Maps between Classifying Spaces

In 1976, Adams & Mahmud 3] published the rst systematic study of the problem of determining the homological properties of maps between classifying spaces of compact connected Lie groups. This was continued in later work by one or both authors: Adams 2] extended some of the results to the case of non-connected Lie groups by using complex K-theory; while Adams & Mahmud 4] identiied further restrictions which could be made using real or symplectic K-theory. Recently, in 21], the three of us developed new techniques for studying maps between classifying spaces: techniques based on new decompositions of BG for any compact Lie group G. The main application in 21] was to the problem of determining self maps of BG for any compact connected simple Lie group G. This problem had earlier been studied by several other people (cf. 26], 28], 16], 18], 19], and 23]). The main missing point was to show that the \unstable Adams operations" k : BG ? ! BG are unique up to homotopy. The main tools which now make possible a more precise study of maps between classifying spaces are a series of consequences of the proof of the Sullivan conjecture by Miller (cf. 13]), Carlsson 11], and Lannes 22]. The principal result which we use directly is the description, by Dwyer & Zabrodsky 14] and Notbohm 24], of map(BP; BG) when G is any compact Lie group and P is p-toral (an extension of a torus by a nite p-group). The idea of our approach was to combine these theorems of Dwyer-Zabrodsky and Notbohm with a new decomposition of BG: a decomposition which approximates BG at any prime p as a homotopy direct limit of classifying spaces of p-toral subgroups of G. In this paper, we show how the same techniques can be used successfully in other situations, to get information about the existence and uniqueness of maps BG ? ! BG 0 when G and G 0 are two distinct compact connected Lie groups. We rst describe the general strategy for doing this, taking as our starting point the work of Adams & Mahmud in 3]. To illustrate how these tools work in practice, we then take those examples listed in 3] involving (potential) maps between classifying spaces of distinct rank; and use our methods to determine exactly which ones actually do exist. We end (Example 3.4) with a description of the set …