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COHOMOLOGY RINGS
GREGORY D. LANDWEBER
• Spheres:
H∗(Sn; Z) ∼= Z[an] /
(a2
n = 0).
• Complex projective spaces:
H∗(CP n; Z) ∼= Z[a2] /
(an+1
2 = 0).
• Real projective spaces:
H∗(RP n; Z2
) ∼= Z2[a1] /(an+1
1 = 0).
• Unitary groups:
H∗(U(n); Z) ∼= ΛZ[a1, a3 . . . , a2n−1].
• Special unitary groups:
H∗(SU(n); Z) ∼= ΛZ[a3, a5 . . . , a2n−1].
• Special orthogonal groups:
H∗(SO(2n + 1); Q) ∼= ΛQ[a3, a7 . . . , a4n−1],
H∗(SO(2n + 2); Q) ∼= ΛQ[a3, a7, . . . , a4n−1, b2n+1],
(Note that SO(n) has 2-torsion for n ≥ 7. See [1].)
H∗(SO(n); Z2
) ∼= ⊗i odd
Z2[βi]/(βpi
i = 0),
where pi is the smallest power of 2 such that ipi ≥ n. See [5, Theorem 3D.2].
• The exceptional Lie group G2:
H∗(G2; Q) ∼= ΛQ[h3, h11],
H∗(G2; Z) ∼= Z[h3, h11] /
(h4
3 = h211 = h2
3h11 = 0).
Additively, we have
H∗(G2; Z) ∼= (
Z, 0, 0, Zh3, 0, 0, Z2h23, 0, 0, Z2h
33, 0, Zh11, 0, 0, Zh3h11
).
See [1, Theoreme 17.2].
• The exceptional Lie group F4:
H∗(F4; Q) ∼= ΛQ[h3, h11, h15, h23],
H∗(F4; Z) ∼= H∗(G2 × S15; Z
)⊗ Z[u8, u23] /
(3u8 = u3
8 = u223 = u8u23 = 0
).
See [1, Theoreme 19.2].
Date: August 21, 2006.1
2 GREGORY D. LANDWEBER
• The exceptional Lie groups E6, E7, E8:
H∗(E6; Q) ∼= ΛQ[h3, h9, h11, h15, h17, h23],
H∗(E7; Q) ∼= ΛQ[h3, h11, h15, h19, h23, h27, h35],
H∗(E8; Q) ∼= ΛQ[h3, h15, h23, h27, h35, h39, h47, h59].
See [4] and [3]. Furthermore, E6, E7, E8 have no p torsion for p ≥ 7, p ≥ 11, and
p ≥ 11, respectively (see [2]).
• A projective K3 surface over C:
H∗(K3; Z)
=
Z in degrees 0, 4,
Z⊕22 in degree 2,
0 otherwise.
The intersection pairing on H2(K3; Z) is even, unimodular, and has signature (3, 19).
References
[1] A. Borel. Sur l’homologie et la cohomologie des groupes de Lie compacts connexes. Amer. J. Math.,76:273–342, 1954.
[2] A. Borel. Sur la torsion des groupes de Lie. J. Math. Pures Appl. (9), 35:127–139, 1956.[3] A. Borel and C. Chevalley. The Betti numbers of the exceptional groups. Mem. Amer. Math. Soc.,
1955(14):1–9, 1955.[4] C. Chevalley. The Betti numbers of the exceptional simple Lie groups. In Proceedings of the International
Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pages 21–24, Providence, R. I., 1952. Amer.Math. Soc.
[5] A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.
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