Mark Mahowald

The image of J in the EHP sequence

The EHP sequence is based on a result of James [J] who showed that there is H a map H such that S _> Q Sn+- > S2n+1 is a fibration when localized at 2. The map Sn ..> Sn+ is usually labeled E. The boundary homomorphism in the homotopy sequence is usually labeled P. For our purposes it will be most convenient to combine all the EHP sequences into one system. This gives the following filtration of O201:S? = Q(S?):

The K-theory localization of an unstable sphere

GIVEN a space or a spectrum X, in [3,4] Bousfield constructs a localization of-X with respect to a generalized homology theory E,( ). An elegant motivation for this construction is presented in [2], Denoting this localization by XE, the homotopy groups of XE should be the target of a generalized Adams spectral sequence based on E,( ). Indeed, this is shown to be true stably. in certain favorable casts, in [4]. If X is a simply connected space. or a conncctivc spectrum (i.e. n,X = 0 for n < 0). and if E, is a connective homology theory, then XE is just the usual arithmetic localization or completion with respect to some set of primes. If E,( ) fails to be connective. then Xf: is more mysterious. The cast where E is periodic (real or complex) K-theory and X is a spectrum. has been studied cxtcnsivcly. The K-theory localization of the sphcrc spectrum was constructed and its homotopy groups wcrc computed in [4]. See also [IS]. In [4] the mod p homotopy groups of Xp for any spectrum X, arc shown to bc csscntially the “modp o,-periodic homotopy groups of X”. Ilcrc ul rcfcrs to the Adams self-map of a Moore spectrum constructed in [I]. SW [6], [IO] for additional reading on stable K-localization. If X is a space, very littlc is known in gcncral about the unstable K-theory localization XK. In [17], Mislin dctcrmincs the K-theory localization of Eilenberg-MacLanc spaces, and proves some general arithmetic results about localization of spaces. In [S], Bousfield determines the localization of an infinite loop space with respect to K-theory. However, if X is not an infinite loop space Xp is not well understood. In this paper we determine Sin+‘, the K-localization of an odd dimensional sphere, n 2 I, that is, construct a space, show that it is the stated localization, and compute its modp homotopy groups. By analogy with the stable result of [4], we show that the modp’ homotopy groups of Sp+ l are essentially the modp’ u,-periodic homotopy groups of Sznc ‘. These mod p’ u,-periodic homotopy groups were defined and computed in [ 133. [ 193, [9]. It is known that for an arbitrary space X, rr,(Xy; Z/p) is not necessarily the mod p u,-periodic homotopy groups of X, even if X is highly connected. See [I43 for a counterexample. Let E bc a spectrum representing a gcncralized homology theory E,( ).

The geometric dimension of some vector bundles over projective spaces

We prove that in many cases the geometric dimension of the p-fold Whitney sum p1Hk of the Hopf bundle Hk over quaternionic projective space QPk is the smallest n such that for all i < k the reduction of the ith symplectic Pontryagin class of pHk to coefficients 74i-1((RP /RRPn1 ) A bo) is zero, where bo is the spectrum for connective KO-theory localized at 2. We immediately obtain new immersions of real projective space RP4k+3 in Euclidean space if the number of Is in the binary expansion of k is between 5 and 8.

The root invariant in homotopy theory

FOR THE last thirty years the EHP sequence has been a major conceptual tool in the attempt to understand the homotopy groups of spheres. It is a collection of long exact sequences of homotopy groups induced by certain fibrations in which all three spaces are loop spaces of spheres. These fibrations are due originally to James, G. W. Whitehead, and Toda. The Freudenthal suspension theorem and the Adams vector field theorem (which is a strengthened form of the Hopf invariant one theorem) can each be interpreted as statements about the EHP sequence. James periodicity, the Hopf invariant and the Whitehead product all fit into the EHP framework in a very simple way. An expository survey of this material is given in the last section of the first chapter of [36]. More recently the work of Morava led the second author and various collaborators to formulate the chromatic approach to stable homotopy theory and the notion of a v,-periodic family (see [32], [35], [29] and the last three chapters of 11361). The recent spectacular work of Devinatz, Hopkins and Smith [l l] is a vindication of this point of view. The purpose of this paper is to describe the partial understanding we have reached on how the chromatic and EHP points of view interact. The central concept here is the root invariant, which is defined in 1.10 using Lin’s theorem. This assigns to each element in the stable homotopy of a finite complex a nonzero coset in a higher stem. The main conjecture (still unproved) in the subject is that this root invariant converts v,-periodic families to v,+ 1-periodic families. The full implications of this are still not understood. In the first section we will recall the relevant properties of the EHP sequence including James periodicity and define the root invariant in the homotopy of the sphere spectrum. Regular and anomalous elements in the EHP sequence will be defined (1.11). In the second section we will generalize the definition to finite complexes, describe Jones’ connection between the root invariant and the quadratic construction and develop various computational tools. In the third section we will indicate the relation between the root invariant and the Greek letter construction. In the fourth section we will consider unstable homotopy and define the progeny (4.1) and the target set (4.3) of an element in the EHP sequence. In the fifth section we will prove that an element is anomalous if and only if it is a root invariant. In the last section we will give a method of improving James periodicity in many cases. Finally we will construct some similar spectral sequences in which the theorem connecting anomalous elements and root invariants (1.12) does not hold; these will be parametrized by the p-adic integers.

