Postnikov Systems Research Articles

Hopf Algebras and Multiplicative Fibrations II

This paper is a continuation of [13] which we refer to as I. In I we introduced some Hopf algebra techniques which together with the spectral sequence of [8] allowed us to deduce various results for stable Postnikov systems. Our objective in this paper is to obtain further results concerning llopf fibre squares and to determine the general form of the differential in the homology spectral sequence of a lopf fibre square. This task divides naturally into two parts. The first part (Sections 1-4) is concerned with the structure of the E2-term of the spectral sequence. More precisely suppose that A, B are homology Hopf algebras over a field k and f: B ->A is a morphism of Hopf algebras. We wish to determine the sructure of CotorA (B, k) as a Hopf algebra and the properties of the morphism

The cohomology of principal bundles, homogeneous spaces, and two-stage Postnikov systems

In Figure 1, Y-+B is an acyclic fibration with fibre F and E—>X is the fibration induced by f: X—>B. In Figure 2, ƒ: X—>B is a Serre fibration with fibre E. We assume (in both cases) that B is pathwise connected and simply connected. Our results concern the cohomology of E. Let A be a commutative Noetherian ring. Cohomology will be taken with coefficients in A except where explicitly stated otherwise. We assume that H*(B) is A-flat and that H*(X) and H*(B) are of finite type as A-modules. Then there is a spectral sequence of differential A-algebras {Er}, defined by Eilenberg and Moore [o], which satisfies the conditions : (i) E2 = TorH*(B)(A, H*(X)), where H*(X) has the structure of left #*(£)-module determined by the m a p / * : H*(B)->H*(X), and (ii) {Er} converges to H*(E)t in the sense that E*, is isomorphic to the associated graded algebra E°H*(E) of H*(E) with respect to a suitable filtration. With these hypotheses and notations, we have the following result.

Higher order Whitehead products and Postnikov systems

Higher order Whitehead products and Postnikov systems

The cohomology of stable two stage Postnikov systems

The cohomology of stable two stage Postnikov systems

A spectrum whose Z p cohomology is the algebra of reduced pth powers

Throughout this paper all cohomology groups will have 2, coefficients unless otherwise stated. All spectra will be O-connected. We will make various constructions on spectra, for example, forming fibrations and Postnikov systems, just as one does with topological spaces. For the details of this see [6]. If one wishes, one may read “spectrum” as iv-connected topological space, iV a large integer, add N to all dimensions in sight, and read all theorems as applicable in dimensions less than 2N.

A note on 𝐻-spaces and Postnikov systems of spheres

In his beautiful paper [l], J. F. Adams settled the question on the nonexistence of maps 52n_1—>S, n = 2k, k>3, of Hopf invariant one. Taken in conjunction with earlier work, this result shows that the only spheres which are iT-spaces are S°, S1, S3 and S7. The present paper grew out of my desire to interpret this result in terms of a Postnikov decomposition for the sphere. There are three sections. The first section contains general results on the Postnikov decompositions of ii-spaces. The second is devoted entirely to Postnikov decompositions of spheres. In the third section, I consider spaces with simple cohomology, which are thus closely related to spheres. Using the information about Postnikov systems obtained in §1, we give some general theorems about these spaces. Throughout this paper, all spaces will be assumed to have the homotopy-type of a countable, 1-connected complex. All spaces will be assumed to have base points, which will be respected by maps and homotopies. For convenience, we shall omit the base points from the notation.

Induced maps for Postnikov systems

Induced maps for Postnikov systems

On the cohomology of two-stage Postnikov systems

On the cohomology of two-stage Postnikov systems

On the cohomology of two-stage Postnikov systems

On the cohomology of two-stage Postnikov systems

Multiplications in Postnikov systems and their applications

Multiplications in Postnikov systems and their applications