W. Stephen Wilson

Periodic phenomena in the Adams-Novikov spectral sequence

The problem of understanding the stable homotopy ring has long been one of the touchstones of algebraic topology. Low dimensional computation has proceeded slowly and has given little insight into the general structure of 7ws(S0). In recent years, however, infinite families of elements of 7rs (S0) have been discovered, generalizing the image of the Whitehead J-homomorphism. In this work we indicate a general program for the detection and description of elements lying in such infinite families. This approach shows that every homotopy class is, in some attenuated sense, a member of such a family. For our algebraic grip on homotopy theory we shall employ S. P. Novikovs analogue of the Adams spectral sequence converging to the stable homotopy ring. Its E2-term can be described algebraically as the cohomology of the Landweber-Novikov algebra of stable operations in complex cobordism. In his seminal work on the subject, Novikov computed the first cohomology group and showed that it was canonically isomorphic to the image of J away from the prime 2. When localized at an odd prime p these elements occur only every 2(p 1) dimensions; so this first cohomology group has a periodic character. Our intention here is to show that the entire cohomology is built up in a very specific way from periodic constituents. Our central application of these ideas is the computation of the second cohomology group at odd primes. By virtue of the Adams-Novikov spectral sequence this information has a number of homotopy-theoretic consequences. The homotopy classes St, t > 1, in the p-component of the (2(p2 1)t 2(p 1) 2)-stem for p > 3, constructed by L. Smith, are detected here. Indeed, it turns out that all elements with Adams-Novikov filtration exactly 2 are closely related to the , family. The lowest dimensional elements of filtration 2 aside from the fi family itself are the elements denoted ej by Toda. The computation of the

THE MORAVA K-THEORIES OF EILENBERG-MACLANE SPACES AND THE CONNER-FLOYD CONJECTURE

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 Moravas 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

PROJECTIVE DIMENSION AND BROWN-PETERSON HOMOLOGY

Abstract BP ∗ (X) is the reduced Brown-Peterson homology of a finite complex X for a fixed prime p. We study a sequence of homology theories BP ∗ (X)→⋯→BP〈n+1〉 ∗ (X) → p(n,n + 1) BP〈n〉 ∗ (X) → p(n - 1,n) BP〈n–1〉 ∗ (X) →⋯→BP〈0〉 ∗ (X) → p( - 1,0) BP〈–1〉 ∗ (X) . Our main result states that there is an n such that ⋯, ϱ(n,n + 1), ϱ(n– 1, n) are all epimorphisms; each of the remaining homomorphism fails to be onto; and n is the projective dimension of BP ∗ (X) as a module over the coefficient ring BP ∗ . The first two sections are introductory in nature while the third contains the core results. The reader should be pleased to know that §4–6 are independent of each other.

Brown–Peterson Cohomology from Morava K ‐Theory, II

We give some structure to the Brown-Peterson cohomology (or its p-completion) of a wide class of spaces. The class of spaces are those with Morava K-theory even dimensional. We can say that the Brown-Peterson cohomology is even dimensional (concentrated in even degrees) and is flat as a BP ∗-module for the category of finitely presented BP ∗(BP )-modules. At first glance this would seem to be a very restricted class of spaces, but the world abounds with naturally occurring examples: Eilenberg-Mac Lane spaces, loops of finite Postnikov systems, classifying spaces of all finite groups whose Morava K-theory is known (including the symmetric groups), QS2n, BO(n), MO(n), BO, ImJ , etc. We finish with an explicit algebraic construction of the Brown-Peterson cohomology of a product of Eilenberg-Mac Lane spaces. ∗Partially supported by the National Science Foundation

On Novikov's ext1 modulo an invariant prime ideal

IN HIS WORK on complex cobordism [ 181 Novikov introduced a spectral sequence converging to the stable homotopy of a space, depending only on the complex cobordism of the space as a module over the ring of primary complex cobordism operations. This spectral sequence can be localized at a prime p and one can work, as Novikov did, with the smaller theory, BP*( ), known as Brown-Peterson cohomology [5]. Adams [4] translated the construction into homology, and we have:

The ER(n)-cohomology of BO(q) and real Johnson–Wilson orientations for vector bundles

Using the Bockstein spectral sequence developed previously by the authors, we compute the ring ER(n) ∗ (BO(q)) explicitly. We then use this calculation to show that the ring spectrum MO[2 n+1 ]i sER(n)-orientable (but not ER(n +1 )-orientable), whereMO[2 n+1 ]i s def ined as the Thom spectrum for the self-map of BO given by multiplication by 2 n+1 .

On fibrations related to real spectra

We consider real spectra, collections of Z=(2)–spaces indexed over Z Z withcompatibility conditions. We produce fibrations connecting the homotopy fixedpointsandthespacesinthesespectra. Wealsoevaluatethemapwhichistheanalogueof the forgetful functor from complex to reals composed with complexification.Our first fibration is used to connect the real 2

Brown-Peterson homology of elementary p-groups II

Abstract We compute the complete Abelian group structure of the Brown-Peterson homology of BV, the classifying space for V=( Z p ) n , the elementary Abelian p-group of rank n.

The projective dimension of the complex bordism of Eilenberg-MacLane spaces

Let MU*X be the complex bordism of the space X and let MU* be the complex bordism coefficient ring [7]. There is a standard conjecture that the projective dimension of MU*K(Z/(p),n) (horn. dim. MUjfMU*K(ZI(p),ri), [4]) should be n. The conjecture was motivated by its truth for w— 0,1 and by the early establishment of the lower bound horn. dim. MU^MU*K(Z/(p)yn)^n[2, 3, 4]. The purpose of this note is to disprove the conjecture in the strongest possible way. Let/) be a prime and let Z/(p) denote the integers modulo p*.

Unstable splittings related to Brown-Peterson cohomology

We give a new and relatively easy proof of the splitting theorem of the second author for the spaces in the Omega spectrum for BP. We then give the first published proofs of our similar theorems for the spectra P(n).