E∞ ring spectra and elements of Hopf invariant 1

Baker, A. (2017) E∞ ring spectra and elements of Hopf invariant 1. Boletín de la Sociedad Matemática Mexicana, 23(1), pp. 195-231. (doi: 10.1007/s40590-016-0096-8)

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The 2-primary Hopf invariant 1 elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper, we explore some properties of the E∞ ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces BSO, BSpin, BString. We show that the homology of these Thom spectra are all extended comodule algebras of the form A∗A(r)∗ P∗ over the dual Steenrod algebra A∗ with A∗A(r)∗F2 as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra HZ, kO or tmf; however, apart from the first case, we have no concrete results on this.

Item Type:Articles
Keywords:Stable homotopy theory, E-infinity ring spectrum, power operations, comodule algberas.
Glasgow Author(s) Enlighten ID:Baker, Dr Andrew
Authors: Baker, A.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Research Group:Geometry & Topology
Journal Name:Boletín de la Sociedad Matemática Mexicana
Publisher:Sociedad Matemática Mexicana
ISSN (Online):2296-4495
Published Online:21 March 2016
Copyright Holders:Copyright © 2016 The Author
First Published:First published in Boletín de la Sociedad Matemática Mexicana 23(1): 195-231
Publisher Policy:Reproduced under a Creative Commons License
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