BP: close encounters of the $$E_\infty $$ E ∞ kind

Abstract

Inspired by Stewart Priddy's cellular model for the p-local Brown-Peterson spectrum BP , we give a construction of a p-local E∞ ring spectrum R which is a close approximation to BP . Indeed we can show that if BP admits an E∞ structure then these are weakly equivalent as E∞ ring spectra. Our inductive cellular construction makes use of power operations on homotopy groups to define homotopy classes which are then killed by attaching E∞ cells.Date: 29/07/2013 version 6 (to appear in Journal of Homotopy and Related Structures) arXiv:1204.4878. 2010 Mathematics Subject Classification. Primary 55N20; Secondary 55N22 55S12 55S15.Key words and phrases. E∞ ring spectrum; Brown-Peterson spectrum; power operation. The author would like to thank Bob Bruner, Mike Mandell, Peter May, Birgit Richter, John Rognes and Markus Szymik; special thanks are due to Tyler Lawson who pointed out the usefulness of coning off Moore spectra rather than spheres. Finally, we note that this paper would not exist without the inspiration provided by Stewart Priddy's elegant cellular construction of the Brown-Peterson spectrum.

From Highly Structured E-Infinity Rings and Transfinite Maximally Symmetric Manifolds to the Dark Energy Density of the Cosmos

“…The result agrees completely with previously obtained results as well as actual cosmological measurements and observations [30]. The mathematical basis of the present analysis is rooted in the theoretical structure of E-infinity theory of highly structured rings [12] [13] of which E-infinity theory of high energy physics and loop quantum gravity are in the meantime two well known applications. This was accomplished in part by integrating number theory, the queen of mathematical science and consequently of science in physics [10].…”

Skeleta and categories of algebras

“…We will show that the T (n) arise quite naturally as skeleta of BP as an associative algebra, and that the necessary cells are detected by topological Hochschild homology with coefficients. (Cellular constructions, in various categories, of objects related to BP are by no means new: see, for example, work of Priddy [Pri80], Hu-Kriz-May [HKM01], or Baker [Bak14].) By similar methods, it is also possible to show that the 2-primary spectra Y (n) of Mahowald-Ravenel-Shick [MRS01] arise as skeleta of the Eilenberg-Mac Lane spectrum for F 2 , although this can be shown more directly using their construction as Thom spectra.…”

Secondary power operations and the Brown--Peterson spectrum at the prime 2

“…More recently Basterra-Mandell showed that BP is a split summand of MU (p) as an E 4 -algebra [BM13], and so the homotopy category of BP-modules has a symmetric monoidal structure; Chadwick-Mandell used idempotent splittings to show that this could be done with the Quillen idempotent as E 2 -algebras [CM15]. Both Hu-Kriz-May [HKM01] and Baker [Bak14] gave iterative constructions by methods that kill torsion, producing two different types of closest possible torsion-free E ∞ -algebra to BP. An unpublished paper of Kriz attempted to prove that BP admits an E ∞ -algebra structure, and Basterra developed the theory of topological André-Quillen (TAQ) cohomology based on his ideas-this theory allows the construction of E ∞ -algebras by systematically lifting the k-invariants in the Postnikov tower from ordinary cohomology to TAQ-cohomology [Kri95,Bas99].…”