Abstract:

In his work on maps between classifying spaces of compact Lie groups, Frank Adams studied the image of the natural homomorphism α: R(G)—>K(BG) for a compact Lie group G. Specififically he proved that the image of α coincides with the subgroup of elements of K(BG) generated by those elements with only fifinitely many non-zero exterior powers when π0(G) is a p-group for any prime p. Bob Oliver proved the real analogue of this result. Atiyah and Segal introduced the notion of Real groups, Real representations and Real vector bundles in late 60s. A few years later, Karoubi studied a wider class of groups which he still called Real groups. They were unaware that Real representations were studied by Eugene Wigner at least 10 years before Atiyah and Segal in the name of corepresentations. We will give a summary of the theory developed by Wigner and then go on to prove that the Real case of the above mentioned result of Frank Adams holds when π0 of the underlying group is a 2-group, but fails to hold when π0 is a p-group for p≠2.