classes so signs don’t
matter) and a7 corresponds to either b1b2b4 or b3b4.
An additive basis for H∗(SO(n); Z2) consists of the products bi1· · · bik
with 0 < i1 <
· · · < ik < n. These classes are in one-to-one correspondence with the cells in a CW
structure on SO(n). There are 2n−1 of these classes, so the size of H∗(SO(n); Z2) grows
exponentially with n, in contrast with the dimension of SO(n) which is n(n − 1)/2, just
quadratic in n. Thus the maximum size of the individual groups Hi(SO(n); Z2) is also
growing exponentially with n, although for fixed i this group is independent of n when
n > i + 1.
M.A.Agosto and J.J. Perez have written a Mathematica program to draw diagrams show-
ing nonzero Bocksteins in H∗(SO(n); Z2) for general n. In the range 5 ≤ n ≤ 12 these are
shown starting on the next page, with a different convention for displaying the picture than
in the figure above, so that Poincare duality appears as a 180 degree rotational symmetry
about the center point of the diagram rather than as reflection across a vertical line. The
labels on the classes are omitted for n > 7 since they become too small to read. There
are some arbitrary choices made in how to draw the diagrams as two-dimensional arrays,
and it might be possible to make different choices so that the diagrams had fewer crossing
edges.
Null 1 2 3
12
4
13
14
23
24
123
34
124 134 234 1234
SO(5)
Null 1 2 3
12
4
13
5
14
23
15
24
123
25
34
124
35
125
134
45
135
234
145
235
1234
245
1235
345
1245 1345 2345 12 345
SO(6)
Null 1 2 3
12
4
13
5
14
23
6
15
24
123
16
25
34
124
26
35
125
134
36
126
45
135
234
46
136
145
235
1234
56
146
236
245
1235
156
246
1236
345
1245
256
346
1246
1345
356
1256
1346
2345
456
1356
2346
12 345
1456
2356
12 346
2456
12 356
3456
12 456 13 456 23 456 123 456
SO(7)
SO(8)
SO(9)
SO(10)
SO(11)
SO(12)
SO(7) ⊂ SO(8) ⊂ SO(9)
SO(5) ⊂ SO(6) ⊂ SO(7) ⊂ SO(8) ⊂ SO(9)
SO(6) ⊂ SO(8) ⊂ SO(10)
Null 1 2 3
12
4
13
5
14
23
6
15
24
123
7
16
25
34
124
8
17
26
35
125
134
9
18
27
36
126
45
135
234
10
19
28
37
127
46
136
145
235
1234
110
29
38
128
47
137
56
146
236
245
1235
210
39
129
48
138
57
147
237
156
246
1236
345
1245
310
1210
49
139
58
148
238
67
157
247
1237
256
346
1246
1345
410
1310
59
149
239
68
158
248
1238
167
257
347
1247
356
1256
1346
2345
510
1410
2310
69
159
249
1239
78
168
258
348
1248
267
357
1257
1347
456
1356
2346
12 345
610
1510
2410
12 310
79
169
259
349
1249
178
268
358
1258
1348
367
1267
457
1357
2347
1456
2356
12 346
710
1610
2510
3410
12 410
89
179
269
359
1259
1349
278
368
1268
458
1358
2348
467
1367
1457
2357
12 347
2456
12 356
810
1710
2610
3510
12 510
13 410
189
279
369
1269
459
1359
2349
378
1278
468
1368
1458
2358
12 348
567
1467
2367
2457
12 357
3456
12 456
910
1810
2710
3610
12 610
4510
13 510
23 410
289
379
1279
469
1369
1459
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12 349
478
1378
568
1468
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12 358
1567
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12 367
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12 457
13 456
1910
2810
3710
12 710
4610
13 610
14 510
23 510
123 410
389
1289
479
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569
1469
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12 359
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12 368
3458
12 458
2567
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12 467
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23 456
2910
3810
12 810
4710
13 710
5610
14 610
23 610
24 510
123 510
489
1389
579
1479
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1569
2469
12 369
3459
12 459
678
1578
2478
12 378
2568
3468
12 468
13 458
3567
12 567
13 467
23 457
123 456
3910
12 910
4810
13 810
5710
14 710
23 710
15 610
24 610
123 610
34 510
124 510
589
1489
2389
679
1579
2479
12 379
2569
3469
12 469
13 459
1678
2578
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12 478
3568
12 568
13 468
23 458
4567
13 567
23 467
123 457
4910
13 910
5810
14 810
23 810
6710
15 710
24 710
123 710
25 610
34 610
124 610
134 510
689
1589
2489
12 389
1679
2579
3479
12 479
3569
12 569
13 469
23 459
2678
3578
12 578
13 478
4568
13 568
23 468
123 458
14 567
23 567
123 467
5910
14 910
23 910
6810
15 810
24 810
123 810
16 710
25 710
34 710
124 710
35 610
125 610
134 610
234 510
789
1689
2589
3489
12 489
2679
3579
12 579
13 479
4569
13 569
23 469
123 459
3678
12 678
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13 578
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14 568
23 568
123 468
24 567
123 567
6910
15 910
24 910
123 910
7810
16 810
25 810
34 810
124 810
26 710
35 710
125 710
134 710
45 610
135 610
234 610
1 234 510
1789
2689
3589
12 589
13 489
3679
12 679
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23 479
14 569
23 569
123 469
4678
13 678
14 578
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123 478
24 568
123 568
34 567
124 567
7910
16 910
25 910
34 910
124 910
17 810
26 810
35 810
125 810
134 810
36 710
126 710
45 710
135 710
234 710
145 610
235 610
1 234 610
2789
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12 689
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4679
13 679
14 579
23 579
123 479
24 569
123 569
5678
14 678
23 678
24 578
123 578
