Maps between classifying spaces revisited

Abstract

In an earlier paper, we developed general techniques which can be used to study the set of homotopy classes of maps between the classifying spaces of two given compact Lie groups. Here, we describe more precisely the general strategy for doing this; and then, as a test of these methods, apply them to determine the existence and uniqueness of (potential) maps BG −→ BG′ studied earlier by Adams and Mahmud. We end with a complete description of the set of homotopy classes of maps from BG2 to BF4. In 1976, Adams & Mahmud [3] published the first 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] identified 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” ψ : 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 1991 Mathematics Subject Classification. Primary 55S37, 55R35; Secondary 55P60. The first author was partly supported by a Polish scientific grant (RP I.10), the second author by a US NSF grant (DMS-8803279), and the third author by a Danish SNF grant (11-7792) c ©0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page