Douglas C. Ravenel

Localization with Respect to Certain Periodic Homology Theories

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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

Generalized group characters and complex oriented cohomology theories

Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E ⁄ , the assignment G 7i! E ⁄ (BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E ⁄ , using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artins Theorem is proved for all complex oriented E ⁄ : the abelian subgroups of G serve as a detecting family for E ⁄ (BG), modulo torsion dividing the order of G. When E ⁄ is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes 1 E ⁄ (BG). The domain of the characters is Gn,p, the set of n-tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E ⁄ n- theory, etc. The nth Morava K-theory Euler characteristic for BG is computed to be the number of G-orbits in Gn.p. For various groups G, including all symmetric groups, we prove that K(n)⁄(BG) concentrated in even degrees. Our results about E⁄(BG) extend to theorems about E⁄(EG◊GX), where X is a finite G-CW complex.

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

The cohomology of the morava stabilizer algebras

In this paper we continue our study of the groups ExtBp, Be(BP,, v 2 t BP,/I,). In [5] it was shown that these groups are essentially isomorphic to the cohomology of a certain Hopf algebra S(n) which we called the Morava stabilizer algebra since it was implicitly introduced in [6]. The structure of S(n) was analyzed in [8] where we defined a filtration on it and described the associated graded Hopf algebra EoS(n ) explicitly. We will use the results of[8] in this paper extensively. Applications to the Novikov spectral sequence will appear in a forthcoming paper with Miller and Wilson. In w 1 we show how the machinery developed by May in [3] can be applied to this situation. As a trivial corollary we show that for n 2. In w we compute H* S(n) at all primes for n 5.

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

The Structure of Morava Stabilizer Algebras

The purpose of this note is to prove some general theorems which will facilitate the computation of Ext*e.sp(BP , , v21BP./I.), where 1.=(p, vl, ..., v,_ 0 is the n-th invariant prime ideal in BP.. Specific calculations and applications to the Novikov spectral sequence will be exposed in [8] and 1-13]. This paper is a sequel to [4] in that we reprove some results of Morava ([10] and [11]) with more conventional algebraic topological methods. Our approach differs from those of Morava and Johnson-Wilson in that no use is made of any cohomology theories other than Brown-Peterson theory. Our results have the advantage of being more directly applicable to homotopy theoretic computations than Moravas were. Although none of his results are actually used here, this paper owes its existence to many inspiring and invaluable conversations with Jack Morava. I would also like to thank Haynes Miller, John Moore, Robert Morris, and Steve Wilson for their interest and help. In w 1 we use the change of rings theorem of [7] to show that computing the above mentioned Ext group is equivalent to computing the cohomology of a certain Hopf algebra S(n), which we call the Morava stabilizer algebra. We describe it explicitly using the results of [12]. In w we describe the relation of S(n) to a certain compact p-adic Lie group S. which Morava called the stabilizer group, as it was the isotropy group of a certain point in a scheme with a certain group action in [10]. This group has been studied to some extent by number theorists but we do not exploit this fact. Its basic cohomological properties were originally found by Morava and the author (very likely not for the first time) by application of the results of Lazard I-5]. The results of w 3, however, make no use of [5] or even of the existence of S., and w 3 is independent of w 2. We do however use this group theoretic interpretation to get a certain splitting (Theorem (2.12)) of S(n) when p does not divide n.

The Segal Conjecture for Cyclic Groups and its Consequences

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THE 7-CONNECTED COBORDISM RING AT p = 3

In this paper, we study the cobordism spectrum MOh8i at the prime 3. This spectrum is important because it is conjectured to play the role for elliptic cohomology that Spin cobordism plays for real K-theory. We show that the torsion is all killed by 3, and that the Adams-Novikov spectral sequence collapses after only 2 dierentials. Many of our methods apply more generally.

EHP sequences in BP theory

We have given the homomorphisms in these sequences their usual names: E for suspension, H for Hopf invariant, and P for Whitehead product. When the prime p = 2, it also happens that 9’” = S2n, and the EHP sequences provide a way of calculating the groups TV, inductive on the sphere dimension n, and on the stem dimension q-n. When p is an odd prime, the space 3’” (which has cells in each dimension 2n, 4n,. . ., 2n(p 1)) replaces the even dimensional sphere in the EHP induction. The homotopy groups of the even-dimensional spheres localized at p may be obtained from the fibration