Haynes R. Miller

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

On relations between Adams spectral sequences, with an application to the stable homotopy of a moore space

abutting to the stable homotopy of X. It has long been recognized that a map A +B of ring-spectra gives rise to information about the differentials in this spectral sequence. The main purpose of this paper is to prove a systematic theorem in this direction, and give some applications. To fix ideas, let p be a prime number, and take B to be the modp EilenbergMacLane spectrum H and A to be the Brown-Peterson spectrum BP at p. For p odd, and X torsion-free (or for example X a Moore-space V= So Up e’), the classical Adams E2-term E2(X;H) may be trigraded; and as such it is E2 of a spectral sequence (which we call the May spectral sequence) converging to the AdamsNovikov Ez-term E2(X; BP). One may say that the classical Adams spectral sequence has been broken in half, with all the “BP-primary” differentials evaluated first. There is in fact a precise relationship between the May spectral sequence and the H-Adams spectral sequence. In a certain sense, the May differentials are the Adams differentials modulo higher BP-filtration. One may say the same for p=2, but in a more attenuated sense. In this paper we restrict attention to dz, although I believe that the machinery developed here sheds light on the higher differentials as well. Assertions similar to these, in case X is torsion-free, have been made by Novikov [24], who however provided only the barest hint of a proof. I have attempted to provide in Section 1 a convenient account of part of the abstract theory of spectral sequences of Adams type, and in Sections 3, 5, and 6, I construct the May spectral sequence and prove the theorem outlined above. The constructions here are

Initial conditions, generalized functions, and the laplace transform troubles at the origin

The unilateral Laplace transform is widely used to analyze signals, linear models, and control systems, and is consequently taught to most engineering undergraduates. In our courses at MIT in electrical engineering and computer science, mathematics, and mechanical engineering, we have found some significant pitfalls associated with teaching students to understand and apply the Laplace transform. We have independently concluded that one reason students find the Laplace transform difficult is that there is significant confusion present in many of the standard textbook presentations of this subject, in all three of our disciplines

Fibrewise completion and unstable Adams spectral sequences

We describe a tower of spaces whose inverse limit is a “fiberwise completion” of a fibrationE →B, and study the resulting spectral sequence converging to the homotopy groups of the space of lifts of a mapX →B. This is used to give a proof of the “generalized Sullivan conjecture”.

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:

VANISHING LINES FOR MODULES OVER THE STEENROD ALGEBRA

The purpose of this paper is to establish criteria for freeness and for the existence of a vanishing line for modules over sub-Hopf algebras of the modp Steenrod algebra. Here, a left module M over a connected K-algebra A is said to have a vanishing fine over A of slope d provided that there exists an intercept -c such that To&K, M) = 0 for all (s, t) satisfying t< u’s c. The criteria for freeness considered here have been dealt with by Adams and Margolis [3] in the case p =2, and by Moore and Peterson 1131 in case p is odd. For p = 2, the existence of these vanishing lines was proved independently by Adams, Anderson, and Mahowald, in special cases, and by Anderson and Davis [5] in general. For p odd, special cases were obtained by Liulevicius [9] and May [unpublished]. The general case has remained open and represents the main new result presented here. To state the result, give the dual Steenrod algebra the basis of monomials in Milnor’s generators [12]. Let P: be dual to <<, and (if p#2) let Q, be dual to rr. Then (Q[)‘=O, and (Pf)p=O for SC f, so we can define for a module M the groups

A localization theorem in homological algebra

Introduction. In (l), J. F. Adams showed that for p odd, the Adams 2? 2-term for a sphere, Ext5*(F p , F p), is zero for s < t < (2p-l)s-l, while Ext^ s (Fj,, F p) is the one-dimensional vector space generated by ql, where g

More on the anti-automorphism of the Steenrod algebra

The relations of Barratt and Miller are shown to include all relations among the elements

A Brief Treatment of Generalized Functions for Use in Teaching the Laplace Transform

P^i\chi P^{n-i}

A Laboratory Course in Mathematics

in the mod