May spectral sequence

May spectral sequence
Name	May spectral sequence
Field	Algebraic topology
Introduced	1964
Introduced by	J. P. May

May spectral sequence The May spectral sequence is a computational tool in algebraic topology and homological algebra used to resolve extensions in spectral sequence computations connecting Adams spectral sequence, Eilenberg–MacLane spectrum, and Steenrod algebra calculations. It refines filtrations arising from resolutions such as the bar resolution and the cobar complex to produce an auxiliary spectral sequence that interacts with the Adams–Novikov spectral sequence, Brown–Peterson cohomology, and constructions in stable homotopy theory. The construction and use of the May spectral sequence feature in work associated with figures and institutions such as J. Peter May, Adams Prize laureates, and research groups at University of Chicago and Massachusetts Institute of Technology.

Introduction

The inception of the May spectral sequence appears in the context of organizing the computations in the Steenrod algebra Ext groups that feed the Adams spectral sequence for the stable homotopy groups of spheres and of spectra such as BP and MU. It leverages filtrations related to the Milnor basis and the lambda algebra seen in work by John Milnor, Adams spectral sequence developers, and subsequent elaborations by researchers connected to Princeton University and Institute for Advanced Study. The May spectral sequence provides an E_r-page machine that clarifies differentials and hidden extensions encountered in computations by Frank Adams, Douglas Ravenel, and collaborators at seminars linked with University of California, Berkeley.

Construction

The construction begins with a filtered differential graded algebra or coalgebra such as the dual Steenrod algebra A_* or a cobar complex for a Hopf algebroid like (BP_*, BP_*BP). One chooses a multiplicative filtration compatible with the Milnor basis or the canonical filtration appearing in the lambda algebra formulation pioneered by researchers in the Oxford University topology community. From this filtration one forms the associated graded object and the spectral sequence of a filtered complex, yielding the initial pages E_0 and E_1. In algebraic terms the construction uses the bar resolution, the cobar complex, and structures studied in the Lannes T-functor program and in the work of groups at Sorbonne University and École Normale Supérieure on Steenrod operations.

Convergence and E2-term

Under suitable boundedness and completeness hypotheses the May spectral sequence converges to Ext groups over the original algebraic object, producing input for the Adams spectral sequence targeted at computing stable homotopy groups of spheres and homotopy of module spectra like BP-modules and E_n-rings. The E_2-term often admits a description in terms of cohomology of an associated graded Hopf algebra, connecting to the Milnor primitive calculations and to algebraic structures studied by Jean-Pierre Serre and colleagues. Convergence proofs reference techniques employed in the study of nilpotence and periodicity as advanced by Devinatz, Hopkins, and Smith and draw on structural results about Hopf algebroids developed at institutions including Northwestern University.

Applications in Homotopy Theory

The May spectral sequence is instrumental in resolving differentials and hidden extensions in the Adams spectral sequence computations of the homotopy groups of spheres, complex cobordism spectra such as MU, and Brown–Peterson theory BP. It played a role in the computational program that informed the Nilpotence Theorem and the chromatic perspective of Ravenel, influencing work at University of Virginia and Rutgers University. The spectral sequence also appears in calculations involving Eilenberg–MacLane spectrum cohomology operations, connections with the Goodwillie calculus program, and investigations into structured ring spectra like E-infinity ring examples studied at University of Illinois Urbana–Champaign.

Computations and Examples

Concrete computations using the May spectral sequence include low-dimensional Ext calculations over the mod p Steenrod algebra that feed Adams charts compiled by research groups at University of Cambridge and University of Chicago. Examples include detecting families such as the h_i and h_j^2 patterns in the Adams E_2-term, and computations related to the Kervaire invariant problem that involved collaborations across Princeton University and SUNY Stony Brook. Algorithmic and computer-aided implementations used by researchers at Cornell University and University of California, San Diego have automated portions of the May spectral sequence for larger chart computations.

Variants and Generalizations

Variants of the May spectral sequence arise when replacing the Steenrod algebra by other Hopf algebras or Hopf algebroids, producing tools applicable to the Adams–Novikov spectral sequence and to Morava K-theory and Lubin–Tate deformation contexts pursued at Harvard University and University of Michigan. Generalizations incorporate filtrations reflecting v_n-periodicity and chromatic layers central to Ravenel’s conjectures and to the work of the Morava stabilizer group community at Institut des Hautes Études Scientifiques. Other adaptations interface with the lambda algebra framework and with categorical approaches in stable ∞-categories developed in seminars at Columbia University and Stanford University.

Category:Algebraic topology