The bo-Adams Spectral Sequence

Due to its relation to the image of the J-homomorphism and first order periodicity (Bott periodicity), connective real K-theory is well suited for problems in 2-local stable homotopy that arise geometrically. On the other hand the use of generalized homology theories in the construction of Adams type spectral sequences has proved to be quite fruitful provided one is able to get a hold on the respective E2-terms. In this paper we make a first attempt to construct an algebraic and computational theory of the E2-term of the bo-Adams spectral sequence. This allows for some concrete computations which are then used to give a proof of the bounded torsion theorem of [8] as used in the geometric application of [2]. The final table of the E2-term for qTs in dim O where B(n) denotes an integral Brown-Gitler spectrum [8]. Received by the editors April 5, 1985 and, in revised form, January 31, 1986 and April 7, 1986. 1980 Mclthemclties Subject Clclssificcltion (1985 Revision). Primary 55T15, 55N15, 55S25, 55Q45.

The spectrum ( P ⋀ bo ) −∞

There are spectra P −k constructed from stunted real projective spaces as in [1] such that H * ( P −k ) is the span in ℤ/2[x, x −1 ] of those x i with i ≥ −k. (All cohomology groups have ℤ/2-coefficients unless specified otherwise.) Using collapsing maps, these form an inverse system which is similar to those of Lin ([15], p. 451). It is a corollary of Lins work that there is an equivalence of spectra where holim is the homotopy inverse limit ([3], ch. 5) and Ŝ –1 the 2-adic completion of a sphere spectrum. One may denote by this holim ( P –κ ), although one must constantly keep in mind that , but rather

On Hopkins’ Picard groups for the prime 3 and chromatic level 2

We give a calculation of Picard groups Pic2 of K(2)-local invertible spectra and Pic(L2) of E(2)-local invertible spectra, both at the prime 3. The main contribution of this paper is to calculation the subgroup κ2 of invertible spectra X with (E2)∗X ∼= (E2)∗S as twisted modules over the Morava stabilizer group G2.

Some generalized brown-gitler spectra

Brown-Gitler spectra for the homology theories associated with the spectra KZp, to, and bu are constructed. Complexes adapted to the new Brown- Gitler spectra are produced and a spectral sequence converging to stable maps into these spectra is constructed and examined.

The image of the stable J-homomorphism

IN THIS PAPER we give detailed proofs of results about the image of the stable J-homomorphism similar to some announced in [24, 25, 26, and 271. The methods are those introduced in those papers; we just take a bit more care here to clarify certain aspects of the proof. The paper could be viewed as a response to Frank Adams’ challenge (Cl]): “for the classical J-homomorphism, we have no published proof independent of the ideas I am now discussing (the Adams conjecture); homotopy theorists know in principle where they should look for an independent proof, but nobody has yet been willing to undertake the heavy task of working it out in detail and writing it down properly.” The J-homomorphism, as introduced by G. W. Whitehead in [41], is the homomorphism X:”

The Homotopy of L 2 V(1) for the Prime 3

Let V(1) be the Toda-Smith complex for the prime 3. We give a complete calculation of the homotopy groups of the L2-localization of V(1) by making use of the higher real K-theory EO 2 of Hopkins and Miller and related homotopy fixed point spectra. In particular we resolve an ambiguity which was left in an earlier approach of Shimomura whose computation was almost complete but left an unspecified parameter still to be determined.