The Morava K-Theories of Eilenberg-MacLane Spaces and the Conner-Floyd Conjecture

Abstract

Introduction. Of the many generalized homology theories available, very few are computable in practice except for the simplest of spaces. Standard homology and K-theory are the only ones which can be considered somewhat accessible. In recent years, complex cobordism, or equivalently, Brown-Peterson homology, has become a useful tool for algebraic topology. The high state of this development is particularly apparent with regard to BP stable operations, which are understood well enough to have many applications to stable homotopy; see for example [16]. Despite this achievement, it is still virtually impossible to compute the Brown-Peterson homology of any but the nicest of spaces; for example: some simple classifying spaces, spaces with no torsion and spaces with few cells. As a replacement for Brown-Peterson homology in this respect, we present the closely related generalized homologies known as the Morava K-theories. These are a sequence of homology theories, K(n) *(-), n > 0, for each prime p. The n = 1 case is essentially standard mod p complex K-theory. These theories are periodic of period 2(pn - 1) and fit together to give Morava's beautiful structure theorem for complex cobordism; see [11]. Because of their close relationship to complex bordism, information about them will sometimes suffice for bordism, and thus geometric, problems. This is the case with our proof of the Conner-Floyd conjecture. The Morava K-theories each possess Kiunneth isomorphisms for all spaces. This feature enhances their computability tremendously. We demonstrate this point by computing the Morava K-theories of the Eilenberg-MacLane spaces. These spaces are difficult to handle even for