K-theory of compact Lie groups

Question

The complex K-theory of a compact connected Lie group $ G $ is computed by Hodgkin in the case that $ G $ has torsion-free fundamental group. The result is that $ K^*(G) $ is an exterior algebra in the fundamental representations of $ G $. Note that every finite dimensional representation gives a canonical $ K^1 $-class.

What happens if $ \pi_1(G) $ has torsion? Are there any general results analogous to Hodgkin's theorem? I am specifically interested in the case of the projective unitary group $ PU(n) = U(n)/center $. In the case $ n = 2 $ this amounts to $ SO(3) $, and identification with $ \mathbb{R}P^3 $ shows that $ K^0 $ has a torsion subgroup $ \mathbb{Z}/2\mathbb{Z} $ in this case. I would expect to find a torsion subgroup $ \mathbb{Z}/n\mathbb{Z} $ inside $ K^0 $ in general - is this true?

Answer 1

Here are some relevant references. First, three papers on different methods to compute the K-theory of compact Lie groups with finite cyclic fundamental group. (The first one also discusses the projective unitary groups as example.)

• R.P. Held and U. Suter. On the unitary K-theory of compact Lie groups with finite fundamental group. Quart. J. Math. Oxford Ser. (2) 24 (1973), 343–356. (link to Quart J. Math)

• L. Hodgkin. The equivariant Künneth theorem in K-theory. In: Topics in K-theory. Lecture Notes in Math., Vol. 496. Springer 1975, 1–101.

• H. Minami. Note on complex K-groups of compact Lie groups with fundamental group of prime order. Osaka J. Math. 35 (1998), no. 3, 547–551. (link to Osaka J. Math)

There is a further paper dealing specifically with projective unitary groups

• T. Petrie. The K theory of the projective unitary groups. Topology 6 (1967), 103–115 (link to Topology)

There are further papers by Minami dealing with the computations for specific projective groups ($PE_6$, $PE_7$, $PSp(n)$, $PO(4l+2)$).