Mahowald proved the height 1 telescope conjecture at prime 2 as an application of his seminal work on bo-resolutions. In this paper we study through lens tmf-resolutions. To end compute structure tmf-resolution for a specifc type complex Z. We find that, analogous to case, E1-page possesses decomposition into v2-periodic summand, and Eilenberg-MacLane summand which consists bounded v2-torsion. ...

We apply an announced result of Blumberg-Cohen-Schlichtkrull to reprove (under restricted hypotheses) a theorem of Mahowald: the connective real and complex K-theory spectra are not Thom spectra. The construction of various bordism theories as Thom spectra served as a motivating example for the development of highly structured ring spectra. Various other examples of Thom spectra followed; for i...

In [AL07] and [AL10], Arone and Lesh constructed and studied spectrum level filtrations that interpolate between connective (topological or algebraic) K-theory and the Eilenberg-MacLane spectrum for the integers. In this paper we consider (global) equivariant generalizations of these filtrations and of another closely related class of filtrations, the modified rank filtrations of the K-theory s...

Introduction. If £ is a group with a normal subgroup K one may form the quotient group E/K^M. Conversely, for preassigned groups K, M, there is the extension problem: to determine (in some sense) all groups E with K as normal subgroup such that E/K^M. Much progress has been made on this problem, particularly through the work of Baer [l, 2, 3] and the cohomology theory of Eilenberg and MacLane [...

Wepresent a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many common spaces are easier to compute in many cases. Cellular cohomology is a special kind of cohomology designed for cell complexes: these are built...

The planarity theorems of MacLane and Whitney are extended to compact graph-like spaces. This generalizes recent results of Bruhn and Stein (MacLane’s Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney’s Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths).

In this note we discuss how the first author came upon the Kervaire invariant question while analyzing the image of the J-homomorphism in the EHP sequence. One of the central projects of algebraic topology is to calculate the homotopy classes of maps between two finite CW complexes. Even in the case of spheres – the smallest non-trivial CW complexes – this project has a long and rich history. L...

The past decade or so has seen considerable development of the idea, originating with Kan and Thurston [12], of modelling a given sequence of abelian groups by the homology of another group. At the heart of such procedures lies the notion of an acyclic group, one having the same homology as the trivial group. (In this note all homology is taken to have trivial integer coefficients.) At some sta...

We prove that the Morava-K-theory-based Eilenberg-Moore spectral sequence has good convergence properties whenever the base space is a p-local finite Postnikov system with vanishing (n + 1)st homotopy group.

This note will present certain algebraic results obtained by Samuel Eilenberg and the author in a study of the relations between homotopy and homology groups of a topological space. These results yield a homology theory for any abelian group II, in which the low dimensional homology and cohomology groups of n correspond to familiar constructions on II. They depend upon the application of the me...