Ohkawa's theorem: There is a set of Bousfield classes
Bousfield classes are a very useful tool in stable homotopy theory. Ten years ago, Ohkawa proved in [5] that there is a set of Bousfield classes, with cardinality at most 2 א0 . This result has some interesting consequences—see [3], for example—but it does not seem to be as well known as it should; we are trying to remedy this. In this short note, we give a simple proof of Ohkawa’s theorem, and we discuss some natural questions that arise from the proof. In particular, the proof leads us to consider “Ohkawa classes”, which provide a weaker equivalence relation than Bousfield classes do. Unfortunately, Ohkawa classes seem unwieldy, and we have not found any uses for them aside from proving Ohkawa’s theorem. We define Ohkawa classes and prove Ohkawa’s theorem (Theorem 1.2) in the Section 1; to prove the result, we show that there is a set O of Ohkawa classes, of cardinality at most 2 א0 , and that this set maps onto the collection of Bousfield classes. This map is in fact a map of partially order sets; in Section 2, we examine the partial ordering on O. It turns out to be rather different from the ordering on the set B of Bousfield classes; we are able to conclude from this that the surjection O −→ B is not one-to-one. In Section 3, we show that the cardinality of O is exactly 2 א0 , but we are unable to deduce anything useful about the cardinality of B from this. Throughout, we ask a number of questions; knowing the answers would shed some light on O and B and the relationship between the two. The authors would like to thank Dan Christensen and Neil Strickland for interesting discussions about this material.