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

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24

123

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34

124

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234

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127

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1234

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128

47

137

56

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245

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SO(9) ⊂ SO(10) ⊂ SO(11)

SO(7) ⊂ SO(8) ⊂ SO(9) ⊂ SO(10) ⊂ SO(11)

SO(10) ⊂ SO(11) ⊂ SO(12)