A new approach to twisted K–theory of compact Lie groups

Abstract

This paper explores further the computation of the twisted K-theory and K-homology of compact simple Lie groups, previously studied by Hopkins, Moore, Maldacena-Moore-Seiberg, Braun, and Douglas, with a focus on groups of rank 2. We give a new method of computation based on the Segal spectral sequence which seems to us appreciably simpler than the methods used previously, at least in many key cases.2010 Mathematics Subject Classification. Primary 19L50. Secondary 81T30, 57T10, 55T15, 55R20.

Computations in higher twisted $K$-theory

“…In particular, when the twist δ can be identified with a cohomology class δ ∈ H 2n+1 (X, Z) we see in Theorem 4.2 that the d 2n+1 differential is of the form d 2n+1 (x) = d 2n+1 (x)−δ ∪x where d 2n+1 is the differential in the ordinary Atiyah-Hirzebruch spectral sequence for topological K-theory, which is an operator whose image is torsion [2]. There is, in fact, a more general Segal spectral sequence that can be applied in this setting and we generalise a result of Rosenberg [48] to obtain this sequence. Letting…”

“…Another particularly useful class of spaces with torsion-free cohomology is formed by certain types of Lie groups. A great deal of work has been done by many mathematicians and physicists in computing the twisted K-theory of Lie groups in the classical setting, including Hopkins, Braun [18], Douglas [29], Rosenberg [48] and Moore [45]. In the case of SU (n), the twisted K-groups were explicitly computed and as a consequence it was shown that the higher differentials in the twisted Atiyah-Hirzebruch spectral sequence are non-zero in general, suggesting that this technique will not yield general results for the higher twisted K-groups of SU (n).…”

“…While an explicit computation of the higher twisted K-theory of SU (n) is difficult in general, we employ techniques of Rosenberg [48] to obtain structural information about these groups. It is at this point that we must turn to the more powerful Segal spectral sequence given in Theorem 4.3 so that Theorem 4.4 may be employed.…”