34 568
124 568
134 567
8910
17 910
26 910
35 910
125 910
134 910
27 810
36 810
126 810
45 810
135 810
234 810
46 710
136 710
145 710
235 710
1 234 710
245 610
1 235 610
3789
12 789
4689
13 689
14 589
23 589
123 489
5679
14 679
23 679
24 579
123 579
34 569
124 569
15 678
24 678
123 678
34 578
124 578
134 568
234 567
18 910
27 910
36 910
126 910
45 910
135 910
234 910
37 810
127 810
46 810
136 810
145 810
235 810
1 234 810
56 710
146 710
236 710
245 710
1 235 710
345 610
1 245 610
4789
13 789
5689
14 689
23 689
24 589
123 589
15 679
24 679
123 679
34 579
124 579
134 569
25 678
34 678
124 678
134 578
234 568
1 234 567
28 910
37 910
127 910
46 910
136 910
145 910
235 910
1 234 910
47 810
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56 810
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1 235 810
156 710
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1 236 710
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1 245 710
1 345 610
5789
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34 589
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25 679
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134 579
234 569
35 678
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134 678
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1 234 568
38 910
128 910
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1 235 910
57 810
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156 810
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1 236 810
345 810
1 245 810
256 710
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1 246 710
1 345 710
2 345 610
6789
15 789
24 789
123 789
25 689
34 689
124 689
134 589
35 679
125 679
134 679
234 579
1 234 569
45 678
135 678
234 678
1 234 578
48 910
138 910
57 910
147 910
237 910
156 910
246 910
1 236 910
345 910
1 245 910
67 810
157 810
247 810
1 237 810
256 810
346 810
1 246 810
1 345 810
356 710
1 256 710
1 346 710
2 345 710
12 345 610
16 789
25 789
34 789
124 789
35 689
125 689
134 689
234 589
45 679
135 679
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1 234 579
145 678
235 678
1 234 678
58 910
148 910
238 910
67 910
157 910
247 910
1 237 910
256 910
346 910
1 246 910
1 345 910
167 810
257 810
347 810
1 247 810
356 810
1 256 810
1 346 810
2 345 810
456 710
1 356 710
2 346 710
12 345 710
26 789
35 789
125 789
134 789
45 689
135 689
234 689
1 234 589
145 679
235 679
1 234 679
245 678
1 235 678
68 910
158 910
248 910
1 238 910
167 910
257 910
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1 247 910
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1 256 910
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267 810
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1 456 710
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36 789
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135 789
234 789
145 689
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1 234 689
245 679
1 235 679
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1 245 678
78 910
168 910
258 910
348 910
1 248 910
267 910
357 910
1 257 910
1 347 910
456 910
1 356 910
2 346 910
12 345 910
367 810
1 267 810
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1 357 810
2 347 810
1 456 810
2 356 810
12 346 810
2 456 710
12 356 710
46 789
136 789
145 789
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1 234 789
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1 235 689
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1 245 679
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178 910
268 910
358 910
1 258 910
1 348 910
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1 267 910
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1 357 910
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467 810
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3 456 710
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56 789
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1 345 679
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278 910
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1 457 910
2 357 910
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2 456 910
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567 810
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12 357 810
3 456 810
12 456 810
13 456 710
156 789
246 789
1 236 789
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1 245 789
1 345 689
2 345 679
12 345 678
378 910
1 278 910
468 910
1 368 910
1 458 910
2 358 910
12 348 910
567 910
1 467 910
2 367 910
2 457 910
12 357 910
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1 567 810
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256 789
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356 789
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578 910
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678 910
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45 678 910
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145 678 910
235 678 910
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245 678 910
1 235 678 910
345 678 910
1 245 678 910 1 345 678 910 2 345 678 910 12 345 678 910
SO(9) ⊂ SO(10) ⊂ SO(11)
SO(7) ⊂ SO(8) ⊂ SO(9) ⊂ SO(10) ⊂ SO(11)
SO(10) ⊂ SO(11) ⊂ SO(12)